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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>From Markov to Laplace: How Mamba In-Context Learns Markov Chains</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2502.10178v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1.SS1" title="In 1 Introduction ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Related Work</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Problem setup</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Input data: Random Markov chains</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS2" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Mamba architecture</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS3" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Learning task: next-token prediction</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS4" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Optimal estimator: Laplacian smoothing</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Does Mamba learn in-context estimators?</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Theoretical results</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS1" title="In 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>MambaZero: Simplified model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS2" title="In 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Main theorem for first-order Markov</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS2.SSS1" title="In 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.1 </span>Proof sketch</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS3" title="In 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Higher order processes (<math alttext="k>1" class="ltx_Math" display="inline"><semantics><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow><annotation-xml encoding="MathML-Content"><apply><gt></gt><ci>𝑘</ci><cn type="integer">1</cn></apply></annotation-xml><annotation encoding="application/x-tex">k>1</annotation><annotation encoding="application/x-llamapun">italic_k > 1</annotation></semantics></math>)</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Beyond Markov</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5.SS1" title="In 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Switching Markov model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5.SS2" title="In 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Natural language modeling</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S6" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A1" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Preliminaries on Laplacian smoothing</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Preliminaries and Proof of Thm. <span class="ltx_text ltx_ref_tag">1</span></span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS1" title="In Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.1 </span>Empirical insights</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS2" title="In Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.2 </span>Proof of Thm. <span class="ltx_text ltx_ref_tag">1</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A3" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Proof of Thm. <span class="ltx_text ltx_ref_tag">2</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A4" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Model architectures and hyper-parameters</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\addbibresource</span> <p class="ltx_p" id="p1.2">main.bib <span class="ltx_ERROR undefined" id="p1.2.1">\AtEveryBibitem</span><span class="ltx_ERROR undefined" id="p1.2.2">\clearfield</span>pages </p> </div> <h1 class="ltx_title ltx_title_document">From Markov to Laplace: How Mamba In-Context Learns Markov Chains</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Marco Bondaschi <br class="ltx_break"/>EPFL </span><span class="ltx_author_notes">Correspondence to <span class="ltx_text ltx_font_typewriter" id="id3.1.id1">marco.bondaschi@epfl.ch</span> and <span class="ltx_text ltx_font_typewriter" id="id4.2.id2">ashok.makkuva@epfl.ch</span>.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Nived Rajaraman <br class="ltx_break"/>UC Berkeley </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Xiuying Wei <br class="ltx_break"/>EPFL </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Kannan Ramchandran <br class="ltx_break"/>UC Berkeley </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Razvan Pascanu <br class="ltx_break"/>Google DeepMind </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Caglar Gulcehre <br class="ltx_break"/>EPFL </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Michael Gastpar <br class="ltx_break"/>EPFL </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ashok Vardhan Makkuva <br class="ltx_break"/>EPFL </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.2">While transformer-based language models have driven the AI revolution thus far, their computational complexity has spurred growing interest in viable alternatives, such as structured state space sequence models (SSMs) and Selective SSMs. Among these, Mamba (S<math alttext="6" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mn id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><cn id="id1.1.m1.1.1.cmml" type="integer" xref="id1.1.m1.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">6</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">6</annotation></semantics></math>) and its variant Mamba-<math alttext="2" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mn id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><cn id="id2.2.m2.1.1.cmml" type="integer" xref="id2.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">2</annotation></semantics></math> have shown remarkable inference speed ups over transformers while achieving comparable or superior performance on complex language modeling tasks. However, despite these architectural innovations and empirical successes, the fundamental learning capabilities of Mamba remain poorly understood. In this paper, we address this gap by studying in-context learning (ICL) on Markov chains and uncovering a surprising phenomenon: unlike transformers, even a single-layer Mamba efficiently learns the in-context Laplacian smoothing estimator, which is both Bayes and minimax optimal, for all Markovian orders. To explain this, we theoretically characterize the representation capacity of Mamba and reveal the fundamental role of convolution in enabling it to represent the optimal Laplacian smoothing. These theoretical insights align strongly with empirical results and, to the best of our knowledge, represent the first formal connection between Mamba and optimal statistical estimators. Finally, we outline promising research directions inspired by these findings. Code is available at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/Bond1995/Markov-Mamba" title="">https://github.com/Bond1995/Markov-Mamba</a>.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1.SS1" title="In 1 Introduction ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Related Work</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Problem setup</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Input data: Random Markov chains</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS2" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Mamba architecture</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS3" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Learning task: next-token prediction</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS4" title="In 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Optimal estimator: Laplacian smoothing</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Does Mamba learn in-context estimators?</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Theoretical results</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS1" title="In 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>MambaZero: Simplified model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS2" title="In 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Main theorem for first-order Markov</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS2.SSS1" title="In 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.1 </span>Proof sketch</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS3" title="In 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Higher order processes (<math alttext="k>1" class="ltx_Math" display="inline"><semantics><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow><annotation-xml encoding="MathML-Content"><apply><gt></gt><ci>𝑘</ci><cn type="integer">1</cn></apply></annotation-xml><annotation encoding="application/x-tex">k>1</annotation><annotation encoding="application/x-llamapun">italic_k > 1</annotation></semantics></math>)</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Beyond Markov</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5.SS1" title="In 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Switching Markov model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5.SS2" title="In 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Natural language modeling</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S6" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A1" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Preliminaries on Laplacian smoothing</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Preliminaries and Proof of Thm. <span class="ltx_text ltx_ref_tag">1</span></span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS1" title="In Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.1 </span>Empirical insights</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS2" title="In Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.2 </span>Proof of Thm. <span class="ltx_text ltx_ref_tag">1</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A3" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Proof of Thm. <span class="ltx_text ltx_ref_tag">2</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A4" title="In From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Model architectures and hyper-parameters</span></a></li> </ol></nav> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Transformers have been at the forefront of recent breakthroughs in language modeling, driving the AI revolution <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">vaswani2017attention</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">Radford2018ImprovingLU</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">devlin18</span>]</cite>. Despite their empirical success, transformers suffer from high computational complexity, such as quadratic scaling in sequence length during training and linear cache size at inference <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">gu2023mamba</span>]</cite>. To address these limitations, there is a growing interest in designing alternative efficient architectures among which structured state space models (SSMs) are the most prominent. In particular, Selective SSMs such as Mamba and Mamba-<math alttext="2" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mn id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><cn id="S1.p1.1.m1.1.1.cmml" type="integer" xref="S1.p1.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">2</annotation></semantics></math>, have achieved state-of-the-art results in various language modeling tasks, while greatly improving the inference throughput <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">cirone2025theoreticalfoundationsdeepselective</span>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Motivated by this success, there is tremendous interest in understanding the sequential modeling abilities of SSMs, especially that of Mamba. In particular, mirroring a theme that has been successful in unraveling fundamental mechanisms (e.g. induction heads) behind transformers <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">makkuva2024attention</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">makkuva2024local</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">rajaraman2024transformersmarkovdataconstant</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">nic2024trans</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">edelman2024evolution</span>]</cite>, a growing body of research explores Mamba through its in-context learning (ICL) capabilities <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">grazzi2024mamba</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">halloran2024mamba</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">akyurek2024context</span>]</cite>. While these works reveal interesting insights about Mamba’s ICL abilities vis-a-vis transformers, they are largely empirical in nature, and we currently lack a fundamental understanding of Mamba and its underlying learning mechanisms. We are thus motivated to ask:</p> <blockquote class="ltx_quote" id="S1.p2.2"> <p class="ltx_p" id="S1.p2.2.1"><span class="ltx_text ltx_font_bold" id="S1.p2.2.1.1">Can we systematically characterize the ICL capabilities of Mamba?</span></p> </blockquote> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">To address this, in this paper we introduce a principled framework for theoretical and empirical analysis of Mamba’s ICL capabilities via random Markov chains. Leveraging this framework, we uncover a surprising phenomenon: unlike transformers, even a single-layer Mamba efficiently learns the in-context Laplacian smoothing estimator, which is both Bayes and minimax optimal, for all Markov orders (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1</span></a>). Towards explaining this, we theoretically characterize the representation capacity of Mamba and demonstrate that the convolution mechanism plays a fundamental role in realizing the Laplacian smoothing. Further we showcase that these theoretical insights align strongly with empirical results, both on Markovian and complex natural language data. To the best of our knowledge, this is the first result of its kind connecting Mamba and optimal statistical estimators.</p> </div> <figure class="ltx_figure" id="S1.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S1.F1.sf1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="626" id="S1.F1.sf1.g1" src="x1.png" width="830"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S1.F1.sf1.1"><svg height="233.67" overflow="visible" version="1.1" width="502.54"><g transform="translate(0,233.67) scale(1,-1)"><g transform="translate(-182.65,-9.69)"><g transform="translate(0,11.2079701120797) scale(1, -1)"><foreignobject height="8.8556800885568" overflow="visible" 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Markov chains, exhibiting in-context learning. A two-layer transformer also learns the same, albeit less precisely. In contrast, a single-layer transformer fails to solve this task. We observe the same phenomenon for various Markov orders.</span></figcaption> </figure> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">In summary, we make the following contributions:</p> </div> <div class="ltx_para" id="S1.p5"> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">We provide a novel framework for a precise theoretical and empirical study of Mamba’s ICL via random Markov chains (Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="2.1 Input data: Random Markov chains ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.1</span></a>).</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">Capitalizing our framework, we uncover the surprising fact that the convolution mechanism plays a pivotal role in Mamba’s learning abilities (Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3" title="3 Does Mamba learn in-context estimators? ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3</span></a>).</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.1">Building upon these insights, we characterize the representational capacity of single-layer Mamba and demonstrate that it can represent the optimal Laplacian smoothing estimator for first-order processes (Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4" title="4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">4</span></a>).</p> </div> </li> <li class="ltx_item" id="S1.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i4.p1"> <p class="ltx_p" id="S1.I1.i4.p1.1">We demonstrate the generality of our findings on non-Markovian data and illustrate the fundamental role of convolution even on complex language-modeling tasks (Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5" title="5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">5</span></a>).</p> </div> </li> </ul> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Related Work</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.1">State Space Models <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">gu2020hippo</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">gu2021combining</span>]</cite> have been recently introduced as an alternative recurrent architecture aimed at rivaling the well established transformer backbone <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">vaswani2017attention</span>]</cite> widely used across a range of domains, from language modeling to vision. The model was originally introduced as a discretization of a time-continuous linear dynamical system <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">gu2021combining</span>]</cite>. Recent works tried to re-frame the architecture from a linear recurrent perspective. For example <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">orvieto2023resurrecting</span>]</cite> tried to understand what components of the architecture are necessary, highlighting the importance of linearity, as well as of the particular parametrization that allows control over the stability of the model.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.1">However, there are still many gaps in understanding this family of models <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">team2024jamba</span>]</cite>, such as questions around expressivity <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">orvieto2023universality</span>]</cite>. This is particularly important given the proliferation of Mamba-inspired architectures that has emerged since the introduction of the model. For example, Mamba-Spike <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">qin2024mamba</span>]</cite> integrates a spiking front-end with the Mamba backbone, leveraging event-driven processing for efficient temporal data handling. The Mixture of Mamba approach combines Mamba’s selective state spaces with the Transformer’s Mixture of Experts (MoE) layer <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">csordas2024moeut</span>]</cite>, potentially offering faster convergence and improved performance scaling. Vision Mamba (Vim) <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">zhu2401vision</span>]</cite> adapts the architecture for visual tasks, employing bidirectional Mamba blocks to process high-resolution images efficiently. Besides specific adaptation of the model into new architectures, the core Mamba block also evolved, for example, by the introduction of gating in Mamba-2 <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">mamba2023gu</span>]</cite> and similar SSM architectures <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">de2024griffin</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">Beck2024xLSTM</span>]</cite> which for example invalidates to a certain extent the convolutional view of the architecture.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.1">Our work squarely focuses on understanding the representation power of Mamba, and ICL, which remains an unanswered question. We believe such insights can help navigate the many flavors of SSM architectures. In this space, recent studies have shown that SSMs can perform gradient-based learning for in-context adaptation, similar to transformers <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">sushma2024state</span>]</cite>, where they show that a single SSM layer, augmented with local self-attention, is able to reproduce outputs of an implicit linear model after one step of gradient descent. While some work suggests that Mamba’s ICL abilities rival state-of-the-art transformer models <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">grazzi2024mamba</span>]</cite>, other research indicates that the pretrained Mamba models perform worse in in-context learning than comparable transformer models <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">halloran2024mamba</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">akyurek2024context</span>]</cite>. SSMs have also demonstrated promising results in in-context reinforcement learning tasks, offering fast and performant solutions <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">lu2024structured</span>]</cite>. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">joseph2024hippo</span>]</cite> has introduced a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system in-context. These findings highlight the potential of SSMs as efficient alternatives to transformers for ICL tasks, particularly those involving long input sequences.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Problem setup</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.23">We formally define the problem setting and provide necessary background. 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id="S2.p1.9.m9.4a"><mrow id="S2.p1.9.m9.4.5" xref="S2.p1.9.m9.4.5.cmml"><mrow id="S2.p1.9.m9.4.5.2.2" xref="S2.p1.9.m9.4.5.2.1.cmml"><mo id="S2.p1.9.m9.4.5.2.2.1" stretchy="false" xref="S2.p1.9.m9.4.5.2.1.1.cmml">[</mo><mi id="S2.p1.9.m9.1.1" xref="S2.p1.9.m9.1.1.cmml">T</mi><mo id="S2.p1.9.m9.4.5.2.2.2" stretchy="false" xref="S2.p1.9.m9.4.5.2.1.1.cmml">]</mo></mrow><mo id="S2.p1.9.m9.4.5.1" xref="S2.p1.9.m9.4.5.1.cmml">≜</mo><mrow id="S2.p1.9.m9.4.5.3.2" xref="S2.p1.9.m9.4.5.3.1.cmml"><mo id="S2.p1.9.m9.4.5.3.2.1" stretchy="false" xref="S2.p1.9.m9.4.5.3.1.cmml">{</mo><mn id="S2.p1.9.m9.2.2" xref="S2.p1.9.m9.2.2.cmml">1</mn><mo id="S2.p1.9.m9.4.5.3.2.2" xref="S2.p1.9.m9.4.5.3.1.cmml">,</mo><mi id="S2.p1.9.m9.3.3" mathvariant="normal" xref="S2.p1.9.m9.3.3.cmml">…</mi><mo id="S2.p1.9.m9.4.5.3.2.3" xref="S2.p1.9.m9.4.5.3.1.cmml">,</mo><mi id="S2.p1.9.m9.4.4" xref="S2.p1.9.m9.4.4.cmml">T</mi><mo id="S2.p1.9.m9.4.5.3.2.4" stretchy="false" xref="S2.p1.9.m9.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m9.4b"><apply id="S2.p1.9.m9.4.5.cmml" xref="S2.p1.9.m9.4.5"><ci id="S2.p1.9.m9.4.5.1.cmml" xref="S2.p1.9.m9.4.5.1">≜</ci><apply id="S2.p1.9.m9.4.5.2.1.cmml" xref="S2.p1.9.m9.4.5.2.2"><csymbol cd="latexml" id="S2.p1.9.m9.4.5.2.1.1.cmml" xref="S2.p1.9.m9.4.5.2.2.1">delimited-[]</csymbol><ci id="S2.p1.9.m9.1.1.cmml" xref="S2.p1.9.m9.1.1">𝑇</ci></apply><set id="S2.p1.9.m9.4.5.3.1.cmml" xref="S2.p1.9.m9.4.5.3.2"><cn id="S2.p1.9.m9.2.2.cmml" type="integer" xref="S2.p1.9.m9.2.2">1</cn><ci id="S2.p1.9.m9.3.3.cmml" xref="S2.p1.9.m9.3.3">…</ci><ci id="S2.p1.9.m9.4.4.cmml" xref="S2.p1.9.m9.4.4">𝑇</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m9.4c">[T]\triangleq\{1,\ldots,T\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m9.4d">[ italic_T ] ≜ { 1 , … , italic_T }</annotation></semantics></math>, and for a sequence <math alttext="(x_{t})_{t\geq 1}" class="ltx_Math" display="inline" id="S2.p1.10.m10.1"><semantics id="S2.p1.10.m10.1a"><msub id="S2.p1.10.m10.1.1" xref="S2.p1.10.m10.1.1.cmml"><mrow id="S2.p1.10.m10.1.1.1.1" xref="S2.p1.10.m10.1.1.1.1.1.cmml"><mo id="S2.p1.10.m10.1.1.1.1.2" stretchy="false" xref="S2.p1.10.m10.1.1.1.1.1.cmml">(</mo><msub id="S2.p1.10.m10.1.1.1.1.1" xref="S2.p1.10.m10.1.1.1.1.1.cmml"><mi id="S2.p1.10.m10.1.1.1.1.1.2" xref="S2.p1.10.m10.1.1.1.1.1.2.cmml">x</mi><mi id="S2.p1.10.m10.1.1.1.1.1.3" xref="S2.p1.10.m10.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.p1.10.m10.1.1.1.1.3" stretchy="false" xref="S2.p1.10.m10.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.p1.10.m10.1.1.3" xref="S2.p1.10.m10.1.1.3.cmml"><mi id="S2.p1.10.m10.1.1.3.2" xref="S2.p1.10.m10.1.1.3.2.cmml">t</mi><mo id="S2.p1.10.m10.1.1.3.1" xref="S2.p1.10.m10.1.1.3.1.cmml">≥</mo><mn id="S2.p1.10.m10.1.1.3.3" xref="S2.p1.10.m10.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p1.10.m10.1b"><apply id="S2.p1.10.m10.1.1.cmml" xref="S2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S2.p1.10.m10.1.1.2.cmml" xref="S2.p1.10.m10.1.1">subscript</csymbol><apply id="S2.p1.10.m10.1.1.1.1.1.cmml" xref="S2.p1.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.10.m10.1.1.1.1.1.1.cmml" xref="S2.p1.10.m10.1.1.1.1">subscript</csymbol><ci id="S2.p1.10.m10.1.1.1.1.1.2.cmml" xref="S2.p1.10.m10.1.1.1.1.1.2">𝑥</ci><ci id="S2.p1.10.m10.1.1.1.1.1.3.cmml" xref="S2.p1.10.m10.1.1.1.1.1.3">𝑡</ci></apply><apply id="S2.p1.10.m10.1.1.3.cmml" xref="S2.p1.10.m10.1.1.3"><geq id="S2.p1.10.m10.1.1.3.1.cmml" xref="S2.p1.10.m10.1.1.3.1"></geq><ci id="S2.p1.10.m10.1.1.3.2.cmml" xref="S2.p1.10.m10.1.1.3.2">𝑡</ci><cn id="S2.p1.10.m10.1.1.3.3.cmml" type="integer" xref="S2.p1.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m10.1c">(x_{t})_{t\geq 1}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m10.1d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t ≥ 1 end_POSTSUBSCRIPT</annotation></semantics></math>, define <math alttext="x_{k}^{t}\triangleq(x_{k},\ldots,x_{t})" class="ltx_Math" display="inline" id="S2.p1.11.m11.3"><semantics id="S2.p1.11.m11.3a"><mrow id="S2.p1.11.m11.3.3" xref="S2.p1.11.m11.3.3.cmml"><msubsup id="S2.p1.11.m11.3.3.4" xref="S2.p1.11.m11.3.3.4.cmml"><mi id="S2.p1.11.m11.3.3.4.2.2" xref="S2.p1.11.m11.3.3.4.2.2.cmml">x</mi><mi id="S2.p1.11.m11.3.3.4.2.3" xref="S2.p1.11.m11.3.3.4.2.3.cmml">k</mi><mi id="S2.p1.11.m11.3.3.4.3" xref="S2.p1.11.m11.3.3.4.3.cmml">t</mi></msubsup><mo id="S2.p1.11.m11.3.3.3" xref="S2.p1.11.m11.3.3.3.cmml">≜</mo><mrow id="S2.p1.11.m11.3.3.2.2" xref="S2.p1.11.m11.3.3.2.3.cmml"><mo id="S2.p1.11.m11.3.3.2.2.3" stretchy="false" xref="S2.p1.11.m11.3.3.2.3.cmml">(</mo><msub id="S2.p1.11.m11.2.2.1.1.1" xref="S2.p1.11.m11.2.2.1.1.1.cmml"><mi id="S2.p1.11.m11.2.2.1.1.1.2" xref="S2.p1.11.m11.2.2.1.1.1.2.cmml">x</mi><mi id="S2.p1.11.m11.2.2.1.1.1.3" xref="S2.p1.11.m11.2.2.1.1.1.3.cmml">k</mi></msub><mo id="S2.p1.11.m11.3.3.2.2.4" xref="S2.p1.11.m11.3.3.2.3.cmml">,</mo><mi id="S2.p1.11.m11.1.1" mathvariant="normal" xref="S2.p1.11.m11.1.1.cmml">…</mi><mo id="S2.p1.11.m11.3.3.2.2.5" xref="S2.p1.11.m11.3.3.2.3.cmml">,</mo><msub id="S2.p1.11.m11.3.3.2.2.2" xref="S2.p1.11.m11.3.3.2.2.2.cmml"><mi id="S2.p1.11.m11.3.3.2.2.2.2" xref="S2.p1.11.m11.3.3.2.2.2.2.cmml">x</mi><mi id="S2.p1.11.m11.3.3.2.2.2.3" xref="S2.p1.11.m11.3.3.2.2.2.3.cmml">t</mi></msub><mo id="S2.p1.11.m11.3.3.2.2.6" stretchy="false" xref="S2.p1.11.m11.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.11.m11.3b"><apply id="S2.p1.11.m11.3.3.cmml" xref="S2.p1.11.m11.3.3"><ci id="S2.p1.11.m11.3.3.3.cmml" xref="S2.p1.11.m11.3.3.3">≜</ci><apply id="S2.p1.11.m11.3.3.4.cmml" xref="S2.p1.11.m11.3.3.4"><csymbol cd="ambiguous" id="S2.p1.11.m11.3.3.4.1.cmml" xref="S2.p1.11.m11.3.3.4">superscript</csymbol><apply id="S2.p1.11.m11.3.3.4.2.cmml" xref="S2.p1.11.m11.3.3.4"><csymbol cd="ambiguous" id="S2.p1.11.m11.3.3.4.2.1.cmml" xref="S2.p1.11.m11.3.3.4">subscript</csymbol><ci id="S2.p1.11.m11.3.3.4.2.2.cmml" xref="S2.p1.11.m11.3.3.4.2.2">𝑥</ci><ci id="S2.p1.11.m11.3.3.4.2.3.cmml" xref="S2.p1.11.m11.3.3.4.2.3">𝑘</ci></apply><ci id="S2.p1.11.m11.3.3.4.3.cmml" xref="S2.p1.11.m11.3.3.4.3">𝑡</ci></apply><vector id="S2.p1.11.m11.3.3.2.3.cmml" xref="S2.p1.11.m11.3.3.2.2"><apply id="S2.p1.11.m11.2.2.1.1.1.cmml" xref="S2.p1.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.p1.11.m11.2.2.1.1.1.1.cmml" xref="S2.p1.11.m11.2.2.1.1.1">subscript</csymbol><ci id="S2.p1.11.m11.2.2.1.1.1.2.cmml" xref="S2.p1.11.m11.2.2.1.1.1.2">𝑥</ci><ci id="S2.p1.11.m11.2.2.1.1.1.3.cmml" xref="S2.p1.11.m11.2.2.1.1.1.3">𝑘</ci></apply><ci id="S2.p1.11.m11.1.1.cmml" xref="S2.p1.11.m11.1.1">…</ci><apply id="S2.p1.11.m11.3.3.2.2.2.cmml" xref="S2.p1.11.m11.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.p1.11.m11.3.3.2.2.2.1.cmml" xref="S2.p1.11.m11.3.3.2.2.2">subscript</csymbol><ci id="S2.p1.11.m11.3.3.2.2.2.2.cmml" xref="S2.p1.11.m11.3.3.2.2.2.2">𝑥</ci><ci id="S2.p1.11.m11.3.3.2.2.2.3.cmml" xref="S2.p1.11.m11.3.3.2.2.2.3">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m11.3c">x_{k}^{t}\triangleq(x_{k},\ldots,x_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m11.3d">italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≜ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> if <math alttext="k\geq 1" class="ltx_Math" display="inline" id="S2.p1.12.m12.1"><semantics id="S2.p1.12.m12.1a"><mrow id="S2.p1.12.m12.1.1" xref="S2.p1.12.m12.1.1.cmml"><mi id="S2.p1.12.m12.1.1.2" xref="S2.p1.12.m12.1.1.2.cmml">k</mi><mo id="S2.p1.12.m12.1.1.1" xref="S2.p1.12.m12.1.1.1.cmml">≥</mo><mn id="S2.p1.12.m12.1.1.3" xref="S2.p1.12.m12.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m12.1b"><apply id="S2.p1.12.m12.1.1.cmml" xref="S2.p1.12.m12.1.1"><geq id="S2.p1.12.m12.1.1.1.cmml" xref="S2.p1.12.m12.1.1.1"></geq><ci id="S2.p1.12.m12.1.1.2.cmml" xref="S2.p1.12.m12.1.1.2">𝑘</ci><cn id="S2.p1.12.m12.1.1.3.cmml" type="integer" xref="S2.p1.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m12.1c">k\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m12.1d">italic_k ≥ 1</annotation></semantics></math> and <math alttext="(x_{1},\ldots,x_{t})" class="ltx_Math" display="inline" id="S2.p1.13.m13.3"><semantics id="S2.p1.13.m13.3a"><mrow id="S2.p1.13.m13.3.3.2" xref="S2.p1.13.m13.3.3.3.cmml"><mo id="S2.p1.13.m13.3.3.2.3" stretchy="false" xref="S2.p1.13.m13.3.3.3.cmml">(</mo><msub id="S2.p1.13.m13.2.2.1.1" xref="S2.p1.13.m13.2.2.1.1.cmml"><mi id="S2.p1.13.m13.2.2.1.1.2" xref="S2.p1.13.m13.2.2.1.1.2.cmml">x</mi><mn id="S2.p1.13.m13.2.2.1.1.3" xref="S2.p1.13.m13.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.p1.13.m13.3.3.2.4" xref="S2.p1.13.m13.3.3.3.cmml">,</mo><mi id="S2.p1.13.m13.1.1" mathvariant="normal" xref="S2.p1.13.m13.1.1.cmml">…</mi><mo id="S2.p1.13.m13.3.3.2.5" xref="S2.p1.13.m13.3.3.3.cmml">,</mo><msub id="S2.p1.13.m13.3.3.2.2" xref="S2.p1.13.m13.3.3.2.2.cmml"><mi id="S2.p1.13.m13.3.3.2.2.2" xref="S2.p1.13.m13.3.3.2.2.2.cmml">x</mi><mi id="S2.p1.13.m13.3.3.2.2.3" xref="S2.p1.13.m13.3.3.2.2.3.cmml">t</mi></msub><mo id="S2.p1.13.m13.3.3.2.6" stretchy="false" xref="S2.p1.13.m13.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.13.m13.3b"><vector id="S2.p1.13.m13.3.3.3.cmml" xref="S2.p1.13.m13.3.3.2"><apply id="S2.p1.13.m13.2.2.1.1.cmml" xref="S2.p1.13.m13.2.2.1.1"><csymbol cd="ambiguous" id="S2.p1.13.m13.2.2.1.1.1.cmml" xref="S2.p1.13.m13.2.2.1.1">subscript</csymbol><ci id="S2.p1.13.m13.2.2.1.1.2.cmml" xref="S2.p1.13.m13.2.2.1.1.2">𝑥</ci><cn id="S2.p1.13.m13.2.2.1.1.3.cmml" type="integer" xref="S2.p1.13.m13.2.2.1.1.3">1</cn></apply><ci id="S2.p1.13.m13.1.1.cmml" xref="S2.p1.13.m13.1.1">…</ci><apply id="S2.p1.13.m13.3.3.2.2.cmml" xref="S2.p1.13.m13.3.3.2.2"><csymbol cd="ambiguous" id="S2.p1.13.m13.3.3.2.2.1.cmml" xref="S2.p1.13.m13.3.3.2.2">subscript</csymbol><ci id="S2.p1.13.m13.3.3.2.2.2.cmml" xref="S2.p1.13.m13.3.3.2.2.2">𝑥</ci><ci id="S2.p1.13.m13.3.3.2.2.3.cmml" xref="S2.p1.13.m13.3.3.2.2.3">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m13.3c">(x_{1},\ldots,x_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m13.3d">( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> otherwise. For <math alttext="z\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.p1.14.m14.1"><semantics id="S2.p1.14.m14.1a"><mrow id="S2.p1.14.m14.1.1" xref="S2.p1.14.m14.1.1.cmml"><mi id="S2.p1.14.m14.1.1.2" xref="S2.p1.14.m14.1.1.2.cmml">z</mi><mo id="S2.p1.14.m14.1.1.1" xref="S2.p1.14.m14.1.1.1.cmml">∈</mo><mi id="S2.p1.14.m14.1.1.3" xref="S2.p1.14.m14.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.14.m14.1b"><apply id="S2.p1.14.m14.1.1.cmml" xref="S2.p1.14.m14.1.1"><in id="S2.p1.14.m14.1.1.1.cmml" xref="S2.p1.14.m14.1.1.1"></in><ci id="S2.p1.14.m14.1.1.2.cmml" xref="S2.p1.14.m14.1.1.2">𝑧</ci><ci id="S2.p1.14.m14.1.1.3.cmml" xref="S2.p1.14.m14.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m14.1c">z\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m14.1d">italic_z ∈ blackboard_R</annotation></semantics></math>, <math alttext="\mathrm{sigmoid}(z)\triangleq 1/(1+e^{-z}),\mathrm{ReLU}(z)\triangleq\max(0,z)" class="ltx_Math" display="inline" id="S2.p1.15.m15.7"><semantics id="S2.p1.15.m15.7a"><mrow id="S2.p1.15.m15.7.7.2" xref="S2.p1.15.m15.7.7.3.cmml"><mrow id="S2.p1.15.m15.6.6.1.1" xref="S2.p1.15.m15.6.6.1.1.cmml"><mrow id="S2.p1.15.m15.6.6.1.1.3" xref="S2.p1.15.m15.6.6.1.1.3.cmml"><mi id="S2.p1.15.m15.6.6.1.1.3.2" xref="S2.p1.15.m15.6.6.1.1.3.2.cmml">sigmoid</mi><mo id="S2.p1.15.m15.6.6.1.1.3.1" xref="S2.p1.15.m15.6.6.1.1.3.1.cmml">⁢</mo><mrow id="S2.p1.15.m15.6.6.1.1.3.3.2" xref="S2.p1.15.m15.6.6.1.1.3.cmml"><mo id="S2.p1.15.m15.6.6.1.1.3.3.2.1" stretchy="false" xref="S2.p1.15.m15.6.6.1.1.3.cmml">(</mo><mi id="S2.p1.15.m15.1.1" xref="S2.p1.15.m15.1.1.cmml">z</mi><mo id="S2.p1.15.m15.6.6.1.1.3.3.2.2" stretchy="false" xref="S2.p1.15.m15.6.6.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.p1.15.m15.6.6.1.1.2" xref="S2.p1.15.m15.6.6.1.1.2.cmml">≜</mo><mrow id="S2.p1.15.m15.6.6.1.1.1" xref="S2.p1.15.m15.6.6.1.1.1.cmml"><mn 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xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3.2.cmml">z</mi></mrow></msup></mrow><mo id="S2.p1.15.m15.6.6.1.1.1.1.1.3" stretchy="false" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.p1.15.m15.7.7.2.3" xref="S2.p1.15.m15.7.7.3a.cmml">,</mo><mrow id="S2.p1.15.m15.7.7.2.2" xref="S2.p1.15.m15.7.7.2.2.cmml"><mrow id="S2.p1.15.m15.7.7.2.2.2" xref="S2.p1.15.m15.7.7.2.2.2.cmml"><mi id="S2.p1.15.m15.7.7.2.2.2.2" xref="S2.p1.15.m15.7.7.2.2.2.2.cmml">ReLU</mi><mo id="S2.p1.15.m15.7.7.2.2.2.1" xref="S2.p1.15.m15.7.7.2.2.2.1.cmml">⁢</mo><mrow id="S2.p1.15.m15.7.7.2.2.2.3.2" xref="S2.p1.15.m15.7.7.2.2.2.cmml"><mo id="S2.p1.15.m15.7.7.2.2.2.3.2.1" stretchy="false" xref="S2.p1.15.m15.7.7.2.2.2.cmml">(</mo><mi id="S2.p1.15.m15.2.2" xref="S2.p1.15.m15.2.2.cmml">z</mi><mo id="S2.p1.15.m15.7.7.2.2.2.3.2.2" stretchy="false" xref="S2.p1.15.m15.7.7.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.15.m15.7.7.2.2.1" xref="S2.p1.15.m15.7.7.2.2.1.cmml">≜</mo><mrow id="S2.p1.15.m15.7.7.2.2.3.2" xref="S2.p1.15.m15.7.7.2.2.3.1.cmml"><mi id="S2.p1.15.m15.3.3" xref="S2.p1.15.m15.3.3.cmml">max</mi><mo id="S2.p1.15.m15.7.7.2.2.3.2a" xref="S2.p1.15.m15.7.7.2.2.3.1.cmml">⁡</mo><mrow id="S2.p1.15.m15.7.7.2.2.3.2.1" xref="S2.p1.15.m15.7.7.2.2.3.1.cmml"><mo id="S2.p1.15.m15.7.7.2.2.3.2.1.1" stretchy="false" xref="S2.p1.15.m15.7.7.2.2.3.1.cmml">(</mo><mn id="S2.p1.15.m15.4.4" xref="S2.p1.15.m15.4.4.cmml">0</mn><mo id="S2.p1.15.m15.7.7.2.2.3.2.1.2" xref="S2.p1.15.m15.7.7.2.2.3.1.cmml">,</mo><mi id="S2.p1.15.m15.5.5" xref="S2.p1.15.m15.5.5.cmml">z</mi><mo id="S2.p1.15.m15.7.7.2.2.3.2.1.3" stretchy="false" xref="S2.p1.15.m15.7.7.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.15.m15.7b"><apply id="S2.p1.15.m15.7.7.3.cmml" xref="S2.p1.15.m15.7.7.2"><csymbol cd="ambiguous" id="S2.p1.15.m15.7.7.3a.cmml" xref="S2.p1.15.m15.7.7.2.3">formulae-sequence</csymbol><apply id="S2.p1.15.m15.6.6.1.1.cmml" xref="S2.p1.15.m15.6.6.1.1"><ci id="S2.p1.15.m15.6.6.1.1.2.cmml" xref="S2.p1.15.m15.6.6.1.1.2">≜</ci><apply id="S2.p1.15.m15.6.6.1.1.3.cmml" xref="S2.p1.15.m15.6.6.1.1.3"><times id="S2.p1.15.m15.6.6.1.1.3.1.cmml" xref="S2.p1.15.m15.6.6.1.1.3.1"></times><ci id="S2.p1.15.m15.6.6.1.1.3.2.cmml" xref="S2.p1.15.m15.6.6.1.1.3.2">sigmoid</ci><ci id="S2.p1.15.m15.1.1.cmml" xref="S2.p1.15.m15.1.1">𝑧</ci></apply><apply id="S2.p1.15.m15.6.6.1.1.1.cmml" xref="S2.p1.15.m15.6.6.1.1.1"><divide id="S2.p1.15.m15.6.6.1.1.1.2.cmml" xref="S2.p1.15.m15.6.6.1.1.1.2"></divide><cn id="S2.p1.15.m15.6.6.1.1.1.3.cmml" type="integer" xref="S2.p1.15.m15.6.6.1.1.1.3">1</cn><apply id="S2.p1.15.m15.6.6.1.1.1.1.1.1.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1"><plus id="S2.p1.15.m15.6.6.1.1.1.1.1.1.1.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.1"></plus><cn id="S2.p1.15.m15.6.6.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.2">1</cn><apply id="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.1.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.2.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.2">𝑒</ci><apply id="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3"><minus id="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3.1.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3"></minus><ci id="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3.2.cmml" xref="S2.p1.15.m15.6.6.1.1.1.1.1.1.3.3.2">𝑧</ci></apply></apply></apply></apply></apply><apply id="S2.p1.15.m15.7.7.2.2.cmml" xref="S2.p1.15.m15.7.7.2.2"><ci id="S2.p1.15.m15.7.7.2.2.1.cmml" xref="S2.p1.15.m15.7.7.2.2.1">≜</ci><apply id="S2.p1.15.m15.7.7.2.2.2.cmml" xref="S2.p1.15.m15.7.7.2.2.2"><times id="S2.p1.15.m15.7.7.2.2.2.1.cmml" xref="S2.p1.15.m15.7.7.2.2.2.1"></times><ci id="S2.p1.15.m15.7.7.2.2.2.2.cmml" xref="S2.p1.15.m15.7.7.2.2.2.2">ReLU</ci><ci id="S2.p1.15.m15.2.2.cmml" xref="S2.p1.15.m15.2.2">𝑧</ci></apply><apply id="S2.p1.15.m15.7.7.2.2.3.1.cmml" xref="S2.p1.15.m15.7.7.2.2.3.2"><max id="S2.p1.15.m15.3.3.cmml" xref="S2.p1.15.m15.3.3"></max><cn id="S2.p1.15.m15.4.4.cmml" type="integer" xref="S2.p1.15.m15.4.4">0</cn><ci id="S2.p1.15.m15.5.5.cmml" xref="S2.p1.15.m15.5.5">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m15.7c">\mathrm{sigmoid}(z)\triangleq 1/(1+e^{-z}),\mathrm{ReLU}(z)\triangleq\max(0,z)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m15.7d">roman_sigmoid ( italic_z ) ≜ 1 / ( 1 + italic_e start_POSTSUPERSCRIPT - italic_z end_POSTSUPERSCRIPT ) , roman_ReLU ( italic_z ) ≜ roman_max ( 0 , italic_z )</annotation></semantics></math>, and <math alttext="\mathrm{softplus}(z)\triangleq\log(1+e^{z})" class="ltx_Math" display="inline" id="S2.p1.16.m16.3"><semantics id="S2.p1.16.m16.3a"><mrow id="S2.p1.16.m16.3.3" xref="S2.p1.16.m16.3.3.cmml"><mrow id="S2.p1.16.m16.3.3.3" xref="S2.p1.16.m16.3.3.3.cmml"><mi id="S2.p1.16.m16.3.3.3.2" xref="S2.p1.16.m16.3.3.3.2.cmml">softplus</mi><mo id="S2.p1.16.m16.3.3.3.1" xref="S2.p1.16.m16.3.3.3.1.cmml">⁢</mo><mrow id="S2.p1.16.m16.3.3.3.3.2" xref="S2.p1.16.m16.3.3.3.cmml"><mo id="S2.p1.16.m16.3.3.3.3.2.1" stretchy="false" xref="S2.p1.16.m16.3.3.3.cmml">(</mo><mi id="S2.p1.16.m16.1.1" xref="S2.p1.16.m16.1.1.cmml">z</mi><mo id="S2.p1.16.m16.3.3.3.3.2.2" stretchy="false" xref="S2.p1.16.m16.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.p1.16.m16.3.3.2" xref="S2.p1.16.m16.3.3.2.cmml">≜</mo><mrow id="S2.p1.16.m16.3.3.1.1" xref="S2.p1.16.m16.3.3.1.2.cmml"><mi id="S2.p1.16.m16.2.2" xref="S2.p1.16.m16.2.2.cmml">log</mi><mo id="S2.p1.16.m16.3.3.1.1a" xref="S2.p1.16.m16.3.3.1.2.cmml">⁡</mo><mrow id="S2.p1.16.m16.3.3.1.1.1" xref="S2.p1.16.m16.3.3.1.2.cmml"><mo id="S2.p1.16.m16.3.3.1.1.1.2" stretchy="false" xref="S2.p1.16.m16.3.3.1.2.cmml">(</mo><mrow id="S2.p1.16.m16.3.3.1.1.1.1" xref="S2.p1.16.m16.3.3.1.1.1.1.cmml"><mn id="S2.p1.16.m16.3.3.1.1.1.1.2" xref="S2.p1.16.m16.3.3.1.1.1.1.2.cmml">1</mn><mo id="S2.p1.16.m16.3.3.1.1.1.1.1" xref="S2.p1.16.m16.3.3.1.1.1.1.1.cmml">+</mo><msup id="S2.p1.16.m16.3.3.1.1.1.1.3" xref="S2.p1.16.m16.3.3.1.1.1.1.3.cmml"><mi id="S2.p1.16.m16.3.3.1.1.1.1.3.2" xref="S2.p1.16.m16.3.3.1.1.1.1.3.2.cmml">e</mi><mi id="S2.p1.16.m16.3.3.1.1.1.1.3.3" xref="S2.p1.16.m16.3.3.1.1.1.1.3.3.cmml">z</mi></msup></mrow><mo id="S2.p1.16.m16.3.3.1.1.1.3" stretchy="false" xref="S2.p1.16.m16.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.16.m16.3b"><apply id="S2.p1.16.m16.3.3.cmml" xref="S2.p1.16.m16.3.3"><ci id="S2.p1.16.m16.3.3.2.cmml" xref="S2.p1.16.m16.3.3.2">≜</ci><apply id="S2.p1.16.m16.3.3.3.cmml" xref="S2.p1.16.m16.3.3.3"><times id="S2.p1.16.m16.3.3.3.1.cmml" xref="S2.p1.16.m16.3.3.3.1"></times><ci id="S2.p1.16.m16.3.3.3.2.cmml" xref="S2.p1.16.m16.3.3.3.2">softplus</ci><ci id="S2.p1.16.m16.1.1.cmml" xref="S2.p1.16.m16.1.1">𝑧</ci></apply><apply id="S2.p1.16.m16.3.3.1.2.cmml" xref="S2.p1.16.m16.3.3.1.1"><log id="S2.p1.16.m16.2.2.cmml" xref="S2.p1.16.m16.2.2"></log><apply id="S2.p1.16.m16.3.3.1.1.1.1.cmml" xref="S2.p1.16.m16.3.3.1.1.1.1"><plus id="S2.p1.16.m16.3.3.1.1.1.1.1.cmml" xref="S2.p1.16.m16.3.3.1.1.1.1.1"></plus><cn id="S2.p1.16.m16.3.3.1.1.1.1.2.cmml" type="integer" xref="S2.p1.16.m16.3.3.1.1.1.1.2">1</cn><apply id="S2.p1.16.m16.3.3.1.1.1.1.3.cmml" xref="S2.p1.16.m16.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.16.m16.3.3.1.1.1.1.3.1.cmml" xref="S2.p1.16.m16.3.3.1.1.1.1.3">superscript</csymbol><ci id="S2.p1.16.m16.3.3.1.1.1.1.3.2.cmml" xref="S2.p1.16.m16.3.3.1.1.1.1.3.2">𝑒</ci><ci id="S2.p1.16.m16.3.3.1.1.1.1.3.3.cmml" xref="S2.p1.16.m16.3.3.1.1.1.1.3.3">𝑧</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m16.3c">\mathrm{softplus}(z)\triangleq\log(1+e^{z})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m16.3d">roman_softplus ( italic_z ) ≜ roman_log ( 1 + italic_e start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT )</annotation></semantics></math>. <math alttext="\mathrm{Unif}(S)" class="ltx_Math" display="inline" id="S2.p1.17.m17.1"><semantics id="S2.p1.17.m17.1a"><mrow id="S2.p1.17.m17.1.2" xref="S2.p1.17.m17.1.2.cmml"><mi id="S2.p1.17.m17.1.2.2" xref="S2.p1.17.m17.1.2.2.cmml">Unif</mi><mo id="S2.p1.17.m17.1.2.1" xref="S2.p1.17.m17.1.2.1.cmml">⁢</mo><mrow id="S2.p1.17.m17.1.2.3.2" xref="S2.p1.17.m17.1.2.cmml"><mo id="S2.p1.17.m17.1.2.3.2.1" stretchy="false" xref="S2.p1.17.m17.1.2.cmml">(</mo><mi id="S2.p1.17.m17.1.1" xref="S2.p1.17.m17.1.1.cmml">S</mi><mo id="S2.p1.17.m17.1.2.3.2.2" stretchy="false" xref="S2.p1.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.17.m17.1b"><apply id="S2.p1.17.m17.1.2.cmml" xref="S2.p1.17.m17.1.2"><times id="S2.p1.17.m17.1.2.1.cmml" xref="S2.p1.17.m17.1.2.1"></times><ci id="S2.p1.17.m17.1.2.2.cmml" xref="S2.p1.17.m17.1.2.2">Unif</ci><ci id="S2.p1.17.m17.1.1.cmml" xref="S2.p1.17.m17.1.1">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m17.1c">\mathrm{Unif}(S)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m17.1d">roman_Unif ( italic_S )</annotation></semantics></math> denotes the uniform distribution over a set <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.18.m18.1"><semantics id="S2.p1.18.m18.1a"><mi id="S2.p1.18.m18.1.1" xref="S2.p1.18.m18.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.18.m18.1b"><ci id="S2.p1.18.m18.1.1.cmml" xref="S2.p1.18.m18.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.18.m18.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.18.m18.1d">italic_S</annotation></semantics></math> and <math alttext="\mathrm{Dir}(\bm{\beta})" class="ltx_Math" display="inline" id="S2.p1.19.m19.1"><semantics id="S2.p1.19.m19.1a"><mrow id="S2.p1.19.m19.1.2" xref="S2.p1.19.m19.1.2.cmml"><mi id="S2.p1.19.m19.1.2.2" xref="S2.p1.19.m19.1.2.2.cmml">Dir</mi><mo id="S2.p1.19.m19.1.2.1" xref="S2.p1.19.m19.1.2.1.cmml">⁢</mo><mrow id="S2.p1.19.m19.1.2.3.2" xref="S2.p1.19.m19.1.2.cmml"><mo id="S2.p1.19.m19.1.2.3.2.1" stretchy="false" xref="S2.p1.19.m19.1.2.cmml">(</mo><mi id="S2.p1.19.m19.1.1" xref="S2.p1.19.m19.1.1.cmml">𝜷</mi><mo id="S2.p1.19.m19.1.2.3.2.2" stretchy="false" xref="S2.p1.19.m19.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.19.m19.1b"><apply id="S2.p1.19.m19.1.2.cmml" xref="S2.p1.19.m19.1.2"><times id="S2.p1.19.m19.1.2.1.cmml" xref="S2.p1.19.m19.1.2.1"></times><ci id="S2.p1.19.m19.1.2.2.cmml" xref="S2.p1.19.m19.1.2.2">Dir</ci><ci id="S2.p1.19.m19.1.1.cmml" xref="S2.p1.19.m19.1.1">𝜷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.19.m19.1c">\mathrm{Dir}(\bm{\beta})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.19.m19.1d">roman_Dir ( bold_italic_β )</annotation></semantics></math> denotes the Dirichlet distribution with parameter <math alttext="\bm{\beta}>0" class="ltx_Math" display="inline" id="S2.p1.20.m20.1"><semantics id="S2.p1.20.m20.1a"><mrow id="S2.p1.20.m20.1.1" xref="S2.p1.20.m20.1.1.cmml"><mi id="S2.p1.20.m20.1.1.2" xref="S2.p1.20.m20.1.1.2.cmml">𝜷</mi><mo id="S2.p1.20.m20.1.1.1" xref="S2.p1.20.m20.1.1.1.cmml">></mo><mn id="S2.p1.20.m20.1.1.3" xref="S2.p1.20.m20.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.20.m20.1b"><apply id="S2.p1.20.m20.1.1.cmml" xref="S2.p1.20.m20.1.1"><gt id="S2.p1.20.m20.1.1.1.cmml" xref="S2.p1.20.m20.1.1.1"></gt><ci id="S2.p1.20.m20.1.1.2.cmml" xref="S2.p1.20.m20.1.1.2">𝜷</ci><cn id="S2.p1.20.m20.1.1.3.cmml" type="integer" xref="S2.p1.20.m20.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.20.m20.1c">\bm{\beta}>0</annotation><annotation encoding="application/x-llamapun" id="S2.p1.20.m20.1d">bold_italic_β > 0</annotation></semantics></math>. <math alttext="D_{\rm KL}\left(P\|Q\right)" class="ltx_Math" display="inline" id="S2.p1.21.m21.1"><semantics id="S2.p1.21.m21.1a"><mrow id="S2.p1.21.m21.1.1" xref="S2.p1.21.m21.1.1.cmml"><msub id="S2.p1.21.m21.1.1.3" xref="S2.p1.21.m21.1.1.3.cmml"><mi id="S2.p1.21.m21.1.1.3.2" xref="S2.p1.21.m21.1.1.3.2.cmml">D</mi><mi id="S2.p1.21.m21.1.1.3.3" xref="S2.p1.21.m21.1.1.3.3.cmml">KL</mi></msub><mo id="S2.p1.21.m21.1.1.2" xref="S2.p1.21.m21.1.1.2.cmml">⁢</mo><mrow id="S2.p1.21.m21.1.1.1.1" xref="S2.p1.21.m21.1.1.1.1.1.cmml"><mo id="S2.p1.21.m21.1.1.1.1.2" xref="S2.p1.21.m21.1.1.1.1.1.cmml">(</mo><mrow id="S2.p1.21.m21.1.1.1.1.1" xref="S2.p1.21.m21.1.1.1.1.1.cmml"><mi id="S2.p1.21.m21.1.1.1.1.1.2" xref="S2.p1.21.m21.1.1.1.1.1.2.cmml">P</mi><mo id="S2.p1.21.m21.1.1.1.1.1.1" xref="S2.p1.21.m21.1.1.1.1.1.1.cmml">∥</mo><mi id="S2.p1.21.m21.1.1.1.1.1.3" xref="S2.p1.21.m21.1.1.1.1.1.3.cmml">Q</mi></mrow><mo id="S2.p1.21.m21.1.1.1.1.3" xref="S2.p1.21.m21.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.21.m21.1b"><apply id="S2.p1.21.m21.1.1.cmml" xref="S2.p1.21.m21.1.1"><times id="S2.p1.21.m21.1.1.2.cmml" xref="S2.p1.21.m21.1.1.2"></times><apply id="S2.p1.21.m21.1.1.3.cmml" xref="S2.p1.21.m21.1.1.3"><csymbol cd="ambiguous" id="S2.p1.21.m21.1.1.3.1.cmml" xref="S2.p1.21.m21.1.1.3">subscript</csymbol><ci id="S2.p1.21.m21.1.1.3.2.cmml" xref="S2.p1.21.m21.1.1.3.2">𝐷</ci><ci id="S2.p1.21.m21.1.1.3.3.cmml" xref="S2.p1.21.m21.1.1.3.3">KL</ci></apply><apply id="S2.p1.21.m21.1.1.1.1.1.cmml" xref="S2.p1.21.m21.1.1.1.1"><csymbol cd="latexml" id="S2.p1.21.m21.1.1.1.1.1.1.cmml" xref="S2.p1.21.m21.1.1.1.1.1.1">conditional</csymbol><ci id="S2.p1.21.m21.1.1.1.1.1.2.cmml" xref="S2.p1.21.m21.1.1.1.1.1.2">𝑃</ci><ci id="S2.p1.21.m21.1.1.1.1.1.3.cmml" xref="S2.p1.21.m21.1.1.1.1.1.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.21.m21.1c">D_{\rm KL}\left(P\|Q\right)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.21.m21.1d">italic_D start_POSTSUBSCRIPT roman_KL end_POSTSUBSCRIPT ( italic_P ∥ italic_Q )</annotation></semantics></math> denotes the KL-divergence between distributions <math alttext="P" class="ltx_Math" display="inline" id="S2.p1.22.m22.1"><semantics id="S2.p1.22.m22.1a"><mi id="S2.p1.22.m22.1.1" xref="S2.p1.22.m22.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.p1.22.m22.1b"><ci id="S2.p1.22.m22.1.1.cmml" xref="S2.p1.22.m22.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.22.m22.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.p1.22.m22.1d">italic_P</annotation></semantics></math> and <math alttext="Q" class="ltx_Math" display="inline" id="S2.p1.23.m23.1"><semantics id="S2.p1.23.m23.1a"><mi id="S2.p1.23.m23.1.1" xref="S2.p1.23.m23.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.p1.23.m23.1b"><ci id="S2.p1.23.m23.1.1.cmml" xref="S2.p1.23.m23.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.23.m23.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.p1.23.m23.1d">italic_Q</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Input data: Random Markov chains</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.3">To investigate the ICL capabilities of Mamba, we let the input tokens to be stochastic and drawn from a random Markov chain of order <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_k</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">edelman2024evolution</span>]</cite>. That is, the token sequence <math alttext="x=(x_{t})_{t=1}^{T}\in\mathcal{X}^{T}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mrow id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">x</mi><mo id="S2.SS1.p1.2.m2.1.1.4" xref="S2.SS1.p1.2.m2.1.1.4.cmml">=</mo><msubsup id="S2.SS1.p1.2.m2.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.cmml"><mrow id="S2.SS1.p1.2.m2.1.1.1.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p1.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p1.2.m2.1.1.1.1.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.3" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.SS1.p1.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS1.p1.2.m2.1.1.1.1.3" xref="S2.SS1.p1.2.m2.1.1.1.1.3.cmml"><mi id="S2.SS1.p1.2.m2.1.1.1.1.3.2" xref="S2.SS1.p1.2.m2.1.1.1.1.3.2.cmml">t</mi><mo id="S2.SS1.p1.2.m2.1.1.1.1.3.1" xref="S2.SS1.p1.2.m2.1.1.1.1.3.1.cmml">=</mo><mn id="S2.SS1.p1.2.m2.1.1.1.1.3.3" xref="S2.SS1.p1.2.m2.1.1.1.1.3.3.cmml">1</mn></mrow><mi id="S2.SS1.p1.2.m2.1.1.1.3" xref="S2.SS1.p1.2.m2.1.1.1.3.cmml">T</mi></msubsup><mo id="S2.SS1.p1.2.m2.1.1.5" xref="S2.SS1.p1.2.m2.1.1.5.cmml">∈</mo><msup id="S2.SS1.p1.2.m2.1.1.6" xref="S2.SS1.p1.2.m2.1.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.2.m2.1.1.6.2" xref="S2.SS1.p1.2.m2.1.1.6.2.cmml">𝒳</mi><mi id="S2.SS1.p1.2.m2.1.1.6.3" xref="S2.SS1.p1.2.m2.1.1.6.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><and id="S2.SS1.p1.2.m2.1.1a.cmml" xref="S2.SS1.p1.2.m2.1.1"></and><apply id="S2.SS1.p1.2.m2.1.1b.cmml" xref="S2.SS1.p1.2.m2.1.1"><eq 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id="S2.SS1.p1.2.m2.1c">x=(x_{t})_{t=1}^{T}\in\mathcal{X}^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_x = ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> on the state space (vocabulary) <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><ci id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">caligraphic_X</annotation></semantics></math> follows the transition dynamics:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx1"> <tbody id="S2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{P}\left(x_{t+1}=\cdot\mid x_{1}^{t}\right)=\mathbb{P}% \left(x_{t+1}=\cdot\mid x_{t-k+1}^{t}\right)," class="ltx_math_unparsed" display="inline" id="S2.E1.m1.1"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1b"><mi id="S2.E1.m1.1.1">ℙ</mi><mrow id="S2.E1.m1.1.2"><mo id="S2.E1.m1.1.2.1">(</mo><msub id="S2.E1.m1.1.2.2"><mi id="S2.E1.m1.1.2.2.2">x</mi><mrow id="S2.E1.m1.1.2.2.3"><mi id="S2.E1.m1.1.2.2.3.2">t</mi><mo id="S2.E1.m1.1.2.2.3.1">+</mo><mn id="S2.E1.m1.1.2.2.3.3">1</mn></mrow></msub><mo id="S2.E1.m1.1.2.3" rspace="0em">=</mo><mo id="S2.E1.m1.1.2.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.E1.m1.1.2.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S2.E1.m1.1.2.6"><mi id="S2.E1.m1.1.2.6.2.2">x</mi><mn id="S2.E1.m1.1.2.6.2.3">1</mn><mi id="S2.E1.m1.1.2.6.3">t</mi></msubsup><mo id="S2.E1.m1.1.2.7">)</mo></mrow><mo id="S2.E1.m1.1.3">=</mo><mi id="S2.E1.m1.1.4">ℙ</mi><mrow id="S2.E1.m1.1.5"><mo id="S2.E1.m1.1.5.1">(</mo><msub id="S2.E1.m1.1.5.2"><mi id="S2.E1.m1.1.5.2.2">x</mi><mrow id="S2.E1.m1.1.5.2.3"><mi id="S2.E1.m1.1.5.2.3.2">t</mi><mo id="S2.E1.m1.1.5.2.3.1">+</mo><mn id="S2.E1.m1.1.5.2.3.3">1</mn></mrow></msub><mo id="S2.E1.m1.1.5.3" rspace="0em">=</mo><mo id="S2.E1.m1.1.5.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.E1.m1.1.5.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S2.E1.m1.1.5.6"><mi id="S2.E1.m1.1.5.6.2.2">x</mi><mrow id="S2.E1.m1.1.5.6.2.3"><mrow id="S2.E1.m1.1.5.6.2.3.2"><mi id="S2.E1.m1.1.5.6.2.3.2.2">t</mi><mo id="S2.E1.m1.1.5.6.2.3.2.1">−</mo><mi id="S2.E1.m1.1.5.6.2.3.2.3">k</mi></mrow><mo id="S2.E1.m1.1.5.6.2.3.1">+</mo><mn id="S2.E1.m1.1.5.6.2.3.3">1</mn></mrow><mi id="S2.E1.m1.1.5.6.3">t</mi></msubsup><mo id="S2.E1.m1.1.5.7">)</mo></mrow><mo id="S2.E1.m1.1.6">,</mo></mrow><annotation encoding="application/x-tex" id="S2.E1.m1.1c">\displaystyle\mathbb{P}\left(x_{t+1}=\cdot\mid x_{1}^{t}\right)=\mathbb{P}% \left(x_{t+1}=\cdot\mid x_{t-k+1}^{t}\right),</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">blackboard_P ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = blackboard_P ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = ⋅ ∣ italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p1.13">almost surely for all <math alttext="t\in[T]" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m1.1"><semantics id="S2.SS1.p1.4.m1.1a"><mrow id="S2.SS1.p1.4.m1.1.2" xref="S2.SS1.p1.4.m1.1.2.cmml"><mi id="S2.SS1.p1.4.m1.1.2.2" xref="S2.SS1.p1.4.m1.1.2.2.cmml">t</mi><mo id="S2.SS1.p1.4.m1.1.2.1" xref="S2.SS1.p1.4.m1.1.2.1.cmml">∈</mo><mrow id="S2.SS1.p1.4.m1.1.2.3.2" xref="S2.SS1.p1.4.m1.1.2.3.1.cmml"><mo id="S2.SS1.p1.4.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.4.m1.1.2.3.1.1.cmml">[</mo><mi id="S2.SS1.p1.4.m1.1.1" xref="S2.SS1.p1.4.m1.1.1.cmml">T</mi><mo id="S2.SS1.p1.4.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.4.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m1.1b"><apply id="S2.SS1.p1.4.m1.1.2.cmml" xref="S2.SS1.p1.4.m1.1.2"><in id="S2.SS1.p1.4.m1.1.2.1.cmml" xref="S2.SS1.p1.4.m1.1.2.1"></in><ci id="S2.SS1.p1.4.m1.1.2.2.cmml" xref="S2.SS1.p1.4.m1.1.2.2">𝑡</ci><apply id="S2.SS1.p1.4.m1.1.2.3.1.cmml" xref="S2.SS1.p1.4.m1.1.2.3.2"><csymbol cd="latexml" id="S2.SS1.p1.4.m1.1.2.3.1.1.cmml" xref="S2.SS1.p1.4.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS1.p1.4.m1.1.1.cmml" xref="S2.SS1.p1.4.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m1.1c">t\in[T]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m1.1d">italic_t ∈ [ italic_T ]</annotation></semantics></math>, and the <math alttext="k^{\text{th}}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m2.1"><semantics id="S2.SS1.p1.5.m2.1a"><msup id="S2.SS1.p1.5.m2.1.1" xref="S2.SS1.p1.5.m2.1.1.cmml"><mi id="S2.SS1.p1.5.m2.1.1.2" xref="S2.SS1.p1.5.m2.1.1.2.cmml">k</mi><mtext id="S2.SS1.p1.5.m2.1.1.3" xref="S2.SS1.p1.5.m2.1.1.3a.cmml">th</mtext></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m2.1b"><apply id="S2.SS1.p1.5.m2.1.1.cmml" xref="S2.SS1.p1.5.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m2.1.1.1.cmml" xref="S2.SS1.p1.5.m2.1.1">superscript</csymbol><ci id="S2.SS1.p1.5.m2.1.1.2.cmml" xref="S2.SS1.p1.5.m2.1.1.2">𝑘</ci><ci id="S2.SS1.p1.5.m2.1.1.3a.cmml" xref="S2.SS1.p1.5.m2.1.1.3"><mtext id="S2.SS1.p1.5.m2.1.1.3.cmml" mathsize="70%" xref="S2.SS1.p1.5.m2.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m2.1c">k^{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m2.1d">italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT</annotation></semantics></math>-order Markov kernels, <math alttext="\mathbb{P}\left(x_{t+1}=\cdot\mid x_{t-k+1}^{t}=i_{t-k+1}^{t}\right)" class="ltx_math_unparsed" display="inline" id="S2.SS1.p1.6.m3.1"><semantics id="S2.SS1.p1.6.m3.1a"><mrow id="S2.SS1.p1.6.m3.1b"><mi id="S2.SS1.p1.6.m3.1.1">ℙ</mi><mrow id="S2.SS1.p1.6.m3.1.2"><mo id="S2.SS1.p1.6.m3.1.2.1">(</mo><msub id="S2.SS1.p1.6.m3.1.2.2"><mi id="S2.SS1.p1.6.m3.1.2.2.2">x</mi><mrow id="S2.SS1.p1.6.m3.1.2.2.3"><mi id="S2.SS1.p1.6.m3.1.2.2.3.2">t</mi><mo id="S2.SS1.p1.6.m3.1.2.2.3.1">+</mo><mn id="S2.SS1.p1.6.m3.1.2.2.3.3">1</mn></mrow></msub><mo id="S2.SS1.p1.6.m3.1.2.3" rspace="0em">=</mo><mo id="S2.SS1.p1.6.m3.1.2.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.SS1.p1.6.m3.1.2.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S2.SS1.p1.6.m3.1.2.6"><mi id="S2.SS1.p1.6.m3.1.2.6.2.2">x</mi><mrow id="S2.SS1.p1.6.m3.1.2.6.2.3"><mrow id="S2.SS1.p1.6.m3.1.2.6.2.3.2"><mi id="S2.SS1.p1.6.m3.1.2.6.2.3.2.2">t</mi><mo id="S2.SS1.p1.6.m3.1.2.6.2.3.2.1">−</mo><mi id="S2.SS1.p1.6.m3.1.2.6.2.3.2.3">k</mi></mrow><mo id="S2.SS1.p1.6.m3.1.2.6.2.3.1">+</mo><mn id="S2.SS1.p1.6.m3.1.2.6.2.3.3">1</mn></mrow><mi id="S2.SS1.p1.6.m3.1.2.6.3">t</mi></msubsup><mo id="S2.SS1.p1.6.m3.1.2.7">=</mo><msubsup id="S2.SS1.p1.6.m3.1.2.8"><mi id="S2.SS1.p1.6.m3.1.2.8.2.2">i</mi><mrow id="S2.SS1.p1.6.m3.1.2.8.2.3"><mrow id="S2.SS1.p1.6.m3.1.2.8.2.3.2"><mi id="S2.SS1.p1.6.m3.1.2.8.2.3.2.2">t</mi><mo id="S2.SS1.p1.6.m3.1.2.8.2.3.2.1">−</mo><mi id="S2.SS1.p1.6.m3.1.2.8.2.3.2.3">k</mi></mrow><mo id="S2.SS1.p1.6.m3.1.2.8.2.3.1">+</mo><mn id="S2.SS1.p1.6.m3.1.2.8.2.3.3">1</mn></mrow><mi id="S2.SS1.p1.6.m3.1.2.8.3">t</mi></msubsup><mo id="S2.SS1.p1.6.m3.1.2.9">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m3.1c">\mathbb{P}\left(x_{t+1}=\cdot\mid x_{t-k+1}^{t}=i_{t-k+1}^{t}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m3.1d">blackboard_P ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = ⋅ ∣ italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = italic_i start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>, are sampled independently for each tuple <math alttext="(i_{t-k+1},\cdots,i_{t})" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m4.3"><semantics id="S2.SS1.p1.7.m4.3a"><mrow id="S2.SS1.p1.7.m4.3.3.2" xref="S2.SS1.p1.7.m4.3.3.3.cmml"><mo id="S2.SS1.p1.7.m4.3.3.2.3" stretchy="false" xref="S2.SS1.p1.7.m4.3.3.3.cmml">(</mo><msub id="S2.SS1.p1.7.m4.2.2.1.1" xref="S2.SS1.p1.7.m4.2.2.1.1.cmml"><mi id="S2.SS1.p1.7.m4.2.2.1.1.2" xref="S2.SS1.p1.7.m4.2.2.1.1.2.cmml">i</mi><mrow id="S2.SS1.p1.7.m4.2.2.1.1.3" xref="S2.SS1.p1.7.m4.2.2.1.1.3.cmml"><mrow id="S2.SS1.p1.7.m4.2.2.1.1.3.2" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.cmml"><mi id="S2.SS1.p1.7.m4.2.2.1.1.3.2.2" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.2.cmml">t</mi><mo id="S2.SS1.p1.7.m4.2.2.1.1.3.2.1" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.1.cmml">−</mo><mi id="S2.SS1.p1.7.m4.2.2.1.1.3.2.3" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.3.cmml">k</mi></mrow><mo id="S2.SS1.p1.7.m4.2.2.1.1.3.1" xref="S2.SS1.p1.7.m4.2.2.1.1.3.1.cmml">+</mo><mn id="S2.SS1.p1.7.m4.2.2.1.1.3.3" xref="S2.SS1.p1.7.m4.2.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p1.7.m4.3.3.2.4" xref="S2.SS1.p1.7.m4.3.3.3.cmml">,</mo><mi id="S2.SS1.p1.7.m4.1.1" mathvariant="normal" xref="S2.SS1.p1.7.m4.1.1.cmml">⋯</mi><mo id="S2.SS1.p1.7.m4.3.3.2.5" xref="S2.SS1.p1.7.m4.3.3.3.cmml">,</mo><msub id="S2.SS1.p1.7.m4.3.3.2.2" xref="S2.SS1.p1.7.m4.3.3.2.2.cmml"><mi id="S2.SS1.p1.7.m4.3.3.2.2.2" xref="S2.SS1.p1.7.m4.3.3.2.2.2.cmml">i</mi><mi id="S2.SS1.p1.7.m4.3.3.2.2.3" xref="S2.SS1.p1.7.m4.3.3.2.2.3.cmml">t</mi></msub><mo id="S2.SS1.p1.7.m4.3.3.2.6" stretchy="false" xref="S2.SS1.p1.7.m4.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m4.3b"><vector id="S2.SS1.p1.7.m4.3.3.3.cmml" xref="S2.SS1.p1.7.m4.3.3.2"><apply id="S2.SS1.p1.7.m4.2.2.1.1.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m4.2.2.1.1.1.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p1.7.m4.2.2.1.1.2.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.2">𝑖</ci><apply id="S2.SS1.p1.7.m4.2.2.1.1.3.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.3"><plus id="S2.SS1.p1.7.m4.2.2.1.1.3.1.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.3.1"></plus><apply id="S2.SS1.p1.7.m4.2.2.1.1.3.2.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2"><minus id="S2.SS1.p1.7.m4.2.2.1.1.3.2.1.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.1"></minus><ci id="S2.SS1.p1.7.m4.2.2.1.1.3.2.2.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.2">𝑡</ci><ci id="S2.SS1.p1.7.m4.2.2.1.1.3.2.3.cmml" xref="S2.SS1.p1.7.m4.2.2.1.1.3.2.3">𝑘</ci></apply><cn id="S2.SS1.p1.7.m4.2.2.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.7.m4.2.2.1.1.3.3">1</cn></apply></apply><ci id="S2.SS1.p1.7.m4.1.1.cmml" xref="S2.SS1.p1.7.m4.1.1">⋯</ci><apply id="S2.SS1.p1.7.m4.3.3.2.2.cmml" xref="S2.SS1.p1.7.m4.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m4.3.3.2.2.1.cmml" xref="S2.SS1.p1.7.m4.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p1.7.m4.3.3.2.2.2.cmml" xref="S2.SS1.p1.7.m4.3.3.2.2.2">𝑖</ci><ci id="S2.SS1.p1.7.m4.3.3.2.2.3.cmml" xref="S2.SS1.p1.7.m4.3.3.2.2.3">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m4.3c">(i_{t-k+1},\cdots,i_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m4.3d">( italic_i start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT , ⋯ , italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> from the Dirichlet prior <math alttext="\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m5.1"><semantics id="S2.SS1.p1.8.m5.1a"><mrow id="S2.SS1.p1.8.m5.1.1" xref="S2.SS1.p1.8.m5.1.1.cmml"><mi id="S2.SS1.p1.8.m5.1.1.3" xref="S2.SS1.p1.8.m5.1.1.3.cmml">Dir</mi><mo id="S2.SS1.p1.8.m5.1.1.2" xref="S2.SS1.p1.8.m5.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.8.m5.1.1.1.1" xref="S2.SS1.p1.8.m5.1.1.1.1.1.cmml"><mo id="S2.SS1.p1.8.m5.1.1.1.1.2" stretchy="false" xref="S2.SS1.p1.8.m5.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p1.8.m5.1.1.1.1.1" xref="S2.SS1.p1.8.m5.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.8.m5.1.1.1.1.1.2" xref="S2.SS1.p1.8.m5.1.1.1.1.1.2.cmml">β</mi><mo id="S2.SS1.p1.8.m5.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.8.m5.1.1.1.1.1.1.cmml">⋅</mo><mn id="S2.SS1.p1.8.m5.1.1.1.1.1.3" xref="S2.SS1.p1.8.m5.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S2.SS1.p1.8.m5.1.1.1.1.3" stretchy="false" xref="S2.SS1.p1.8.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m5.1b"><apply id="S2.SS1.p1.8.m5.1.1.cmml" xref="S2.SS1.p1.8.m5.1.1"><times id="S2.SS1.p1.8.m5.1.1.2.cmml" xref="S2.SS1.p1.8.m5.1.1.2"></times><ci id="S2.SS1.p1.8.m5.1.1.3.cmml" xref="S2.SS1.p1.8.m5.1.1.3">Dir</ci><apply id="S2.SS1.p1.8.m5.1.1.1.1.1.cmml" xref="S2.SS1.p1.8.m5.1.1.1.1"><ci id="S2.SS1.p1.8.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.8.m5.1.1.1.1.1.1">⋅</ci><ci id="S2.SS1.p1.8.m5.1.1.1.1.1.2.cmml" xref="S2.SS1.p1.8.m5.1.1.1.1.1.2">𝛽</ci><cn id="S2.SS1.p1.8.m5.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.8.m5.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m5.1c">\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m5.1d">roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math>, with <math alttext="\beta>0" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m6.1"><semantics id="S2.SS1.p1.9.m6.1a"><mrow id="S2.SS1.p1.9.m6.1.1" xref="S2.SS1.p1.9.m6.1.1.cmml"><mi id="S2.SS1.p1.9.m6.1.1.2" xref="S2.SS1.p1.9.m6.1.1.2.cmml">β</mi><mo id="S2.SS1.p1.9.m6.1.1.1" xref="S2.SS1.p1.9.m6.1.1.1.cmml">></mo><mn id="S2.SS1.p1.9.m6.1.1.3" xref="S2.SS1.p1.9.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m6.1b"><apply id="S2.SS1.p1.9.m6.1.1.cmml" xref="S2.SS1.p1.9.m6.1.1"><gt id="S2.SS1.p1.9.m6.1.1.1.cmml" xref="S2.SS1.p1.9.m6.1.1.1"></gt><ci id="S2.SS1.p1.9.m6.1.1.2.cmml" xref="S2.SS1.p1.9.m6.1.1.2">𝛽</ci><cn id="S2.SS1.p1.9.m6.1.1.3.cmml" type="integer" xref="S2.SS1.p1.9.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m6.1c">\beta>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m6.1d">italic_β > 0</annotation></semantics></math>. When <math alttext="\beta=1" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m7.1"><semantics id="S2.SS1.p1.10.m7.1a"><mrow id="S2.SS1.p1.10.m7.1.1" xref="S2.SS1.p1.10.m7.1.1.cmml"><mi id="S2.SS1.p1.10.m7.1.1.2" xref="S2.SS1.p1.10.m7.1.1.2.cmml">β</mi><mo id="S2.SS1.p1.10.m7.1.1.1" xref="S2.SS1.p1.10.m7.1.1.1.cmml">=</mo><mn id="S2.SS1.p1.10.m7.1.1.3" xref="S2.SS1.p1.10.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m7.1b"><apply id="S2.SS1.p1.10.m7.1.1.cmml" xref="S2.SS1.p1.10.m7.1.1"><eq id="S2.SS1.p1.10.m7.1.1.1.cmml" xref="S2.SS1.p1.10.m7.1.1.1"></eq><ci id="S2.SS1.p1.10.m7.1.1.2.cmml" xref="S2.SS1.p1.10.m7.1.1.2">𝛽</ci><cn id="S2.SS1.p1.10.m7.1.1.3.cmml" type="integer" xref="S2.SS1.p1.10.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m7.1c">\beta=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m7.1d">italic_β = 1</annotation></semantics></math>, this corresponds to the uniform distribution on the <math alttext="S" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m8.1"><semantics id="S2.SS1.p1.11.m8.1a"><mi id="S2.SS1.p1.11.m8.1.1" xref="S2.SS1.p1.11.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m8.1b"><ci id="S2.SS1.p1.11.m8.1.1.cmml" xref="S2.SS1.p1.11.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m8.1d">italic_S</annotation></semantics></math>-dimensional simplex <math alttext="\Delta_{1}^{S}" class="ltx_Math" display="inline" id="S2.SS1.p1.12.m9.1"><semantics id="S2.SS1.p1.12.m9.1a"><msubsup id="S2.SS1.p1.12.m9.1.1" xref="S2.SS1.p1.12.m9.1.1.cmml"><mi id="S2.SS1.p1.12.m9.1.1.2.2" mathvariant="normal" xref="S2.SS1.p1.12.m9.1.1.2.2.cmml">Δ</mi><mn id="S2.SS1.p1.12.m9.1.1.2.3" xref="S2.SS1.p1.12.m9.1.1.2.3.cmml">1</mn><mi id="S2.SS1.p1.12.m9.1.1.3" xref="S2.SS1.p1.12.m9.1.1.3.cmml">S</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.12.m9.1b"><apply id="S2.SS1.p1.12.m9.1.1.cmml" xref="S2.SS1.p1.12.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m9.1.1.1.cmml" xref="S2.SS1.p1.12.m9.1.1">superscript</csymbol><apply id="S2.SS1.p1.12.m9.1.1.2.cmml" xref="S2.SS1.p1.12.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m9.1.1.2.1.cmml" xref="S2.SS1.p1.12.m9.1.1">subscript</csymbol><ci id="S2.SS1.p1.12.m9.1.1.2.2.cmml" xref="S2.SS1.p1.12.m9.1.1.2.2">Δ</ci><cn id="S2.SS1.p1.12.m9.1.1.2.3.cmml" type="integer" xref="S2.SS1.p1.12.m9.1.1.2.3">1</cn></apply><ci id="S2.SS1.p1.12.m9.1.1.3.cmml" xref="S2.SS1.p1.12.m9.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.12.m9.1c">\Delta_{1}^{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.12.m9.1d">roman_Δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>, where size <math alttext="S=|\mathcal{X}|" class="ltx_Math" display="inline" id="S2.SS1.p1.13.m10.1"><semantics id="S2.SS1.p1.13.m10.1a"><mrow id="S2.SS1.p1.13.m10.1.2" xref="S2.SS1.p1.13.m10.1.2.cmml"><mi id="S2.SS1.p1.13.m10.1.2.2" xref="S2.SS1.p1.13.m10.1.2.2.cmml">S</mi><mo id="S2.SS1.p1.13.m10.1.2.1" xref="S2.SS1.p1.13.m10.1.2.1.cmml">=</mo><mrow id="S2.SS1.p1.13.m10.1.2.3.2" xref="S2.SS1.p1.13.m10.1.2.3.1.cmml"><mo id="S2.SS1.p1.13.m10.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.13.m10.1.2.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.13.m10.1.1" xref="S2.SS1.p1.13.m10.1.1.cmml">𝒳</mi><mo id="S2.SS1.p1.13.m10.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.13.m10.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.13.m10.1b"><apply id="S2.SS1.p1.13.m10.1.2.cmml" xref="S2.SS1.p1.13.m10.1.2"><eq id="S2.SS1.p1.13.m10.1.2.1.cmml" xref="S2.SS1.p1.13.m10.1.2.1"></eq><ci id="S2.SS1.p1.13.m10.1.2.2.cmml" xref="S2.SS1.p1.13.m10.1.2.2">𝑆</ci><apply id="S2.SS1.p1.13.m10.1.2.3.1.cmml" xref="S2.SS1.p1.13.m10.1.2.3.2"><abs id="S2.SS1.p1.13.m10.1.2.3.1.1.cmml" xref="S2.SS1.p1.13.m10.1.2.3.2.1"></abs><ci id="S2.SS1.p1.13.m10.1.1.cmml" xref="S2.SS1.p1.13.m10.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.13.m10.1c">S=|\mathcal{X}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.13.m10.1d">italic_S = | caligraphic_X |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.8">The transition matrix <math alttext="P=(P_{i_{1}^{k}})_{i_{1}^{k}\in\mathcal{X}^{k}},P_{i_{1}^{k}}\in[0,1]^{S}" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.4"><semantics id="S2.SS1.p2.1.m1.4a"><mrow id="S2.SS1.p2.1.m1.4.4.2" xref="S2.SS1.p2.1.m1.4.4.3.cmml"><mrow id="S2.SS1.p2.1.m1.3.3.1.1" xref="S2.SS1.p2.1.m1.3.3.1.1.cmml"><mi id="S2.SS1.p2.1.m1.3.3.1.1.3" xref="S2.SS1.p2.1.m1.3.3.1.1.3.cmml">P</mi><mo id="S2.SS1.p2.1.m1.3.3.1.1.2" xref="S2.SS1.p2.1.m1.3.3.1.1.2.cmml">=</mo><msub id="S2.SS1.p2.1.m1.3.3.1.1.1" xref="S2.SS1.p2.1.m1.3.3.1.1.1.cmml"><mrow id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.2" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.2.cmml">P</mi><msubsup id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.2.2" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.2.2.cmml">i</mi><mn id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.2.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></msub><mo id="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.1.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS1.p2.1.m1.3.3.1.1.1.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.cmml"><msubsup id="S2.SS1.p2.1.m1.3.3.1.1.1.3.2" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.cmml"><mi id="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.2.2" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.2.2.cmml">i</mi><mn id="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.2.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.2.3.cmml">1</mn><mi id="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.2.3.cmml">k</mi></msubsup><mo id="S2.SS1.p2.1.m1.3.3.1.1.1.3.1" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.1.cmml">∈</mo><msup id="S2.SS1.p2.1.m1.3.3.1.1.1.3.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.3.3.1.1.1.3.3.2" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.3.2.cmml">𝒳</mi><mi id="S2.SS1.p2.1.m1.3.3.1.1.1.3.3.3" xref="S2.SS1.p2.1.m1.3.3.1.1.1.3.3.3.cmml">k</mi></msup></mrow></msub></mrow><mo id="S2.SS1.p2.1.m1.4.4.2.3" xref="S2.SS1.p2.1.m1.4.4.3a.cmml">,</mo><mrow id="S2.SS1.p2.1.m1.4.4.2.2" xref="S2.SS1.p2.1.m1.4.4.2.2.cmml"><msub id="S2.SS1.p2.1.m1.4.4.2.2.2" xref="S2.SS1.p2.1.m1.4.4.2.2.2.cmml"><mi id="S2.SS1.p2.1.m1.4.4.2.2.2.2" xref="S2.SS1.p2.1.m1.4.4.2.2.2.2.cmml">P</mi><msubsup id="S2.SS1.p2.1.m1.4.4.2.2.2.3" xref="S2.SS1.p2.1.m1.4.4.2.2.2.3.cmml"><mi id="S2.SS1.p2.1.m1.4.4.2.2.2.3.2.2" xref="S2.SS1.p2.1.m1.4.4.2.2.2.3.2.2.cmml">i</mi><mn id="S2.SS1.p2.1.m1.4.4.2.2.2.3.2.3" xref="S2.SS1.p2.1.m1.4.4.2.2.2.3.2.3.cmml">1</mn><mi id="S2.SS1.p2.1.m1.4.4.2.2.2.3.3" xref="S2.SS1.p2.1.m1.4.4.2.2.2.3.3.cmml">k</mi></msubsup></msub><mo id="S2.SS1.p2.1.m1.4.4.2.2.1" xref="S2.SS1.p2.1.m1.4.4.2.2.1.cmml">∈</mo><msup id="S2.SS1.p2.1.m1.4.4.2.2.3" xref="S2.SS1.p2.1.m1.4.4.2.2.3.cmml"><mrow id="S2.SS1.p2.1.m1.4.4.2.2.3.2.2" xref="S2.SS1.p2.1.m1.4.4.2.2.3.2.1.cmml"><mo 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xref="S2.SS1.p2.1.m1.4.4.2.2.3">superscript</csymbol><interval closure="closed" id="S2.SS1.p2.1.m1.4.4.2.2.3.2.1.cmml" xref="S2.SS1.p2.1.m1.4.4.2.2.3.2.2"><cn id="S2.SS1.p2.1.m1.1.1.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1">0</cn><cn id="S2.SS1.p2.1.m1.2.2.cmml" type="integer" xref="S2.SS1.p2.1.m1.2.2">1</cn></interval><ci id="S2.SS1.p2.1.m1.4.4.2.2.3.3.cmml" xref="S2.SS1.p2.1.m1.4.4.2.2.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.4c">P=(P_{i_{1}^{k}})_{i_{1}^{k}\in\mathcal{X}^{k}},P_{i_{1}^{k}}\in[0,1]^{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.4d">italic_P = ( italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>, encapsulates the set of all <math alttext="S^{k}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><msup id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml"><mi id="S2.SS1.p2.2.m2.1.1.2" xref="S2.SS1.p2.2.m2.1.1.2.cmml">S</mi><mi id="S2.SS1.p2.2.m2.1.1.3" xref="S2.SS1.p2.2.m2.1.1.3.cmml">k</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.1.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">superscript</csymbol><ci id="S2.SS1.p2.2.m2.1.1.2.cmml" xref="S2.SS1.p2.2.m2.1.1.2">𝑆</ci><ci id="S2.SS1.p2.2.m2.1.1.3.cmml" xref="S2.SS1.p2.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">S^{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_S start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> conditional probabilities of the chain, each row corresponding to one of them. While this transition matrix governs the generation of each token <math alttext="x_{t}" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><msub id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml">x</mi><mi id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2">𝑥</ci><ci id="S2.SS1.p2.3.m3.1.1.3.cmml" xref="S2.SS1.p2.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="t>k" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mrow id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml"><mi id="S2.SS1.p2.4.m4.1.1.2" xref="S2.SS1.p2.4.m4.1.1.2.cmml">t</mi><mo id="S2.SS1.p2.4.m4.1.1.1" xref="S2.SS1.p2.4.m4.1.1.1.cmml">></mo><mi id="S2.SS1.p2.4.m4.1.1.3" xref="S2.SS1.p2.4.m4.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1"><gt id="S2.SS1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1.1"></gt><ci id="S2.SS1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2">𝑡</ci><ci id="S2.SS1.p2.4.m4.1.1.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">t>k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_t > italic_k</annotation></semantics></math>, the first <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mi id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><ci id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_k</annotation></semantics></math>-tokens <math alttext="x_{1},\ldots,x_{k}" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.3"><semantics id="S2.SS1.p2.6.m6.3a"><mrow id="S2.SS1.p2.6.m6.3.3.2" xref="S2.SS1.p2.6.m6.3.3.3.cmml"><msub id="S2.SS1.p2.6.m6.2.2.1.1" xref="S2.SS1.p2.6.m6.2.2.1.1.cmml"><mi id="S2.SS1.p2.6.m6.2.2.1.1.2" xref="S2.SS1.p2.6.m6.2.2.1.1.2.cmml">x</mi><mn id="S2.SS1.p2.6.m6.2.2.1.1.3" xref="S2.SS1.p2.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p2.6.m6.3.3.2.3" xref="S2.SS1.p2.6.m6.3.3.3.cmml">,</mo><mi id="S2.SS1.p2.6.m6.1.1" mathvariant="normal" xref="S2.SS1.p2.6.m6.1.1.cmml">…</mi><mo id="S2.SS1.p2.6.m6.3.3.2.4" xref="S2.SS1.p2.6.m6.3.3.3.cmml">,</mo><msub id="S2.SS1.p2.6.m6.3.3.2.2" xref="S2.SS1.p2.6.m6.3.3.2.2.cmml"><mi id="S2.SS1.p2.6.m6.3.3.2.2.2" xref="S2.SS1.p2.6.m6.3.3.2.2.2.cmml">x</mi><mi id="S2.SS1.p2.6.m6.3.3.2.2.3" xref="S2.SS1.p2.6.m6.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m6.3b"><list id="S2.SS1.p2.6.m6.3.3.3.cmml" xref="S2.SS1.p2.6.m6.3.3.2"><apply id="S2.SS1.p2.6.m6.2.2.1.1.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.2.2.1.1.1.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p2.6.m6.2.2.1.1.2.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1.2">𝑥</ci><cn id="S2.SS1.p2.6.m6.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p2.6.m6.2.2.1.1.3">1</cn></apply><ci id="S2.SS1.p2.6.m6.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1">…</ci><apply id="S2.SS1.p2.6.m6.3.3.2.2.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.3.3.2.2.1.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p2.6.m6.3.3.2.2.2.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2.2">𝑥</ci><ci id="S2.SS1.p2.6.m6.3.3.2.2.3.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.3c">x_{1},\ldots,x_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.3d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> are drawn i.i.d. from <math alttext="\mathrm{Unif}(\mathcal{X})" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.1"><semantics id="S2.SS1.p2.7.m7.1a"><mrow id="S2.SS1.p2.7.m7.1.2" xref="S2.SS1.p2.7.m7.1.2.cmml"><mi id="S2.SS1.p2.7.m7.1.2.2" xref="S2.SS1.p2.7.m7.1.2.2.cmml">Unif</mi><mo id="S2.SS1.p2.7.m7.1.2.1" xref="S2.SS1.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.7.m7.1.2.3.2" xref="S2.SS1.p2.7.m7.1.2.cmml"><mo id="S2.SS1.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.7.m7.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml">𝒳</mi><mo id="S2.SS1.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.1b"><apply id="S2.SS1.p2.7.m7.1.2.cmml" xref="S2.SS1.p2.7.m7.1.2"><times id="S2.SS1.p2.7.m7.1.2.1.cmml" xref="S2.SS1.p2.7.m7.1.2.1"></times><ci id="S2.SS1.p2.7.m7.1.2.2.cmml" xref="S2.SS1.p2.7.m7.1.2.2">Unif</ci><ci id="S2.SS1.p2.7.m7.1.1.cmml" xref="S2.SS1.p2.7.m7.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.1c">\mathrm{Unif}(\mathcal{X})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.1d">roman_Unif ( caligraphic_X )</annotation></semantics></math>. This constitutes the joint law of the random variables <math alttext="(P,x)" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.2"><semantics id="S2.SS1.p2.8.m8.2a"><mrow id="S2.SS1.p2.8.m8.2.3.2" xref="S2.SS1.p2.8.m8.2.3.1.cmml"><mo id="S2.SS1.p2.8.m8.2.3.2.1" stretchy="false" xref="S2.SS1.p2.8.m8.2.3.1.cmml">(</mo><mi id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml">P</mi><mo id="S2.SS1.p2.8.m8.2.3.2.2" xref="S2.SS1.p2.8.m8.2.3.1.cmml">,</mo><mi id="S2.SS1.p2.8.m8.2.2" xref="S2.SS1.p2.8.m8.2.2.cmml">x</mi><mo id="S2.SS1.p2.8.m8.2.3.2.3" stretchy="false" xref="S2.SS1.p2.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.2b"><interval closure="open" id="S2.SS1.p2.8.m8.2.3.1.cmml" xref="S2.SS1.p2.8.m8.2.3.2"><ci id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1">𝑃</ci><ci id="S2.SS1.p2.8.m8.2.2.cmml" xref="S2.SS1.p2.8.m8.2.2">𝑥</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.2c">(P,x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.2d">( italic_P , italic_x )</annotation></semantics></math>, termed random Markov distribution henceforth. More succinctly,</p> </div> <div class="ltx_para ltx_noindent" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1"><span class="ltx_text ltx_font_bold" id="S2.SS1.p3.1.1">Data generation</span> (Random Markov sequences).</p> <ol class="ltx_enumerate" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.2">Draw <math alttext="P" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mi id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><ci id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">italic_P</annotation></semantics></math> with each row sampled i.i.d. from <math alttext="\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.1"><semantics id="S2.I1.i1.p1.2.m2.1a"><mrow id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml"><mi id="S2.I1.i1.p1.2.m2.1.1.3" xref="S2.I1.i1.p1.2.m2.1.1.3.cmml">Dir</mi><mo id="S2.I1.i1.p1.2.m2.1.1.2" xref="S2.I1.i1.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.I1.i1.p1.2.m2.1.1.1.1" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S2.I1.i1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S2.I1.i1.p1.2.m2.1.1.1.1.1" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.cmml"><mi id="S2.I1.i1.p1.2.m2.1.1.1.1.1.2" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.2.cmml">β</mi><mo id="S2.I1.i1.p1.2.m2.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml">⋅</mo><mn id="S2.I1.i1.p1.2.m2.1.1.1.1.1.3" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S2.I1.i1.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><apply id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1"><times id="S2.I1.i1.p1.2.m2.1.1.2.cmml" xref="S2.I1.i1.p1.2.m2.1.1.2"></times><ci id="S2.I1.i1.p1.2.m2.1.1.3.cmml" xref="S2.I1.i1.p1.2.m2.1.1.3">Dir</ci><apply id="S2.I1.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1.1.1"><ci id="S2.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.1">⋅</ci><ci id="S2.I1.i1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.2">𝛽</ci><cn id="S2.I1.i1.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S2.I1.i1.p1.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.1d">roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.2">For <math alttext="t=1,\ldots,k" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.3"><semantics id="S2.I1.i2.p1.1.m1.3a"><mrow id="S2.I1.i2.p1.1.m1.3.4" xref="S2.I1.i2.p1.1.m1.3.4.cmml"><mi id="S2.I1.i2.p1.1.m1.3.4.2" xref="S2.I1.i2.p1.1.m1.3.4.2.cmml">t</mi><mo id="S2.I1.i2.p1.1.m1.3.4.1" xref="S2.I1.i2.p1.1.m1.3.4.1.cmml">=</mo><mrow id="S2.I1.i2.p1.1.m1.3.4.3.2" xref="S2.I1.i2.p1.1.m1.3.4.3.1.cmml"><mn id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">1</mn><mo id="S2.I1.i2.p1.1.m1.3.4.3.2.1" xref="S2.I1.i2.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.I1.i2.p1.1.m1.2.2" mathvariant="normal" xref="S2.I1.i2.p1.1.m1.2.2.cmml">…</mi><mo id="S2.I1.i2.p1.1.m1.3.4.3.2.2" xref="S2.I1.i2.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.I1.i2.p1.1.m1.3.3" xref="S2.I1.i2.p1.1.m1.3.3.cmml">k</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.3b"><apply id="S2.I1.i2.p1.1.m1.3.4.cmml" xref="S2.I1.i2.p1.1.m1.3.4"><eq id="S2.I1.i2.p1.1.m1.3.4.1.cmml" xref="S2.I1.i2.p1.1.m1.3.4.1"></eq><ci id="S2.I1.i2.p1.1.m1.3.4.2.cmml" xref="S2.I1.i2.p1.1.m1.3.4.2">𝑡</ci><list id="S2.I1.i2.p1.1.m1.3.4.3.1.cmml" xref="S2.I1.i2.p1.1.m1.3.4.3.2"><cn id="S2.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S2.I1.i2.p1.1.m1.1.1">1</cn><ci id="S2.I1.i2.p1.1.m1.2.2.cmml" xref="S2.I1.i2.p1.1.m1.2.2">…</ci><ci id="S2.I1.i2.p1.1.m1.3.3.cmml" xref="S2.I1.i2.p1.1.m1.3.3">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.3c">t=1,\ldots,k</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.3d">italic_t = 1 , … , italic_k</annotation></semantics></math>, sample <math alttext="x_{t}\sim\mathrm{Unif}(\mathcal{X})" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.1"><semantics id="S2.I1.i2.p1.2.m2.1a"><mrow id="S2.I1.i2.p1.2.m2.1.2" xref="S2.I1.i2.p1.2.m2.1.2.cmml"><msub id="S2.I1.i2.p1.2.m2.1.2.2" xref="S2.I1.i2.p1.2.m2.1.2.2.cmml"><mi id="S2.I1.i2.p1.2.m2.1.2.2.2" xref="S2.I1.i2.p1.2.m2.1.2.2.2.cmml">x</mi><mi id="S2.I1.i2.p1.2.m2.1.2.2.3" xref="S2.I1.i2.p1.2.m2.1.2.2.3.cmml">t</mi></msub><mo id="S2.I1.i2.p1.2.m2.1.2.1" xref="S2.I1.i2.p1.2.m2.1.2.1.cmml">∼</mo><mrow id="S2.I1.i2.p1.2.m2.1.2.3" xref="S2.I1.i2.p1.2.m2.1.2.3.cmml"><mi id="S2.I1.i2.p1.2.m2.1.2.3.2" xref="S2.I1.i2.p1.2.m2.1.2.3.2.cmml">Unif</mi><mo id="S2.I1.i2.p1.2.m2.1.2.3.1" xref="S2.I1.i2.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S2.I1.i2.p1.2.m2.1.2.3.3.2" xref="S2.I1.i2.p1.2.m2.1.2.3.cmml"><mo id="S2.I1.i2.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S2.I1.i2.p1.2.m2.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml">𝒳</mi><mo id="S2.I1.i2.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S2.I1.i2.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.1b"><apply id="S2.I1.i2.p1.2.m2.1.2.cmml" xref="S2.I1.i2.p1.2.m2.1.2"><csymbol cd="latexml" id="S2.I1.i2.p1.2.m2.1.2.1.cmml" xref="S2.I1.i2.p1.2.m2.1.2.1">similar-to</csymbol><apply id="S2.I1.i2.p1.2.m2.1.2.2.cmml" xref="S2.I1.i2.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.1.2.2.1.cmml" xref="S2.I1.i2.p1.2.m2.1.2.2">subscript</csymbol><ci id="S2.I1.i2.p1.2.m2.1.2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.1.2.2.2">𝑥</ci><ci id="S2.I1.i2.p1.2.m2.1.2.2.3.cmml" xref="S2.I1.i2.p1.2.m2.1.2.2.3">𝑡</ci></apply><apply id="S2.I1.i2.p1.2.m2.1.2.3.cmml" xref="S2.I1.i2.p1.2.m2.1.2.3"><times id="S2.I1.i2.p1.2.m2.1.2.3.1.cmml" xref="S2.I1.i2.p1.2.m2.1.2.3.1"></times><ci id="S2.I1.i2.p1.2.m2.1.2.3.2.cmml" xref="S2.I1.i2.p1.2.m2.1.2.3.2">Unif</ci><ci id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.1c">x_{t}\sim\mathrm{Unif}(\mathcal{X})</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∼ roman_Unif ( caligraphic_X )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S2.I1.i3.p1"> <p class="ltx_p" id="S2.I1.i3.p1.2">For <math alttext="t=k,\ldots,T" class="ltx_Math" display="inline" id="S2.I1.i3.p1.1.m1.3"><semantics id="S2.I1.i3.p1.1.m1.3a"><mrow id="S2.I1.i3.p1.1.m1.3.4" xref="S2.I1.i3.p1.1.m1.3.4.cmml"><mi id="S2.I1.i3.p1.1.m1.3.4.2" xref="S2.I1.i3.p1.1.m1.3.4.2.cmml">t</mi><mo id="S2.I1.i3.p1.1.m1.3.4.1" xref="S2.I1.i3.p1.1.m1.3.4.1.cmml">=</mo><mrow id="S2.I1.i3.p1.1.m1.3.4.3.2" xref="S2.I1.i3.p1.1.m1.3.4.3.1.cmml"><mi id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml">k</mi><mo id="S2.I1.i3.p1.1.m1.3.4.3.2.1" xref="S2.I1.i3.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.I1.i3.p1.1.m1.2.2" mathvariant="normal" xref="S2.I1.i3.p1.1.m1.2.2.cmml">…</mi><mo id="S2.I1.i3.p1.1.m1.3.4.3.2.2" xref="S2.I1.i3.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.I1.i3.p1.1.m1.3.3" xref="S2.I1.i3.p1.1.m1.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.3b"><apply id="S2.I1.i3.p1.1.m1.3.4.cmml" xref="S2.I1.i3.p1.1.m1.3.4"><eq id="S2.I1.i3.p1.1.m1.3.4.1.cmml" xref="S2.I1.i3.p1.1.m1.3.4.1"></eq><ci id="S2.I1.i3.p1.1.m1.3.4.2.cmml" xref="S2.I1.i3.p1.1.m1.3.4.2">𝑡</ci><list id="S2.I1.i3.p1.1.m1.3.4.3.1.cmml" xref="S2.I1.i3.p1.1.m1.3.4.3.2"><ci id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1">𝑘</ci><ci id="S2.I1.i3.p1.1.m1.2.2.cmml" xref="S2.I1.i3.p1.1.m1.2.2">…</ci><ci id="S2.I1.i3.p1.1.m1.3.3.cmml" xref="S2.I1.i3.p1.1.m1.3.3">𝑇</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.3c">t=k,\ldots,T</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.1.m1.3d">italic_t = italic_k , … , italic_T</annotation></semantics></math>, sample <math alttext="x_{t+1}\sim P_{x_{t-k+1}^{t}}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.2.m2.1"><semantics id="S2.I1.i3.p1.2.m2.1a"><mrow id="S2.I1.i3.p1.2.m2.1.1" xref="S2.I1.i3.p1.2.m2.1.1.cmml"><msub id="S2.I1.i3.p1.2.m2.1.1.2" xref="S2.I1.i3.p1.2.m2.1.1.2.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1.2.2" xref="S2.I1.i3.p1.2.m2.1.1.2.2.cmml">x</mi><mrow id="S2.I1.i3.p1.2.m2.1.1.2.3" xref="S2.I1.i3.p1.2.m2.1.1.2.3.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1.2.3.2" xref="S2.I1.i3.p1.2.m2.1.1.2.3.2.cmml">t</mi><mo id="S2.I1.i3.p1.2.m2.1.1.2.3.1" xref="S2.I1.i3.p1.2.m2.1.1.2.3.1.cmml">+</mo><mn id="S2.I1.i3.p1.2.m2.1.1.2.3.3" xref="S2.I1.i3.p1.2.m2.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.I1.i3.p1.2.m2.1.1.1" xref="S2.I1.i3.p1.2.m2.1.1.1.cmml">∼</mo><msub id="S2.I1.i3.p1.2.m2.1.1.3" xref="S2.I1.i3.p1.2.m2.1.1.3.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1.3.2" xref="S2.I1.i3.p1.2.m2.1.1.3.2.cmml">P</mi><msubsup id="S2.I1.i3.p1.2.m2.1.1.3.3" xref="S2.I1.i3.p1.2.m2.1.1.3.3.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1.3.3.2.2" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.2.cmml">x</mi><mrow id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.cmml"><mrow id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.2" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.2.cmml">t</mi><mo id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.1" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.1.cmml">−</mo><mi id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.3" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.3.cmml">k</mi></mrow><mo id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.1" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.1.cmml">+</mo><mn id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.3" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.3.cmml">1</mn></mrow><mi id="S2.I1.i3.p1.2.m2.1.1.3.3.3" xref="S2.I1.i3.p1.2.m2.1.1.3.3.3.cmml">t</mi></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.2.m2.1b"><apply id="S2.I1.i3.p1.2.m2.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.I1.i3.p1.2.m2.1.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.1">similar-to</csymbol><apply id="S2.I1.i3.p1.2.m2.1.1.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I1.i3.p1.2.m2.1.1.2.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.I1.i3.p1.2.m2.1.1.2.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2.2">𝑥</ci><apply id="S2.I1.i3.p1.2.m2.1.1.2.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2.3"><plus id="S2.I1.i3.p1.2.m2.1.1.2.3.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2.3.1"></plus><ci id="S2.I1.i3.p1.2.m2.1.1.2.3.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2.3.2">𝑡</ci><cn id="S2.I1.i3.p1.2.m2.1.1.2.3.3.cmml" type="integer" xref="S2.I1.i3.p1.2.m2.1.1.2.3.3">1</cn></apply></apply><apply id="S2.I1.i3.p1.2.m2.1.1.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S2.I1.i3.p1.2.m2.1.1.3.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.2">𝑃</ci><apply id="S2.I1.i3.p1.2.m2.1.1.3.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.I1.i3.p1.2.m2.1.1.3.3.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3">superscript</csymbol><apply id="S2.I1.i3.p1.2.m2.1.1.3.3.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.I1.i3.p1.2.m2.1.1.3.3.2.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.I1.i3.p1.2.m2.1.1.3.3.2.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.2">𝑥</ci><apply id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3"><plus id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.1"></plus><apply id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2"><minus id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.1"></minus><ci id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.2">𝑡</ci><ci id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.2.3">𝑘</ci></apply><cn id="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.3.cmml" type="integer" xref="S2.I1.i3.p1.2.m2.1.1.3.3.2.3.3">1</cn></apply></apply><ci id="S2.I1.i3.p1.2.m2.1.1.3.3.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.2.m2.1c">x_{t+1}\sim P_{x_{t-k+1}^{t}}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ∼ italic_P start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S2.I1.i4.p1"> <p class="ltx_p" id="S2.I1.i4.p1.1">Return the input <math alttext="x=(x_{t})_{t=1}^{T}" class="ltx_Math" display="inline" id="S2.I1.i4.p1.1.m1.1"><semantics id="S2.I1.i4.p1.1.m1.1a"><mrow id="S2.I1.i4.p1.1.m1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.cmml"><mi id="S2.I1.i4.p1.1.m1.1.1.3" xref="S2.I1.i4.p1.1.m1.1.1.3.cmml">x</mi><mo id="S2.I1.i4.p1.1.m1.1.1.2" xref="S2.I1.i4.p1.1.m1.1.1.2.cmml">=</mo><msubsup id="S2.I1.i4.p1.1.m1.1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.1.cmml"><mrow id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.2" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.3" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo 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xref="S2.I1.i4.p1.1.m1.1.1.1">superscript</csymbol><apply id="S2.I1.i4.p1.1.m1.1.1.1.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i4.p1.1.m1.1.1.1.1.2.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1">subscript</csymbol><apply id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S2.I1.i4.p1.1.m1.1.1.1.1.3.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.3"><eq id="S2.I1.i4.p1.1.m1.1.1.1.1.3.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.3.1"></eq><ci id="S2.I1.i4.p1.1.m1.1.1.1.1.3.2.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.1.3.2">𝑡</ci><cn id="S2.I1.i4.p1.1.m1.1.1.1.1.3.3.cmml" type="integer" xref="S2.I1.i4.p1.1.m1.1.1.1.1.3.3">1</cn></apply></apply><ci id="S2.I1.i4.p1.1.m1.1.1.1.3.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i4.p1.1.m1.1c">x=(x_{t})_{t=1}^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i4.p1.1.m1.1d">italic_x = ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">5.</span> <div class="ltx_para" id="S2.I1.i5.p1"> <p class="ltx_p" id="S2.I1.i5.p1.1">Repeat the above steps to generate a batch <math alttext="\{x^{(b)}\}_{b\in[B]}" class="ltx_Math" display="inline" id="S2.I1.i5.p1.1.m1.3"><semantics id="S2.I1.i5.p1.1.m1.3a"><msub id="S2.I1.i5.p1.1.m1.3.3" xref="S2.I1.i5.p1.1.m1.3.3.cmml"><mrow id="S2.I1.i5.p1.1.m1.3.3.1.1" 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class="ltx_para ltx_noindent" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.5"><span class="ltx_text ltx_font_bold" id="S2.SS1.p4.5.1">Relation to ICL.</span> As a consequence of the generation process, no two sequences <math alttext="x" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mi id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">italic_x</annotation></semantics></math> share the same transition matrix <math alttext="P" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mi id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><ci id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_P</annotation></semantics></math> and hence every sequence follows a different Markov distribution. Owing to this fact, even when a model is trained on this random Markovian data, at inference, for every test sequence it has to estimate the next token in-context. Hence this data class serves as a good sandbox to gauge the ICL capabilities of Mamba, which was also used in a similar context for transformers <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">nic2024trans</span>]</cite>. In this paper, we let the state space <math alttext="\mathcal{X}=\{0,1\}" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.2"><semantics id="S2.SS1.p4.3.m3.2a"><mrow id="S2.SS1.p4.3.m3.2.3" xref="S2.SS1.p4.3.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.3.m3.2.3.2" xref="S2.SS1.p4.3.m3.2.3.2.cmml">𝒳</mi><mo id="S2.SS1.p4.3.m3.2.3.1" xref="S2.SS1.p4.3.m3.2.3.1.cmml">=</mo><mrow id="S2.SS1.p4.3.m3.2.3.3.2" xref="S2.SS1.p4.3.m3.2.3.3.1.cmml"><mo id="S2.SS1.p4.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS1.p4.3.m3.2.3.3.1.cmml">{</mo><mn id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml">0</mn><mo id="S2.SS1.p4.3.m3.2.3.3.2.2" xref="S2.SS1.p4.3.m3.2.3.3.1.cmml">,</mo><mn id="S2.SS1.p4.3.m3.2.2" xref="S2.SS1.p4.3.m3.2.2.cmml">1</mn><mo id="S2.SS1.p4.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS1.p4.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.2b"><apply id="S2.SS1.p4.3.m3.2.3.cmml" xref="S2.SS1.p4.3.m3.2.3"><eq id="S2.SS1.p4.3.m3.2.3.1.cmml" xref="S2.SS1.p4.3.m3.2.3.1"></eq><ci id="S2.SS1.p4.3.m3.2.3.2.cmml" xref="S2.SS1.p4.3.m3.2.3.2">𝒳</ci><set id="S2.SS1.p4.3.m3.2.3.3.1.cmml" xref="S2.SS1.p4.3.m3.2.3.3.2"><cn id="S2.SS1.p4.3.m3.1.1.cmml" type="integer" xref="S2.SS1.p4.3.m3.1.1">0</cn><cn id="S2.SS1.p4.3.m3.2.2.cmml" type="integer" xref="S2.SS1.p4.3.m3.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.2c">\mathcal{X}=\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.2d">caligraphic_X = { 0 , 1 }</annotation></semantics></math> for the ease of exposition and the order <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.1"><semantics id="S2.SS1.p4.4.m4.1a"><mi id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.1b"><ci id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.1d">italic_k</annotation></semantics></math> to be any <math alttext="k\geq 1" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.1"><semantics id="S2.SS1.p4.5.m5.1a"><mrow id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml"><mi id="S2.SS1.p4.5.m5.1.1.2" xref="S2.SS1.p4.5.m5.1.1.2.cmml">k</mi><mo id="S2.SS1.p4.5.m5.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.cmml">≥</mo><mn id="S2.SS1.p4.5.m5.1.1.3" xref="S2.SS1.p4.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.1b"><apply id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1"><geq id="S2.SS1.p4.5.m5.1.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1.1"></geq><ci id="S2.SS1.p4.5.m5.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.2">𝑘</ci><cn id="S2.SS1.p4.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p4.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.1c">k\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.1d">italic_k ≥ 1</annotation></semantics></math>. Our results and insights can be readily extended to any finite vocabulary, as demonstrated for the natural language in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5.SS2" title="5.2 Natural language modeling ‣ 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">5.2</span></a>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Mamba architecture</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.9">Selective SSMs such as Mamba and Mamba-2 are a class of sequence-to-sequence models that are closely related to RNNs and classical state space models <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">mamba2023gu</span>]</cite>. A key feature underpinning these models is the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.9.1">selectivity mechanism</em>, enabling them to selectively choose inputs at every timestep, as opposed to linear time-invariant (LTI) systems. While we believe our work captures the behavior of all selective SSMs, we will specifically focus on the state-of-the-art Mamba-2 model to simplify exposition. By slight abuse of terminology, henceforth we will also refer to this model simply as Mamba. 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xref="S2.SS2.p1.2.m2.1.1.6.3"><times id="S2.SS2.p1.2.m2.1.1.6.3.1.cmml" xref="S2.SS2.p1.2.m2.1.1.6.3.1"></times><ci id="S2.SS2.p1.2.m2.1.1.6.3.2.cmml" xref="S2.SS2.p1.2.m2.1.1.6.3.2">𝑑</ci><ci id="S2.SS2.p1.2.m2.1.1.6.3.3.cmml" xref="S2.SS2.p1.2.m2.1.1.6.3.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\bm{x}=(\bm{x}_{t})_{t=1}^{T}\in\mathbb{R}^{d\times T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">bold_italic_x = ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d × italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> of dimension <math alttext="d" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><mi id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><ci id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">italic_d</annotation></semantics></math>, it outputs the corresponding output embeddings <math alttext="\bm{o}=(\bm{o}_{t})_{t=1}^{T}\in\mathbb{R}^{d\times T}" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m4.1"><semantics id="S2.SS2.p1.4.m4.1a"><mrow id="S2.SS2.p1.4.m4.1.1" xref="S2.SS2.p1.4.m4.1.1.cmml"><mi id="S2.SS2.p1.4.m4.1.1.3" xref="S2.SS2.p1.4.m4.1.1.3.cmml">𝒐</mi><mo id="S2.SS2.p1.4.m4.1.1.4" xref="S2.SS2.p1.4.m4.1.1.4.cmml">=</mo><msubsup id="S2.SS2.p1.4.m4.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.cmml"><mrow id="S2.SS2.p1.4.m4.1.1.1.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.2" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.2.cmml">𝒐</mi><mi id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.3" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.SS2.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS2.p1.4.m4.1.1.1.1.3" xref="S2.SS2.p1.4.m4.1.1.1.1.3.cmml"><mi id="S2.SS2.p1.4.m4.1.1.1.1.3.2" xref="S2.SS2.p1.4.m4.1.1.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p1.4.m4.1.1.1.1.3.1" xref="S2.SS2.p1.4.m4.1.1.1.1.3.1.cmml">=</mo><mn id="S2.SS2.p1.4.m4.1.1.1.1.3.3" xref="S2.SS2.p1.4.m4.1.1.1.1.3.3.cmml">1</mn></mrow><mi id="S2.SS2.p1.4.m4.1.1.1.3" xref="S2.SS2.p1.4.m4.1.1.1.3.cmml">T</mi></msubsup><mo id="S2.SS2.p1.4.m4.1.1.5" xref="S2.SS2.p1.4.m4.1.1.5.cmml">∈</mo><msup id="S2.SS2.p1.4.m4.1.1.6" xref="S2.SS2.p1.4.m4.1.1.6.cmml"><mi id="S2.SS2.p1.4.m4.1.1.6.2" xref="S2.SS2.p1.4.m4.1.1.6.2.cmml">ℝ</mi><mrow id="S2.SS2.p1.4.m4.1.1.6.3" xref="S2.SS2.p1.4.m4.1.1.6.3.cmml"><mi id="S2.SS2.p1.4.m4.1.1.6.3.2" xref="S2.SS2.p1.4.m4.1.1.6.3.2.cmml">d</mi><mo id="S2.SS2.p1.4.m4.1.1.6.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p1.4.m4.1.1.6.3.1.cmml">×</mo><mi id="S2.SS2.p1.4.m4.1.1.6.3.3" xref="S2.SS2.p1.4.m4.1.1.6.3.3.cmml">T</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m4.1b"><apply id="S2.SS2.p1.4.m4.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1"><and id="S2.SS2.p1.4.m4.1.1a.cmml" xref="S2.SS2.p1.4.m4.1.1"></and><apply id="S2.SS2.p1.4.m4.1.1b.cmml" xref="S2.SS2.p1.4.m4.1.1"><eq id="S2.SS2.p1.4.m4.1.1.4.cmml" xref="S2.SS2.p1.4.m4.1.1.4"></eq><ci id="S2.SS2.p1.4.m4.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.3">𝒐</ci><apply id="S2.SS2.p1.4.m4.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.1">superscript</csymbol><apply id="S2.SS2.p1.4.m4.1.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.1">subscript</csymbol><apply id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.2">𝒐</ci><ci id="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S2.SS2.p1.4.m4.1.1.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.3"><eq id="S2.SS2.p1.4.m4.1.1.1.1.3.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.3.1"></eq><ci id="S2.SS2.p1.4.m4.1.1.1.1.3.2.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.3.2">𝑡</ci><cn id="S2.SS2.p1.4.m4.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p1.4.m4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S2.SS2.p1.4.m4.1.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.1.3">𝑇</ci></apply></apply><apply id="S2.SS2.p1.4.m4.1.1c.cmml" xref="S2.SS2.p1.4.m4.1.1"><in id="S2.SS2.p1.4.m4.1.1.5.cmml" xref="S2.SS2.p1.4.m4.1.1.5"></in><share href="https://arxiv.org/html/2502.10178v1#S2.SS2.p1.4.m4.1.1.1.cmml" id="S2.SS2.p1.4.m4.1.1d.cmml" xref="S2.SS2.p1.4.m4.1.1"></share><apply id="S2.SS2.p1.4.m4.1.1.6.cmml" xref="S2.SS2.p1.4.m4.1.1.6"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.6.1.cmml" xref="S2.SS2.p1.4.m4.1.1.6">superscript</csymbol><ci id="S2.SS2.p1.4.m4.1.1.6.2.cmml" xref="S2.SS2.p1.4.m4.1.1.6.2">ℝ</ci><apply id="S2.SS2.p1.4.m4.1.1.6.3.cmml" xref="S2.SS2.p1.4.m4.1.1.6.3"><times id="S2.SS2.p1.4.m4.1.1.6.3.1.cmml" xref="S2.SS2.p1.4.m4.1.1.6.3.1"></times><ci id="S2.SS2.p1.4.m4.1.1.6.3.2.cmml" xref="S2.SS2.p1.4.m4.1.1.6.3.2">𝑑</ci><ci id="S2.SS2.p1.4.m4.1.1.6.3.3.cmml" xref="S2.SS2.p1.4.m4.1.1.6.3.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m4.1c">\bm{o}=(\bm{o}_{t})_{t=1}^{T}\in\mathbb{R}^{d\times T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m4.1d">bold_italic_o = ( bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d × italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> of the same dimension with <math alttext="\bm{o}=\mathsf{Mamba}(\bm{x})" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m5.1"><semantics id="S2.SS2.p1.5.m5.1a"><mrow id="S2.SS2.p1.5.m5.1.2" xref="S2.SS2.p1.5.m5.1.2.cmml"><mi id="S2.SS2.p1.5.m5.1.2.2" xref="S2.SS2.p1.5.m5.1.2.2.cmml">𝒐</mi><mo id="S2.SS2.p1.5.m5.1.2.1" xref="S2.SS2.p1.5.m5.1.2.1.cmml">=</mo><mrow id="S2.SS2.p1.5.m5.1.2.3" xref="S2.SS2.p1.5.m5.1.2.3.cmml"><mi id="S2.SS2.p1.5.m5.1.2.3.2" xref="S2.SS2.p1.5.m5.1.2.3.2.cmml">𝖬𝖺𝗆𝖻𝖺</mi><mo id="S2.SS2.p1.5.m5.1.2.3.1" xref="S2.SS2.p1.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p1.5.m5.1.2.3.3.2" xref="S2.SS2.p1.5.m5.1.2.3.cmml"><mo id="S2.SS2.p1.5.m5.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p1.5.m5.1.2.3.cmml">(</mo><mi id="S2.SS2.p1.5.m5.1.1" xref="S2.SS2.p1.5.m5.1.1.cmml">𝒙</mi><mo id="S2.SS2.p1.5.m5.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p1.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m5.1b"><apply id="S2.SS2.p1.5.m5.1.2.cmml" xref="S2.SS2.p1.5.m5.1.2"><eq id="S2.SS2.p1.5.m5.1.2.1.cmml" xref="S2.SS2.p1.5.m5.1.2.1"></eq><ci id="S2.SS2.p1.5.m5.1.2.2.cmml" xref="S2.SS2.p1.5.m5.1.2.2">𝒐</ci><apply id="S2.SS2.p1.5.m5.1.2.3.cmml" xref="S2.SS2.p1.5.m5.1.2.3"><times id="S2.SS2.p1.5.m5.1.2.3.1.cmml" xref="S2.SS2.p1.5.m5.1.2.3.1"></times><ci id="S2.SS2.p1.5.m5.1.2.3.2.cmml" xref="S2.SS2.p1.5.m5.1.2.3.2">𝖬𝖺𝗆𝖻𝖺</ci><ci id="S2.SS2.p1.5.m5.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1">𝒙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m5.1c">\bm{o}=\mathsf{Mamba}(\bm{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m5.1d">bold_italic_o = sansserif_Mamba ( bold_italic_x )</annotation></semantics></math>. More precisely, fix <math alttext="t\in[T]" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m6.1"><semantics id="S2.SS2.p1.6.m6.1a"><mrow id="S2.SS2.p1.6.m6.1.2" xref="S2.SS2.p1.6.m6.1.2.cmml"><mi id="S2.SS2.p1.6.m6.1.2.2" xref="S2.SS2.p1.6.m6.1.2.2.cmml">t</mi><mo id="S2.SS2.p1.6.m6.1.2.1" xref="S2.SS2.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="S2.SS2.p1.6.m6.1.2.3.2" xref="S2.SS2.p1.6.m6.1.2.3.1.cmml"><mo id="S2.SS2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS2.p1.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.p1.6.m6.1.1" xref="S2.SS2.p1.6.m6.1.1.cmml">T</mi><mo id="S2.SS2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS2.p1.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m6.1b"><apply id="S2.SS2.p1.6.m6.1.2.cmml" xref="S2.SS2.p1.6.m6.1.2"><in id="S2.SS2.p1.6.m6.1.2.1.cmml" xref="S2.SS2.p1.6.m6.1.2.1"></in><ci id="S2.SS2.p1.6.m6.1.2.2.cmml" xref="S2.SS2.p1.6.m6.1.2.2">𝑡</ci><apply id="S2.SS2.p1.6.m6.1.2.3.1.cmml" xref="S2.SS2.p1.6.m6.1.2.3.2"><csymbol cd="latexml" id="S2.SS2.p1.6.m6.1.2.3.1.1.cmml" xref="S2.SS2.p1.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.p1.6.m6.1.1.cmml" xref="S2.SS2.p1.6.m6.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m6.1c">t\in[T]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m6.1d">italic_t ∈ [ italic_T ]</annotation></semantics></math>. Then the output <math alttext="\bm{o}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m7.1"><semantics id="S2.SS2.p1.7.m7.1a"><msub id="S2.SS2.p1.7.m7.1.1" xref="S2.SS2.p1.7.m7.1.1.cmml"><mi id="S2.SS2.p1.7.m7.1.1.2" xref="S2.SS2.p1.7.m7.1.1.2.cmml">𝒐</mi><mi id="S2.SS2.p1.7.m7.1.1.3" xref="S2.SS2.p1.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m7.1b"><apply id="S2.SS2.p1.7.m7.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.7.m7.1.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS2.p1.7.m7.1.1.2.cmml" xref="S2.SS2.p1.7.m7.1.1.2">𝒐</ci><ci id="S2.SS2.p1.7.m7.1.1.3.cmml" xref="S2.SS2.p1.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m7.1c">\bm{o}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m7.1d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p1.8.m8.1"><semantics id="S2.SS2.p1.8.m8.1a"><mi id="S2.SS2.p1.8.m8.1.1" xref="S2.SS2.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.8.m8.1b"><ci id="S2.SS2.p1.8.m8.1.1.cmml" xref="S2.SS2.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.8.m8.1d">italic_t</annotation></semantics></math> is computed as <math alttext="\bm{o}_{t}=\mathsf{Mamba}(\bm{x}_{1}^{t})" class="ltx_Math" display="inline" id="S2.SS2.p1.9.m9.1"><semantics id="S2.SS2.p1.9.m9.1a"><mrow id="S2.SS2.p1.9.m9.1.1" xref="S2.SS2.p1.9.m9.1.1.cmml"><msub id="S2.SS2.p1.9.m9.1.1.3" xref="S2.SS2.p1.9.m9.1.1.3.cmml"><mi id="S2.SS2.p1.9.m9.1.1.3.2" xref="S2.SS2.p1.9.m9.1.1.3.2.cmml">𝒐</mi><mi id="S2.SS2.p1.9.m9.1.1.3.3" xref="S2.SS2.p1.9.m9.1.1.3.3.cmml">t</mi></msub><mo id="S2.SS2.p1.9.m9.1.1.2" xref="S2.SS2.p1.9.m9.1.1.2.cmml">=</mo><mrow id="S2.SS2.p1.9.m9.1.1.1" xref="S2.SS2.p1.9.m9.1.1.1.cmml"><mi id="S2.SS2.p1.9.m9.1.1.1.3" xref="S2.SS2.p1.9.m9.1.1.1.3.cmml">𝖬𝖺𝗆𝖻𝖺</mi><mo id="S2.SS2.p1.9.m9.1.1.1.2" xref="S2.SS2.p1.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.9.m9.1.1.1.1.1" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.9.m9.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.cmml">(</mo><msubsup id="S2.SS2.p1.9.m9.1.1.1.1.1.1" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.2" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mn id="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.3" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S2.SS2.p1.9.m9.1.1.1.1.1.1.3" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S2.SS2.p1.9.m9.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.9.m9.1b"><apply id="S2.SS2.p1.9.m9.1.1.cmml" xref="S2.SS2.p1.9.m9.1.1"><eq id="S2.SS2.p1.9.m9.1.1.2.cmml" xref="S2.SS2.p1.9.m9.1.1.2"></eq><apply id="S2.SS2.p1.9.m9.1.1.3.cmml" xref="S2.SS2.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m9.1.1.3.1.cmml" xref="S2.SS2.p1.9.m9.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.9.m9.1.1.3.2.cmml" xref="S2.SS2.p1.9.m9.1.1.3.2">𝒐</ci><ci id="S2.SS2.p1.9.m9.1.1.3.3.cmml" xref="S2.SS2.p1.9.m9.1.1.3.3">𝑡</ci></apply><apply id="S2.SS2.p1.9.m9.1.1.1.cmml" xref="S2.SS2.p1.9.m9.1.1.1"><times id="S2.SS2.p1.9.m9.1.1.1.2.cmml" xref="S2.SS2.p1.9.m9.1.1.1.2"></times><ci id="S2.SS2.p1.9.m9.1.1.1.3.cmml" xref="S2.SS2.p1.9.m9.1.1.1.3">𝖬𝖺𝗆𝖻𝖺</ci><apply id="S2.SS2.p1.9.m9.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m9.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.9.m9.1.1.1.1.1">superscript</csymbol><apply id="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p1.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.2">𝒙</ci><cn id="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S2.SS2.p1.9.m9.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.9.m9.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.9.m9.1c">\bm{o}_{t}=\mathsf{Mamba}(\bm{x}_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.9.m9.1d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = sansserif_Mamba ( bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> using the following recurrence equations <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">dao2024transformers</span>]</cite>:</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}H_{t}&=a_{t}\,H_{t-1}+\widetilde{\bm{x}}_{t}\,\bm{b}_{t}^{\top}% \in\mathbb{R}^{ed\times N},\\ \bm{y}_{t}&=H_{t}\,\bm{c}_{t}\in\mathbb{R}^{ed},\\ \bm{z}_{t}&=\bm{y}_{t}\odot\mathrm{ReLU}(W_{z}\,\bm{x}_{t})\in\mathbb{R}^{ed},% \\ \bm{o}_{t}&=W_{o}\,\bm{z}_{t}\in\mathbb{R}^{d},\end{split}" class="ltx_Math" display="block" id="S2.Ex1.m1.61"><semantics id="S2.Ex1.m1.61a"><mtable columnspacing="0pt" displaystyle="true" id="S2.Ex1.m1.61.61.5" rowspacing="0pt"><mtr id="S2.Ex1.m1.61.61.5a"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex1.m1.61.61.5b"><msub id="S2.Ex1.m1.2.2.2.2.2"><mi id="S2.Ex1.m1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.cmml">H</mi><mi id="S2.Ex1.m1.2.2.2.2.2.2.1" xref="S2.Ex1.m1.2.2.2.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.61.61.5c"><mrow id="S2.Ex1.m1.58.58.2.57.18.16.16"><mrow id="S2.Ex1.m1.58.58.2.57.18.16.16.1"><mi id="S2.Ex1.m1.58.58.2.57.18.16.16.1.2"></mi><mo id="S2.Ex1.m1.3.3.3.3.1.1" xref="S2.Ex1.m1.3.3.3.3.1.1.cmml">=</mo><mrow id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3"><mrow id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.1"><msub id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.1.2"><mi id="S2.Ex1.m1.4.4.4.4.2.2" xref="S2.Ex1.m1.4.4.4.4.2.2.cmml">a</mi><mi id="S2.Ex1.m1.5.5.5.5.3.3.1" xref="S2.Ex1.m1.5.5.5.5.3.3.1.cmml">t</mi></msub><mo id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.1.1">⁢</mo><msub id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.1.3"><mi id="S2.Ex1.m1.6.6.6.6.4.4" xref="S2.Ex1.m1.6.6.6.6.4.4.cmml">H</mi><mrow id="S2.Ex1.m1.7.7.7.7.5.5.1" xref="S2.Ex1.m1.7.7.7.7.5.5.1.cmml"><mi id="S2.Ex1.m1.7.7.7.7.5.5.1.2" xref="S2.Ex1.m1.7.7.7.7.5.5.1.2.cmml">t</mi><mo id="S2.Ex1.m1.7.7.7.7.5.5.1.1" xref="S2.Ex1.m1.7.7.7.7.5.5.1.1.cmml">−</mo><mn id="S2.Ex1.m1.7.7.7.7.5.5.1.3" xref="S2.Ex1.m1.7.7.7.7.5.5.1.3.cmml">1</mn></mrow></msub></mrow><mo id="S2.Ex1.m1.8.8.8.8.6.6" xref="S2.Ex1.m1.8.8.8.8.6.6.cmml">+</mo><mrow id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.2"><msub id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.2.2"><mover accent="true" id="S2.Ex1.m1.9.9.9.9.7.7" xref="S2.Ex1.m1.9.9.9.9.7.7.cmml"><mi id="S2.Ex1.m1.9.9.9.9.7.7.2" xref="S2.Ex1.m1.9.9.9.9.7.7.2.cmml">𝒙</mi><mo id="S2.Ex1.m1.9.9.9.9.7.7.1" xref="S2.Ex1.m1.9.9.9.9.7.7.1.cmml">~</mo></mover><mi id="S2.Ex1.m1.10.10.10.10.8.8.1" xref="S2.Ex1.m1.10.10.10.10.8.8.1.cmml">t</mi></msub><mo id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.2.1">⁢</mo><msubsup id="S2.Ex1.m1.58.58.2.57.18.16.16.1.3.2.3"><mi id="S2.Ex1.m1.11.11.11.11.9.9" xref="S2.Ex1.m1.11.11.11.11.9.9.cmml">𝒃</mi><mi id="S2.Ex1.m1.12.12.12.12.10.10.1" xref="S2.Ex1.m1.12.12.12.12.10.10.1.cmml">t</mi><mo id="S2.Ex1.m1.13.13.13.13.11.11.1" xref="S2.Ex1.m1.13.13.13.13.11.11.1.cmml">⊤</mo></msubsup></mrow></mrow><mo id="S2.Ex1.m1.14.14.14.14.12.12" xref="S2.Ex1.m1.14.14.14.14.12.12.cmml">∈</mo><msup id="S2.Ex1.m1.58.58.2.57.18.16.16.1.4"><mi id="S2.Ex1.m1.15.15.15.15.13.13" xref="S2.Ex1.m1.15.15.15.15.13.13.cmml">ℝ</mi><mrow id="S2.Ex1.m1.16.16.16.16.14.14.1" xref="S2.Ex1.m1.16.16.16.16.14.14.1.cmml"><mrow id="S2.Ex1.m1.16.16.16.16.14.14.1.2" xref="S2.Ex1.m1.16.16.16.16.14.14.1.2.cmml"><mi id="S2.Ex1.m1.16.16.16.16.14.14.1.2.2" xref="S2.Ex1.m1.16.16.16.16.14.14.1.2.2.cmml">e</mi><mo id="S2.Ex1.m1.16.16.16.16.14.14.1.2.1" xref="S2.Ex1.m1.16.16.16.16.14.14.1.2.1.cmml">⁢</mo><mi id="S2.Ex1.m1.16.16.16.16.14.14.1.2.3" xref="S2.Ex1.m1.16.16.16.16.14.14.1.2.3.cmml">d</mi></mrow><mo id="S2.Ex1.m1.16.16.16.16.14.14.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex1.m1.16.16.16.16.14.14.1.1.cmml">×</mo><mi id="S2.Ex1.m1.16.16.16.16.14.14.1.3" xref="S2.Ex1.m1.16.16.16.16.14.14.1.3.cmml">N</mi></mrow></msup></mrow><mo id="S2.Ex1.m1.17.17.17.17.15.15">,</mo></mrow></mtd></mtr><mtr id="S2.Ex1.m1.61.61.5d"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex1.m1.61.61.5e"><msub id="S2.Ex1.m1.19.19.19.2.2"><mi id="S2.Ex1.m1.18.18.18.1.1.1" xref="S2.Ex1.m1.18.18.18.1.1.1.cmml">𝒚</mi><mi id="S2.Ex1.m1.19.19.19.2.2.2.1" xref="S2.Ex1.m1.19.19.19.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.61.61.5f"><mrow id="S2.Ex1.m1.59.59.3.58.12.10.10"><mrow id="S2.Ex1.m1.59.59.3.58.12.10.10.1"><mi id="S2.Ex1.m1.59.59.3.58.12.10.10.1.2"></mi><mo id="S2.Ex1.m1.20.20.20.3.1.1" xref="S2.Ex1.m1.20.20.20.3.1.1.cmml">=</mo><mrow id="S2.Ex1.m1.59.59.3.58.12.10.10.1.3"><msub id="S2.Ex1.m1.59.59.3.58.12.10.10.1.3.2"><mi id="S2.Ex1.m1.21.21.21.4.2.2" xref="S2.Ex1.m1.21.21.21.4.2.2.cmml">H</mi><mi id="S2.Ex1.m1.22.22.22.5.3.3.1" xref="S2.Ex1.m1.22.22.22.5.3.3.1.cmml">t</mi></msub><mo id="S2.Ex1.m1.59.59.3.58.12.10.10.1.3.1">⁢</mo><msub id="S2.Ex1.m1.59.59.3.58.12.10.10.1.3.3"><mi id="S2.Ex1.m1.23.23.23.6.4.4" xref="S2.Ex1.m1.23.23.23.6.4.4.cmml">𝒄</mi><mi id="S2.Ex1.m1.24.24.24.7.5.5.1" xref="S2.Ex1.m1.24.24.24.7.5.5.1.cmml">t</mi></msub></mrow><mo id="S2.Ex1.m1.25.25.25.8.6.6" xref="S2.Ex1.m1.25.25.25.8.6.6.cmml">∈</mo><msup id="S2.Ex1.m1.59.59.3.58.12.10.10.1.4"><mi id="S2.Ex1.m1.26.26.26.9.7.7" xref="S2.Ex1.m1.26.26.26.9.7.7.cmml">ℝ</mi><mrow id="S2.Ex1.m1.27.27.27.10.8.8.1" xref="S2.Ex1.m1.27.27.27.10.8.8.1.cmml"><mi id="S2.Ex1.m1.27.27.27.10.8.8.1.2" xref="S2.Ex1.m1.27.27.27.10.8.8.1.2.cmml">e</mi><mo id="S2.Ex1.m1.27.27.27.10.8.8.1.1" xref="S2.Ex1.m1.27.27.27.10.8.8.1.1.cmml">⁢</mo><mi id="S2.Ex1.m1.27.27.27.10.8.8.1.3" xref="S2.Ex1.m1.27.27.27.10.8.8.1.3.cmml">d</mi></mrow></msup></mrow><mo id="S2.Ex1.m1.28.28.28.11.9.9">,</mo></mrow></mtd></mtr><mtr id="S2.Ex1.m1.61.61.5g"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex1.m1.61.61.5h"><msub id="S2.Ex1.m1.30.30.30.2.2"><mi id="S2.Ex1.m1.29.29.29.1.1.1" xref="S2.Ex1.m1.29.29.29.1.1.1.cmml">𝒛</mi><mi id="S2.Ex1.m1.30.30.30.2.2.2.1" xref="S2.Ex1.m1.30.30.30.2.2.2.1.cmml">t</mi></msub></mtd><mtd 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start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL = bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⊙ roman_ReLU ( italic_W start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_e italic_d end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL = italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT bold_italic_z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Mamba)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.12">where the initial state 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xref="S2.SS2.p1.11.m2.1.1.3.3.2.3">𝑑</ci></apply><ci id="S2.SS2.p1.11.m2.1.1.3.3.3.cmml" xref="S2.SS2.p1.11.m2.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.11.m2.1c">W_{z}\in\mathbb{R}^{ed\times d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.11.m2.1d">italic_W start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_e italic_d × italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="W_{o}\in\mathbb{R}^{d\times ed}" class="ltx_Math" display="inline" id="S2.SS2.p1.12.m3.1"><semantics id="S2.SS2.p1.12.m3.1a"><mrow id="S2.SS2.p1.12.m3.1.1" xref="S2.SS2.p1.12.m3.1.1.cmml"><msub id="S2.SS2.p1.12.m3.1.1.2" xref="S2.SS2.p1.12.m3.1.1.2.cmml"><mi id="S2.SS2.p1.12.m3.1.1.2.2" xref="S2.SS2.p1.12.m3.1.1.2.2.cmml">W</mi><mi id="S2.SS2.p1.12.m3.1.1.2.3" xref="S2.SS2.p1.12.m3.1.1.2.3.cmml">o</mi></msub><mo id="S2.SS2.p1.12.m3.1.1.1" xref="S2.SS2.p1.12.m3.1.1.1.cmml">∈</mo><msup id="S2.SS2.p1.12.m3.1.1.3" xref="S2.SS2.p1.12.m3.1.1.3.cmml"><mi id="S2.SS2.p1.12.m3.1.1.3.2" xref="S2.SS2.p1.12.m3.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS2.p1.12.m3.1.1.3.3" xref="S2.SS2.p1.12.m3.1.1.3.3.cmml"><mrow id="S2.SS2.p1.12.m3.1.1.3.3.2" xref="S2.SS2.p1.12.m3.1.1.3.3.2.cmml"><mi id="S2.SS2.p1.12.m3.1.1.3.3.2.2" xref="S2.SS2.p1.12.m3.1.1.3.3.2.2.cmml">d</mi><mo id="S2.SS2.p1.12.m3.1.1.3.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p1.12.m3.1.1.3.3.2.1.cmml">×</mo><mi id="S2.SS2.p1.12.m3.1.1.3.3.2.3" xref="S2.SS2.p1.12.m3.1.1.3.3.2.3.cmml">e</mi></mrow><mo id="S2.SS2.p1.12.m3.1.1.3.3.1" xref="S2.SS2.p1.12.m3.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p1.12.m3.1.1.3.3.3" xref="S2.SS2.p1.12.m3.1.1.3.3.3.cmml">d</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.12.m3.1b"><apply id="S2.SS2.p1.12.m3.1.1.cmml" xref="S2.SS2.p1.12.m3.1.1"><in id="S2.SS2.p1.12.m3.1.1.1.cmml" xref="S2.SS2.p1.12.m3.1.1.1"></in><apply id="S2.SS2.p1.12.m3.1.1.2.cmml" xref="S2.SS2.p1.12.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.12.m3.1.1.2.1.cmml" xref="S2.SS2.p1.12.m3.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.12.m3.1.1.2.2.cmml" xref="S2.SS2.p1.12.m3.1.1.2.2">𝑊</ci><ci id="S2.SS2.p1.12.m3.1.1.2.3.cmml" xref="S2.SS2.p1.12.m3.1.1.2.3">𝑜</ci></apply><apply id="S2.SS2.p1.12.m3.1.1.3.cmml" xref="S2.SS2.p1.12.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.12.m3.1.1.3.1.cmml" xref="S2.SS2.p1.12.m3.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.12.m3.1.1.3.2.cmml" xref="S2.SS2.p1.12.m3.1.1.3.2">ℝ</ci><apply id="S2.SS2.p1.12.m3.1.1.3.3.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3"><times id="S2.SS2.p1.12.m3.1.1.3.3.1.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3.1"></times><apply id="S2.SS2.p1.12.m3.1.1.3.3.2.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3.2"><times id="S2.SS2.p1.12.m3.1.1.3.3.2.1.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3.2.1"></times><ci id="S2.SS2.p1.12.m3.1.1.3.3.2.2.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3.2.2">𝑑</ci><ci id="S2.SS2.p1.12.m3.1.1.3.3.2.3.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3.2.3">𝑒</ci></apply><ci id="S2.SS2.p1.12.m3.1.1.3.3.3.cmml" xref="S2.SS2.p1.12.m3.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.12.m3.1c">W_{o}\in\mathbb{R}^{d\times ed}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.12.m3.1d">italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d × italic_e italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> are learnable parameters, and the input-selective</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}a_{t}&\triangleq\exp(-a\cdot\Delta_{t})\in(0,1),\,\text{with}\\ \Delta_{t}&\triangleq\mathrm{softplus}(\langle{\bm{w}_{\Delta}},{\bm{x}_{t}}% \rangle+\delta)\in\mathbb{R},\\ \widetilde{\bm{x}}_{t}&\triangleq\mathrm{ReLU}(\mathrm{conv}_{X}(W_{X}\,\bm{x}% _{t-w+1}^{t}))\cdot\Delta_{t}\in\mathbb{R}^{ed},\\ \bm{b}_{t}&\triangleq\mathrm{ReLU}(\mathrm{conv}_{B}(W_{B}\,\bm{x}_{t-w+1}^{t}% ))\in\mathbb{R}^{N},\\ \bm{c}_{t}&\triangleq\mathrm{ReLU}(\mathrm{conv}_{C}(W_{C}\,\bm{x}_{t-w+1}^{t}% ))\in\mathbb{R}^{N},\end{split}" class="ltx_Math" display="block" id="S2.Ex2.m1.103"><semantics id="S2.Ex2.m1.103a"><mtable columnspacing="0pt" displaystyle="true" id="S2.Ex2.m1.103.103.6" rowspacing="0pt"><mtr id="S2.Ex2.m1.103.103.6a"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex2.m1.103.103.6b"><msub id="S2.Ex2.m1.2.2.2.2.2"><mi id="S2.Ex2.m1.1.1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml">a</mi><mi id="S2.Ex2.m1.2.2.2.2.2.2.1" xref="S2.Ex2.m1.2.2.2.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex2.m1.103.103.6c"><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18"><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18.1"><mi id="S2.Ex2.m1.99.99.2.98.20.18.18.1.3"></mi><mo id="S2.Ex2.m1.3.3.3.3.1.1" xref="S2.Ex2.m1.3.3.3.3.1.1.cmml">≜</mo><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1"><mi id="S2.Ex2.m1.4.4.4.4.2.2" xref="S2.Ex2.m1.4.4.4.4.2.2.cmml">exp</mi><mo id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1a">⁡</mo><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1.1"><mo id="S2.Ex2.m1.5.5.5.5.3.3" stretchy="false">(</mo><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1.1.1"><mo id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1.1.1a">−</mo><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1.1.1.1"><mi id="S2.Ex2.m1.7.7.7.7.5.5" xref="S2.Ex2.m1.7.7.7.7.5.5.cmml">a</mi><mo id="S2.Ex2.m1.8.8.8.8.6.6" lspace="0.222em" rspace="0.222em" xref="S2.Ex2.m1.8.8.8.8.6.6.cmml">⋅</mo><msub id="S2.Ex2.m1.99.99.2.98.20.18.18.1.1.1.1.1.1.1"><mi id="S2.Ex2.m1.9.9.9.9.7.7" mathvariant="normal" xref="S2.Ex2.m1.9.9.9.9.7.7.cmml">Δ</mi><mi id="S2.Ex2.m1.10.10.10.10.8.8.1" xref="S2.Ex2.m1.10.10.10.10.8.8.1.cmml">t</mi></msub></mrow></mrow><mo id="S2.Ex2.m1.11.11.11.11.9.9" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.12.12.12.12.10.10" xref="S2.Ex2.m1.12.12.12.12.10.10.cmml">∈</mo><mrow id="S2.Ex2.m1.99.99.2.98.20.18.18.1.4"><mo id="S2.Ex2.m1.13.13.13.13.11.11" stretchy="false">(</mo><mn id="S2.Ex2.m1.14.14.14.14.12.12" xref="S2.Ex2.m1.14.14.14.14.12.12.cmml">0</mn><mo id="S2.Ex2.m1.15.15.15.15.13.13">,</mo><mn id="S2.Ex2.m1.16.16.16.16.14.14" xref="S2.Ex2.m1.16.16.16.16.14.14.cmml">1</mn><mo id="S2.Ex2.m1.17.17.17.17.15.15" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.18.18.18.18.16.16" rspace="0.337em">,</mo><mtext id="S2.Ex2.m1.19.19.19.19.17.17" xref="S2.Ex2.m1.19.19.19.19.17.17a.cmml">with</mtext></mrow></mtd></mtr><mtr id="S2.Ex2.m1.103.103.6d"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex2.m1.103.103.6e"><msub id="S2.Ex2.m1.21.21.21.2.2"><mi id="S2.Ex2.m1.20.20.20.1.1.1" mathvariant="normal" xref="S2.Ex2.m1.20.20.20.1.1.1.cmml">Δ</mi><mi id="S2.Ex2.m1.21.21.21.2.2.2.1" xref="S2.Ex2.m1.21.21.21.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex2.m1.103.103.6f"><mrow id="S2.Ex2.m1.100.100.3.99.19.17.17"><mrow id="S2.Ex2.m1.100.100.3.99.19.17.17.1"><mi id="S2.Ex2.m1.100.100.3.99.19.17.17.1.3"></mi><mo id="S2.Ex2.m1.22.22.22.3.1.1" xref="S2.Ex2.m1.22.22.22.3.1.1.cmml">≜</mo><mrow id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1"><mi id="S2.Ex2.m1.23.23.23.4.2.2" xref="S2.Ex2.m1.23.23.23.4.2.2.cmml">softplus</mi><mo id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1.1.1"><mo id="S2.Ex2.m1.24.24.24.5.3.3" stretchy="false">(</mo><mrow id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1.1.1.1"><mrow id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1.1.1.1.2.2"><mo id="S2.Ex2.m1.25.25.25.6.4.4" stretchy="false">⟨</mo><msub id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1.1.1.1.1.1.1"><mi id="S2.Ex2.m1.26.26.26.7.5.5" xref="S2.Ex2.m1.26.26.26.7.5.5.cmml">𝒘</mi><mi id="S2.Ex2.m1.27.27.27.8.6.6.1" mathvariant="normal" xref="S2.Ex2.m1.27.27.27.8.6.6.1.cmml">Δ</mi></msub><mo id="S2.Ex2.m1.28.28.28.9.7.7">,</mo><msub id="S2.Ex2.m1.100.100.3.99.19.17.17.1.1.1.1.1.2.2.2"><mi id="S2.Ex2.m1.29.29.29.10.8.8" xref="S2.Ex2.m1.29.29.29.10.8.8.cmml">𝒙</mi><mi id="S2.Ex2.m1.30.30.30.11.9.9.1" xref="S2.Ex2.m1.30.30.30.11.9.9.1.cmml">t</mi></msub><mo id="S2.Ex2.m1.31.31.31.12.10.10" stretchy="false">⟩</mo></mrow><mo id="S2.Ex2.m1.32.32.32.13.11.11" xref="S2.Ex2.m1.32.32.32.13.11.11.cmml">+</mo><mi id="S2.Ex2.m1.33.33.33.14.12.12" xref="S2.Ex2.m1.33.33.33.14.12.12.cmml">δ</mi></mrow><mo id="S2.Ex2.m1.34.34.34.15.13.13" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.35.35.35.16.14.14" xref="S2.Ex2.m1.35.35.35.16.14.14.cmml">∈</mo><mi id="S2.Ex2.m1.36.36.36.17.15.15" xref="S2.Ex2.m1.36.36.36.17.15.15.cmml">ℝ</mi></mrow><mo id="S2.Ex2.m1.37.37.37.18.16.16">,</mo></mrow></mtd></mtr><mtr id="S2.Ex2.m1.103.103.6g"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex2.m1.103.103.6h"><msub id="S2.Ex2.m1.39.39.39.2.2"><mover accent="true" id="S2.Ex2.m1.38.38.38.1.1.1" xref="S2.Ex2.m1.38.38.38.1.1.1.cmml"><mi id="S2.Ex2.m1.38.38.38.1.1.1.2" xref="S2.Ex2.m1.38.38.38.1.1.1.2.cmml">𝒙</mi><mo id="S2.Ex2.m1.38.38.38.1.1.1.1" xref="S2.Ex2.m1.38.38.38.1.1.1.1.cmml">~</mo></mover><mi id="S2.Ex2.m1.39.39.39.2.2.2.1" xref="S2.Ex2.m1.39.39.39.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex2.m1.103.103.6i"><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21"><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1"><mi id="S2.Ex2.m1.101.101.4.100.23.21.21.1.3"></mi><mo id="S2.Ex2.m1.40.40.40.3.1.1" xref="S2.Ex2.m1.40.40.40.3.1.1.cmml">≜</mo><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1"><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1"><mi id="S2.Ex2.m1.41.41.41.4.2.2" xref="S2.Ex2.m1.41.41.41.4.2.2.cmml">ReLU</mi><mo id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1"><mo id="S2.Ex2.m1.42.42.42.5.3.3" stretchy="false">(</mo><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1"><msub id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.3"><mi id="S2.Ex2.m1.43.43.43.6.4.4" xref="S2.Ex2.m1.43.43.43.6.4.4.cmml">conv</mi><mi id="S2.Ex2.m1.44.44.44.7.5.5.1" xref="S2.Ex2.m1.44.44.44.7.5.5.1.cmml">X</mi></msub><mo id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.1.1"><mo id="S2.Ex2.m1.45.45.45.8.6.6" stretchy="false">(</mo><mrow id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.1.1.1"><msub id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.1.1.1.2"><mi id="S2.Ex2.m1.46.46.46.9.7.7" xref="S2.Ex2.m1.46.46.46.9.7.7.cmml">W</mi><mi id="S2.Ex2.m1.47.47.47.10.8.8.1" xref="S2.Ex2.m1.47.47.47.10.8.8.1.cmml">X</mi></msub><mo id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.1.1.1.1">⁢</mo><msubsup id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.1.1.1.1.1.1.1.3"><mi id="S2.Ex2.m1.48.48.48.11.9.9" xref="S2.Ex2.m1.48.48.48.11.9.9.cmml">𝒙</mi><mrow id="S2.Ex2.m1.49.49.49.12.10.10.1" xref="S2.Ex2.m1.49.49.49.12.10.10.1.cmml"><mrow id="S2.Ex2.m1.49.49.49.12.10.10.1.2" xref="S2.Ex2.m1.49.49.49.12.10.10.1.2.cmml"><mi id="S2.Ex2.m1.49.49.49.12.10.10.1.2.2" xref="S2.Ex2.m1.49.49.49.12.10.10.1.2.2.cmml">t</mi><mo id="S2.Ex2.m1.49.49.49.12.10.10.1.2.1" xref="S2.Ex2.m1.49.49.49.12.10.10.1.2.1.cmml">−</mo><mi id="S2.Ex2.m1.49.49.49.12.10.10.1.2.3" xref="S2.Ex2.m1.49.49.49.12.10.10.1.2.3.cmml">w</mi></mrow><mo id="S2.Ex2.m1.49.49.49.12.10.10.1.1" xref="S2.Ex2.m1.49.49.49.12.10.10.1.1.cmml">+</mo><mn id="S2.Ex2.m1.49.49.49.12.10.10.1.3" xref="S2.Ex2.m1.49.49.49.12.10.10.1.3.cmml">1</mn></mrow><mi id="S2.Ex2.m1.50.50.50.13.11.11.1" xref="S2.Ex2.m1.50.50.50.13.11.11.1.cmml">t</mi></msubsup></mrow><mo id="S2.Ex2.m1.51.51.51.14.12.12" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.52.52.52.15.13.13" rspace="0.055em" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.53.53.53.16.14.14" rspace="0.222em" xref="S2.Ex2.m1.53.53.53.16.14.14.cmml">⋅</mo><msub id="S2.Ex2.m1.101.101.4.100.23.21.21.1.1.2"><mi id="S2.Ex2.m1.54.54.54.17.15.15" mathvariant="normal" xref="S2.Ex2.m1.54.54.54.17.15.15.cmml">Δ</mi><mi id="S2.Ex2.m1.55.55.55.18.16.16.1" xref="S2.Ex2.m1.55.55.55.18.16.16.1.cmml">t</mi></msub></mrow><mo id="S2.Ex2.m1.56.56.56.19.17.17" xref="S2.Ex2.m1.56.56.56.19.17.17.cmml">∈</mo><msup id="S2.Ex2.m1.101.101.4.100.23.21.21.1.4"><mi id="S2.Ex2.m1.57.57.57.20.18.18" xref="S2.Ex2.m1.57.57.57.20.18.18.cmml">ℝ</mi><mrow id="S2.Ex2.m1.58.58.58.21.19.19.1" xref="S2.Ex2.m1.58.58.58.21.19.19.1.cmml"><mi id="S2.Ex2.m1.58.58.58.21.19.19.1.2" xref="S2.Ex2.m1.58.58.58.21.19.19.1.2.cmml">e</mi><mo id="S2.Ex2.m1.58.58.58.21.19.19.1.1" xref="S2.Ex2.m1.58.58.58.21.19.19.1.1.cmml">⁢</mo><mi id="S2.Ex2.m1.58.58.58.21.19.19.1.3" xref="S2.Ex2.m1.58.58.58.21.19.19.1.3.cmml">d</mi></mrow></msup></mrow><mo id="S2.Ex2.m1.59.59.59.22.20.20">,</mo></mrow></mtd></mtr><mtr id="S2.Ex2.m1.103.103.6j"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex2.m1.103.103.6k"><msub id="S2.Ex2.m1.61.61.61.2.2"><mi id="S2.Ex2.m1.60.60.60.1.1.1" xref="S2.Ex2.m1.60.60.60.1.1.1.cmml">𝒃</mi><mi id="S2.Ex2.m1.61.61.61.2.2.2.1" xref="S2.Ex2.m1.61.61.61.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex2.m1.103.103.6l"><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18"><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18.1"><mi id="S2.Ex2.m1.102.102.5.101.20.18.18.1.3"></mi><mo id="S2.Ex2.m1.62.62.62.3.1.1" xref="S2.Ex2.m1.62.62.62.3.1.1.cmml">≜</mo><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1"><mi id="S2.Ex2.m1.63.63.63.4.2.2" xref="S2.Ex2.m1.63.63.63.4.2.2.cmml">ReLU</mi><mo id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1"><mo id="S2.Ex2.m1.64.64.64.5.3.3" stretchy="false">(</mo><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1"><msub id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.3"><mi id="S2.Ex2.m1.65.65.65.6.4.4" xref="S2.Ex2.m1.65.65.65.6.4.4.cmml">conv</mi><mi id="S2.Ex2.m1.66.66.66.7.5.5.1" xref="S2.Ex2.m1.66.66.66.7.5.5.1.cmml">B</mi></msub><mo id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.1.1"><mo id="S2.Ex2.m1.67.67.67.8.6.6" stretchy="false">(</mo><mrow id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.1.1.1"><msub id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.1.1.1.2"><mi id="S2.Ex2.m1.68.68.68.9.7.7" xref="S2.Ex2.m1.68.68.68.9.7.7.cmml">W</mi><mi id="S2.Ex2.m1.69.69.69.10.8.8.1" xref="S2.Ex2.m1.69.69.69.10.8.8.1.cmml">B</mi></msub><mo id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.1.1.1.1">⁢</mo><msubsup id="S2.Ex2.m1.102.102.5.101.20.18.18.1.1.1.1.1.1.1.1.3"><mi id="S2.Ex2.m1.70.70.70.11.9.9" xref="S2.Ex2.m1.70.70.70.11.9.9.cmml">𝒙</mi><mrow id="S2.Ex2.m1.71.71.71.12.10.10.1" xref="S2.Ex2.m1.71.71.71.12.10.10.1.cmml"><mrow id="S2.Ex2.m1.71.71.71.12.10.10.1.2" xref="S2.Ex2.m1.71.71.71.12.10.10.1.2.cmml"><mi id="S2.Ex2.m1.71.71.71.12.10.10.1.2.2" xref="S2.Ex2.m1.71.71.71.12.10.10.1.2.2.cmml">t</mi><mo id="S2.Ex2.m1.71.71.71.12.10.10.1.2.1" xref="S2.Ex2.m1.71.71.71.12.10.10.1.2.1.cmml">−</mo><mi id="S2.Ex2.m1.71.71.71.12.10.10.1.2.3" xref="S2.Ex2.m1.71.71.71.12.10.10.1.2.3.cmml">w</mi></mrow><mo id="S2.Ex2.m1.71.71.71.12.10.10.1.1" xref="S2.Ex2.m1.71.71.71.12.10.10.1.1.cmml">+</mo><mn id="S2.Ex2.m1.71.71.71.12.10.10.1.3" xref="S2.Ex2.m1.71.71.71.12.10.10.1.3.cmml">1</mn></mrow><mi id="S2.Ex2.m1.72.72.72.13.11.11.1" xref="S2.Ex2.m1.72.72.72.13.11.11.1.cmml">t</mi></msubsup></mrow><mo id="S2.Ex2.m1.73.73.73.14.12.12" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.74.74.74.15.13.13" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.75.75.75.16.14.14" xref="S2.Ex2.m1.75.75.75.16.14.14.cmml">∈</mo><msup id="S2.Ex2.m1.102.102.5.101.20.18.18.1.4"><mi id="S2.Ex2.m1.76.76.76.17.15.15" xref="S2.Ex2.m1.76.76.76.17.15.15.cmml">ℝ</mi><mi id="S2.Ex2.m1.77.77.77.18.16.16.1" xref="S2.Ex2.m1.77.77.77.18.16.16.1.cmml">N</mi></msup></mrow><mo id="S2.Ex2.m1.78.78.78.19.17.17">,</mo></mrow></mtd></mtr><mtr id="S2.Ex2.m1.103.103.6m"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex2.m1.103.103.6n"><msub id="S2.Ex2.m1.80.80.80.2.2"><mi id="S2.Ex2.m1.79.79.79.1.1.1" xref="S2.Ex2.m1.79.79.79.1.1.1.cmml">𝒄</mi><mi id="S2.Ex2.m1.80.80.80.2.2.2.1" xref="S2.Ex2.m1.80.80.80.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex2.m1.103.103.6o"><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18"><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18.1"><mi id="S2.Ex2.m1.103.103.6.102.20.18.18.1.3"></mi><mo id="S2.Ex2.m1.81.81.81.3.1.1" xref="S2.Ex2.m1.81.81.81.3.1.1.cmml">≜</mo><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1"><mi id="S2.Ex2.m1.82.82.82.4.2.2" xref="S2.Ex2.m1.82.82.82.4.2.2.cmml">ReLU</mi><mo id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1"><mo id="S2.Ex2.m1.83.83.83.5.3.3" stretchy="false">(</mo><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1"><msub id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.3"><mi id="S2.Ex2.m1.84.84.84.6.4.4" xref="S2.Ex2.m1.84.84.84.6.4.4.cmml">conv</mi><mi id="S2.Ex2.m1.85.85.85.7.5.5.1" xref="S2.Ex2.m1.85.85.85.7.5.5.1.cmml">C</mi></msub><mo id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.2">⁢</mo><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.1.1"><mo id="S2.Ex2.m1.86.86.86.8.6.6" stretchy="false">(</mo><mrow id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.1.1.1"><msub id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.1.1.1.2"><mi id="S2.Ex2.m1.87.87.87.9.7.7" xref="S2.Ex2.m1.87.87.87.9.7.7.cmml">W</mi><mi id="S2.Ex2.m1.88.88.88.10.8.8.1" xref="S2.Ex2.m1.88.88.88.10.8.8.1.cmml">C</mi></msub><mo id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.1.1.1.1">⁢</mo><msubsup id="S2.Ex2.m1.103.103.6.102.20.18.18.1.1.1.1.1.1.1.1.3"><mi id="S2.Ex2.m1.89.89.89.11.9.9" xref="S2.Ex2.m1.89.89.89.11.9.9.cmml">𝒙</mi><mrow id="S2.Ex2.m1.90.90.90.12.10.10.1" xref="S2.Ex2.m1.90.90.90.12.10.10.1.cmml"><mrow id="S2.Ex2.m1.90.90.90.12.10.10.1.2" xref="S2.Ex2.m1.90.90.90.12.10.10.1.2.cmml"><mi id="S2.Ex2.m1.90.90.90.12.10.10.1.2.2" xref="S2.Ex2.m1.90.90.90.12.10.10.1.2.2.cmml">t</mi><mo id="S2.Ex2.m1.90.90.90.12.10.10.1.2.1" xref="S2.Ex2.m1.90.90.90.12.10.10.1.2.1.cmml">−</mo><mi id="S2.Ex2.m1.90.90.90.12.10.10.1.2.3" xref="S2.Ex2.m1.90.90.90.12.10.10.1.2.3.cmml">w</mi></mrow><mo id="S2.Ex2.m1.90.90.90.12.10.10.1.1" xref="S2.Ex2.m1.90.90.90.12.10.10.1.1.cmml">+</mo><mn id="S2.Ex2.m1.90.90.90.12.10.10.1.3" xref="S2.Ex2.m1.90.90.90.12.10.10.1.3.cmml">1</mn></mrow><mi id="S2.Ex2.m1.91.91.91.13.11.11.1" xref="S2.Ex2.m1.91.91.91.13.11.11.1.cmml">t</mi></msubsup></mrow><mo id="S2.Ex2.m1.92.92.92.14.12.12" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.93.93.93.15.13.13" stretchy="false">)</mo></mrow></mrow><mo id="S2.Ex2.m1.94.94.94.16.14.14" xref="S2.Ex2.m1.94.94.94.16.14.14.cmml">∈</mo><msup id="S2.Ex2.m1.103.103.6.102.20.18.18.1.4"><mi id="S2.Ex2.m1.95.95.95.17.15.15" xref="S2.Ex2.m1.95.95.95.17.15.15.cmml">ℝ</mi><mi id="S2.Ex2.m1.96.96.96.18.16.16.1" xref="S2.Ex2.m1.96.96.96.18.16.16.1.cmml">N</mi></msup></mrow><mo id="S2.Ex2.m1.97.97.97.19.17.17">,</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.103b"><apply id="S2.Ex2.m1.98.98.1.1.1.6.cmml"><csymbol cd="ambiguous" id="S2.Ex2.m1.98.98.1.1.1.6a.cmml">formulae-sequence</csymbol><apply id="S2.Ex2.m1.98.98.1.1.1.1.1.cmml"><and id="S2.Ex2.m1.98.98.1.1.1.1.1a.cmml"></and><apply id="S2.Ex2.m1.98.98.1.1.1.1.1b.cmml"><ci id="S2.Ex2.m1.3.3.3.3.1.1.cmml" xref="S2.Ex2.m1.3.3.3.3.1.1">≜</ci><apply id="S2.Ex2.m1.98.98.1.1.1.1.1.3.cmml"><csymbol cd="ambiguous" 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id="S2.Ex2.m1.98.98.1.1.1.5.5.6.1.cmml">superscript</csymbol><ci id="S2.Ex2.m1.95.95.95.17.15.15.cmml" xref="S2.Ex2.m1.95.95.95.17.15.15">ℝ</ci><ci id="S2.Ex2.m1.96.96.96.18.16.16.1.cmml" xref="S2.Ex2.m1.96.96.96.18.16.16.1">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.103c">\begin{split}a_{t}&\triangleq\exp(-a\cdot\Delta_{t})\in(0,1),\,\text{with}\\ \Delta_{t}&\triangleq\mathrm{softplus}(\langle{\bm{w}_{\Delta}},{\bm{x}_{t}}% \rangle+\delta)\in\mathbb{R},\\ \widetilde{\bm{x}}_{t}&\triangleq\mathrm{ReLU}(\mathrm{conv}_{X}(W_{X}\,\bm{x}% _{t-w+1}^{t}))\cdot\Delta_{t}\in\mathbb{R}^{ed},\\ \bm{b}_{t}&\triangleq\mathrm{ReLU}(\mathrm{conv}_{B}(W_{B}\,\bm{x}_{t-w+1}^{t}% ))\in\mathbb{R}^{N},\\ \bm{c}_{t}&\triangleq\mathrm{ReLU}(\mathrm{conv}_{C}(W_{C}\,\bm{x}_{t-w+1}^{t}% ))\in\mathbb{R}^{N},\end{split}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.103d">start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL ≜ roman_exp ( - italic_a ⋅ roman_Δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ∈ ( 0 , 1 ) , with end_CELL end_ROW start_ROW start_CELL roman_Δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL ≜ roman_softplus ( ⟨ bold_italic_w start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT , bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⟩ + italic_δ ) ∈ blackboard_R , end_CELL end_ROW start_ROW start_CELL over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL ≜ roman_ReLU ( roman_conv start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT bold_italic_x start_POSTSUBSCRIPT italic_t - italic_w + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) ⋅ roman_Δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_e italic_d end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL ≜ roman_ReLU ( roman_conv start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT bold_italic_x start_POSTSUBSCRIPT italic_t - italic_w + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL ≜ roman_ReLU ( roman_conv start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT bold_italic_x start_POSTSUBSCRIPT italic_t - italic_w + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Input selectivity)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.19">where <math alttext="a\geq 0,\bm{w}_{\Delta}\in\mathbb{R}^{d},\delta\in\mathbb{R},W_{X}\in\mathbb{R% }^{ed\times d},W_{B}\in\mathbb{R}^{N\times d}" class="ltx_Math" display="inline" id="S2.SS2.p1.13.m1.2"><semantics id="S2.SS2.p1.13.m1.2a"><mrow id="S2.SS2.p1.13.m1.2.2.2" xref="S2.SS2.p1.13.m1.2.2.3.cmml"><mrow id="S2.SS2.p1.13.m1.1.1.1.1" xref="S2.SS2.p1.13.m1.1.1.1.1.cmml"><mi id="S2.SS2.p1.13.m1.1.1.1.1.2" xref="S2.SS2.p1.13.m1.1.1.1.1.2.cmml">a</mi><mo id="S2.SS2.p1.13.m1.1.1.1.1.1" xref="S2.SS2.p1.13.m1.1.1.1.1.1.cmml">≥</mo><mn id="S2.SS2.p1.13.m1.1.1.1.1.3" xref="S2.SS2.p1.13.m1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S2.SS2.p1.13.m1.2.2.2.3" xref="S2.SS2.p1.13.m1.2.2.3a.cmml">,</mo><mrow id="S2.SS2.p1.13.m1.2.2.2.2.2" 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xref="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.2">ℝ</ci><apply id="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.cmml" xref="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3"><times id="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.1.cmml" xref="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.1"></times><ci id="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.2.cmml" xref="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.2">𝑁</ci><ci id="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.3.cmml" xref="S2.SS2.p1.13.m1.2.2.2.2.2.2.2.2.2.2.3.3.3">𝑑</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.13.m1.2c">a\geq 0,\bm{w}_{\Delta}\in\mathbb{R}^{d},\delta\in\mathbb{R},W_{X}\in\mathbb{R% }^{ed\times d},W_{B}\in\mathbb{R}^{N\times d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.13.m1.2d">italic_a ≥ 0 , bold_italic_w start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_δ ∈ blackboard_R , italic_W start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_e italic_d × italic_d end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="W_{C}\in\mathbb{R}^{N\times d}" class="ltx_Math" display="inline" id="S2.SS2.p1.14.m2.1"><semantics id="S2.SS2.p1.14.m2.1a"><mrow id="S2.SS2.p1.14.m2.1.1" xref="S2.SS2.p1.14.m2.1.1.cmml"><msub id="S2.SS2.p1.14.m2.1.1.2" xref="S2.SS2.p1.14.m2.1.1.2.cmml"><mi id="S2.SS2.p1.14.m2.1.1.2.2" xref="S2.SS2.p1.14.m2.1.1.2.2.cmml">W</mi><mi id="S2.SS2.p1.14.m2.1.1.2.3" xref="S2.SS2.p1.14.m2.1.1.2.3.cmml">C</mi></msub><mo id="S2.SS2.p1.14.m2.1.1.1" xref="S2.SS2.p1.14.m2.1.1.1.cmml">∈</mo><msup id="S2.SS2.p1.14.m2.1.1.3" xref="S2.SS2.p1.14.m2.1.1.3.cmml"><mi id="S2.SS2.p1.14.m2.1.1.3.2" xref="S2.SS2.p1.14.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS2.p1.14.m2.1.1.3.3" xref="S2.SS2.p1.14.m2.1.1.3.3.cmml"><mi id="S2.SS2.p1.14.m2.1.1.3.3.2" xref="S2.SS2.p1.14.m2.1.1.3.3.2.cmml">N</mi><mo id="S2.SS2.p1.14.m2.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p1.14.m2.1.1.3.3.1.cmml">×</mo><mi id="S2.SS2.p1.14.m2.1.1.3.3.3" xref="S2.SS2.p1.14.m2.1.1.3.3.3.cmml">d</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.14.m2.1b"><apply id="S2.SS2.p1.14.m2.1.1.cmml" xref="S2.SS2.p1.14.m2.1.1"><in id="S2.SS2.p1.14.m2.1.1.1.cmml" xref="S2.SS2.p1.14.m2.1.1.1"></in><apply id="S2.SS2.p1.14.m2.1.1.2.cmml" xref="S2.SS2.p1.14.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.14.m2.1.1.2.1.cmml" xref="S2.SS2.p1.14.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.14.m2.1.1.2.2.cmml" xref="S2.SS2.p1.14.m2.1.1.2.2">𝑊</ci><ci id="S2.SS2.p1.14.m2.1.1.2.3.cmml" xref="S2.SS2.p1.14.m2.1.1.2.3">𝐶</ci></apply><apply id="S2.SS2.p1.14.m2.1.1.3.cmml" xref="S2.SS2.p1.14.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.14.m2.1.1.3.1.cmml" xref="S2.SS2.p1.14.m2.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.14.m2.1.1.3.2.cmml" xref="S2.SS2.p1.14.m2.1.1.3.2">ℝ</ci><apply id="S2.SS2.p1.14.m2.1.1.3.3.cmml" xref="S2.SS2.p1.14.m2.1.1.3.3"><times id="S2.SS2.p1.14.m2.1.1.3.3.1.cmml" xref="S2.SS2.p1.14.m2.1.1.3.3.1"></times><ci id="S2.SS2.p1.14.m2.1.1.3.3.2.cmml" xref="S2.SS2.p1.14.m2.1.1.3.3.2">𝑁</ci><ci id="S2.SS2.p1.14.m2.1.1.3.3.3.cmml" xref="S2.SS2.p1.14.m2.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.14.m2.1c">W_{C}\in\mathbb{R}^{N\times d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.14.m2.1d">italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> are all learnable parameters and <math alttext="\mathrm{conv}(\bm{z}_{t-w+1}^{t})" class="ltx_Math" display="inline" id="S2.SS2.p1.15.m3.1"><semantics id="S2.SS2.p1.15.m3.1a"><mrow id="S2.SS2.p1.15.m3.1.1" xref="S2.SS2.p1.15.m3.1.1.cmml"><mi id="S2.SS2.p1.15.m3.1.1.3" xref="S2.SS2.p1.15.m3.1.1.3.cmml">conv</mi><mo id="S2.SS2.p1.15.m3.1.1.2" xref="S2.SS2.p1.15.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.15.m3.1.1.1.1" xref="S2.SS2.p1.15.m3.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.15.m3.1.1.1.1.2" stretchy="false" xref="S2.SS2.p1.15.m3.1.1.1.1.1.cmml">(</mo><msubsup id="S2.SS2.p1.15.m3.1.1.1.1.1" xref="S2.SS2.p1.15.m3.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.15.m3.1.1.1.1.1.2.2" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.2.cmml">𝒛</mi><mrow id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.cmml"><mrow id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.cmml"><mi id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.2" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.2.cmml">t</mi><mo id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.1" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.1.cmml">−</mo><mi id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.3" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.3.cmml">w</mi></mrow><mo id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.1" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.3" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S2.SS2.p1.15.m3.1.1.1.1.1.3" xref="S2.SS2.p1.15.m3.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S2.SS2.p1.15.m3.1.1.1.1.3" stretchy="false" xref="S2.SS2.p1.15.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.15.m3.1b"><apply id="S2.SS2.p1.15.m3.1.1.cmml" xref="S2.SS2.p1.15.m3.1.1"><times id="S2.SS2.p1.15.m3.1.1.2.cmml" xref="S2.SS2.p1.15.m3.1.1.2"></times><ci id="S2.SS2.p1.15.m3.1.1.3.cmml" xref="S2.SS2.p1.15.m3.1.1.3">conv</ci><apply id="S2.SS2.p1.15.m3.1.1.1.1.1.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m3.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1">superscript</csymbol><apply id="S2.SS2.p1.15.m3.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m3.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.15.m3.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.2">𝒛</ci><apply id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3"><plus id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.1.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.1"></plus><apply id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2"><minus id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.1.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.1"></minus><ci id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.2.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.2">𝑡</ci><ci id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.3.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.2.3">𝑤</ci></apply><cn id="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS2.p1.15.m3.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S2.SS2.p1.15.m3.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.15.m3.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.15.m3.1c">\mathrm{conv}(\bm{z}_{t-w+1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.15.m3.1d">roman_conv ( bold_italic_z start_POSTSUBSCRIPT italic_t - italic_w + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> is a (learnable) time-wise convolution of window <math alttext="w\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.SS2.p1.16.m4.1"><semantics id="S2.SS2.p1.16.m4.1a"><mrow id="S2.SS2.p1.16.m4.1.1" xref="S2.SS2.p1.16.m4.1.1.cmml"><mi id="S2.SS2.p1.16.m4.1.1.2" xref="S2.SS2.p1.16.m4.1.1.2.cmml">w</mi><mo id="S2.SS2.p1.16.m4.1.1.1" xref="S2.SS2.p1.16.m4.1.1.1.cmml">∈</mo><mi id="S2.SS2.p1.16.m4.1.1.3" xref="S2.SS2.p1.16.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.16.m4.1b"><apply id="S2.SS2.p1.16.m4.1.1.cmml" xref="S2.SS2.p1.16.m4.1.1"><in id="S2.SS2.p1.16.m4.1.1.1.cmml" xref="S2.SS2.p1.16.m4.1.1.1"></in><ci id="S2.SS2.p1.16.m4.1.1.2.cmml" xref="S2.SS2.p1.16.m4.1.1.2">𝑤</ci><ci id="S2.SS2.p1.16.m4.1.1.3.cmml" xref="S2.SS2.p1.16.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.16.m4.1c">w\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.16.m4.1d">italic_w ∈ blackboard_N</annotation></semantics></math> with distinct kernels per dimension. Here <math alttext="e\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.SS2.p1.17.m5.1"><semantics id="S2.SS2.p1.17.m5.1a"><mrow id="S2.SS2.p1.17.m5.1.1" xref="S2.SS2.p1.17.m5.1.1.cmml"><mi id="S2.SS2.p1.17.m5.1.1.2" xref="S2.SS2.p1.17.m5.1.1.2.cmml">e</mi><mo id="S2.SS2.p1.17.m5.1.1.1" xref="S2.SS2.p1.17.m5.1.1.1.cmml">∈</mo><mi id="S2.SS2.p1.17.m5.1.1.3" xref="S2.SS2.p1.17.m5.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.17.m5.1b"><apply id="S2.SS2.p1.17.m5.1.1.cmml" xref="S2.SS2.p1.17.m5.1.1"><in id="S2.SS2.p1.17.m5.1.1.1.cmml" xref="S2.SS2.p1.17.m5.1.1.1"></in><ci id="S2.SS2.p1.17.m5.1.1.2.cmml" xref="S2.SS2.p1.17.m5.1.1.2">𝑒</ci><ci id="S2.SS2.p1.17.m5.1.1.3.cmml" xref="S2.SS2.p1.17.m5.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.17.m5.1c">e\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.17.m5.1d">italic_e ∈ blackboard_N</annotation></semantics></math> denotes the expansion factor for the features, typically <math alttext="2" class="ltx_Math" display="inline" id="S2.SS2.p1.18.m6.1"><semantics id="S2.SS2.p1.18.m6.1a"><mn id="S2.SS2.p1.18.m6.1.1" xref="S2.SS2.p1.18.m6.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.18.m6.1b"><cn id="S2.SS2.p1.18.m6.1.1.cmml" type="integer" xref="S2.SS2.p1.18.m6.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.18.m6.1c">2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.18.m6.1d">2</annotation></semantics></math>. Let <math alttext="\bm{\theta}_{\mathsf{Mamba}}" class="ltx_Math" display="inline" id="S2.SS2.p1.19.m7.1"><semantics id="S2.SS2.p1.19.m7.1a"><msub id="S2.SS2.p1.19.m7.1.1" xref="S2.SS2.p1.19.m7.1.1.cmml"><mi id="S2.SS2.p1.19.m7.1.1.2" xref="S2.SS2.p1.19.m7.1.1.2.cmml">𝜽</mi><mi id="S2.SS2.p1.19.m7.1.1.3" xref="S2.SS2.p1.19.m7.1.1.3.cmml">𝖬𝖺𝗆𝖻𝖺</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.19.m7.1b"><apply id="S2.SS2.p1.19.m7.1.1.cmml" xref="S2.SS2.p1.19.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.19.m7.1.1.1.cmml" xref="S2.SS2.p1.19.m7.1.1">subscript</csymbol><ci id="S2.SS2.p1.19.m7.1.1.2.cmml" xref="S2.SS2.p1.19.m7.1.1.2">𝜽</ci><ci id="S2.SS2.p1.19.m7.1.1.3.cmml" xref="S2.SS2.p1.19.m7.1.1.3">𝖬𝖺𝗆𝖻𝖺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.19.m7.1c">\bm{\theta}_{\mathsf{Mamba}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.19.m7.1d">bold_italic_θ start_POSTSUBSCRIPT sansserif_Mamba end_POSTSUBSCRIPT</annotation></semantics></math> denote the set of all these parameters.</p> </div> <div class="ltx_para ltx_noindent" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.15"><span class="ltx_text ltx_font_bold" id="S2.SS2.p2.15.1">Intuition behind Mamba.</span> While the update equations in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex1" title="Equation Mamba ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Mamba</span></a> might seem complicated at a first glance, the underlying intuition is simple: given a sequence of input embeddings <math alttext="(\bm{x}_{t})" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.1.1" xref="S2.SS2.p2.1.m1.1.1.1.1.cmml"><mo id="S2.SS2.p2.1.m1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.1.m1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p2.1.m1.1.1.1.1" xref="S2.SS2.p2.1.m1.1.1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.1.1.2.cmml">𝒙</mi><mi id="S2.SS2.p2.1.m1.1.1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.SS2.p2.1.m1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.1.1.2">𝒙</ci><ci id="S2.SS2.p2.1.m1.1.1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">(\bm{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>, we first capture their local temporal information using three separate convolutions to compute <math alttext="\widetilde{\bm{x}}_{t},\bm{b}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.2"><semantics id="S2.SS2.p2.2.m2.2a"><mrow id="S2.SS2.p2.2.m2.2.2.2" xref="S2.SS2.p2.2.m2.2.2.3.cmml"><msub id="S2.SS2.p2.2.m2.1.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.1.cmml"><mover accent="true" id="S2.SS2.p2.2.m2.1.1.1.1.2" xref="S2.SS2.p2.2.m2.1.1.1.1.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.1.1.2.2" xref="S2.SS2.p2.2.m2.1.1.1.1.2.2.cmml">𝒙</mi><mo id="S2.SS2.p2.2.m2.1.1.1.1.2.1" xref="S2.SS2.p2.2.m2.1.1.1.1.2.1.cmml">~</mo></mover><mi id="S2.SS2.p2.2.m2.1.1.1.1.3" xref="S2.SS2.p2.2.m2.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.SS2.p2.2.m2.2.2.2.3" xref="S2.SS2.p2.2.m2.2.2.3.cmml">,</mo><msub id="S2.SS2.p2.2.m2.2.2.2.2" xref="S2.SS2.p2.2.m2.2.2.2.2.cmml"><mi id="S2.SS2.p2.2.m2.2.2.2.2.2" xref="S2.SS2.p2.2.m2.2.2.2.2.2.cmml">𝒃</mi><mi id="S2.SS2.p2.2.m2.2.2.2.2.3" xref="S2.SS2.p2.2.m2.2.2.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.2b"><list id="S2.SS2.p2.2.m2.2.2.3.cmml" xref="S2.SS2.p2.2.m2.2.2.2"><apply id="S2.SS2.p2.2.m2.1.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m2.1.1.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1.1">subscript</csymbol><apply id="S2.SS2.p2.2.m2.1.1.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.1.1.2"><ci id="S2.SS2.p2.2.m2.1.1.1.1.2.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1.1.2.1">~</ci><ci id="S2.SS2.p2.2.m2.1.1.1.1.2.2.cmml" xref="S2.SS2.p2.2.m2.1.1.1.1.2.2">𝒙</ci></apply><ci id="S2.SS2.p2.2.m2.1.1.1.1.3.cmml" xref="S2.SS2.p2.2.m2.1.1.1.1.3">𝑡</ci></apply><apply id="S2.SS2.p2.2.m2.2.2.2.2.cmml" xref="S2.SS2.p2.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m2.2.2.2.2.1.cmml" xref="S2.SS2.p2.2.m2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.p2.2.m2.2.2.2.2.2.cmml" xref="S2.SS2.p2.2.m2.2.2.2.2.2">𝒃</ci><ci id="S2.SS2.p2.2.m2.2.2.2.2.3.cmml" xref="S2.SS2.p2.2.m2.2.2.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.2c">\widetilde{\bm{x}}_{t},\bm{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.2d">over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.1"><semantics id="S2.SS2.p2.3.m3.1a"><msub id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml"><mi id="S2.SS2.p2.3.m3.1.1.2" xref="S2.SS2.p2.3.m3.1.1.2.cmml">𝒄</mi><mi id="S2.SS2.p2.3.m3.1.1.3" xref="S2.SS2.p2.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.1b"><apply id="S2.SS2.p2.3.m3.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m3.1.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.p2.3.m3.1.1.2.cmml" xref="S2.SS2.p2.3.m3.1.1.2">𝒄</ci><ci id="S2.SS2.p2.3.m3.1.1.3.cmml" xref="S2.SS2.p2.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> respectively (<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a>). Equipped with this local memory, we perform the classical linear state update to compute the current state <math alttext="H_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><msub id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml"><mi id="S2.SS2.p2.4.m4.1.1.2" xref="S2.SS2.p2.4.m4.1.1.2.cmml">H</mi><mi id="S2.SS2.p2.4.m4.1.1.3" xref="S2.SS2.p2.4.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><apply id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m4.1.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.p2.4.m4.1.1.2.cmml" xref="S2.SS2.p2.4.m4.1.1.2">𝐻</ci><ci id="S2.SS2.p2.4.m4.1.1.3.cmml" xref="S2.SS2.p2.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> from the past <math alttext="H_{t-1}" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m5.1"><semantics id="S2.SS2.p2.5.m5.1a"><msub id="S2.SS2.p2.5.m5.1.1" xref="S2.SS2.p2.5.m5.1.1.cmml"><mi id="S2.SS2.p2.5.m5.1.1.2" xref="S2.SS2.p2.5.m5.1.1.2.cmml">H</mi><mrow id="S2.SS2.p2.5.m5.1.1.3" xref="S2.SS2.p2.5.m5.1.1.3.cmml"><mi id="S2.SS2.p2.5.m5.1.1.3.2" xref="S2.SS2.p2.5.m5.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p2.5.m5.1.1.3.1" xref="S2.SS2.p2.5.m5.1.1.3.1.cmml">−</mo><mn id="S2.SS2.p2.5.m5.1.1.3.3" xref="S2.SS2.p2.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m5.1b"><apply id="S2.SS2.p2.5.m5.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m5.1.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1">subscript</csymbol><ci id="S2.SS2.p2.5.m5.1.1.2.cmml" xref="S2.SS2.p2.5.m5.1.1.2">𝐻</ci><apply id="S2.SS2.p2.5.m5.1.1.3.cmml" xref="S2.SS2.p2.5.m5.1.1.3"><minus id="S2.SS2.p2.5.m5.1.1.3.1.cmml" xref="S2.SS2.p2.5.m5.1.1.3.1"></minus><ci id="S2.SS2.p2.5.m5.1.1.3.2.cmml" xref="S2.SS2.p2.5.m5.1.1.3.2">𝑡</ci><cn id="S2.SS2.p2.5.m5.1.1.3.3.cmml" type="integer" xref="S2.SS2.p2.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m5.1c">H_{t-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m5.1d">italic_H start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT</annotation></semantics></math>, weighed by an input-dependent decay factor <math alttext="a_{t}\in(0,1)" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m6.2"><semantics id="S2.SS2.p2.6.m6.2a"><mrow id="S2.SS2.p2.6.m6.2.3" xref="S2.SS2.p2.6.m6.2.3.cmml"><msub id="S2.SS2.p2.6.m6.2.3.2" xref="S2.SS2.p2.6.m6.2.3.2.cmml"><mi id="S2.SS2.p2.6.m6.2.3.2.2" xref="S2.SS2.p2.6.m6.2.3.2.2.cmml">a</mi><mi id="S2.SS2.p2.6.m6.2.3.2.3" xref="S2.SS2.p2.6.m6.2.3.2.3.cmml">t</mi></msub><mo id="S2.SS2.p2.6.m6.2.3.1" xref="S2.SS2.p2.6.m6.2.3.1.cmml">∈</mo><mrow id="S2.SS2.p2.6.m6.2.3.3.2" xref="S2.SS2.p2.6.m6.2.3.3.1.cmml"><mo id="S2.SS2.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S2.SS2.p2.6.m6.2.3.3.1.cmml">(</mo><mn id="S2.SS2.p2.6.m6.1.1" xref="S2.SS2.p2.6.m6.1.1.cmml">0</mn><mo id="S2.SS2.p2.6.m6.2.3.3.2.2" xref="S2.SS2.p2.6.m6.2.3.3.1.cmml">,</mo><mn id="S2.SS2.p2.6.m6.2.2" xref="S2.SS2.p2.6.m6.2.2.cmml">1</mn><mo id="S2.SS2.p2.6.m6.2.3.3.2.3" stretchy="false" xref="S2.SS2.p2.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m6.2b"><apply id="S2.SS2.p2.6.m6.2.3.cmml" xref="S2.SS2.p2.6.m6.2.3"><in id="S2.SS2.p2.6.m6.2.3.1.cmml" xref="S2.SS2.p2.6.m6.2.3.1"></in><apply id="S2.SS2.p2.6.m6.2.3.2.cmml" xref="S2.SS2.p2.6.m6.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.6.m6.2.3.2.1.cmml" xref="S2.SS2.p2.6.m6.2.3.2">subscript</csymbol><ci id="S2.SS2.p2.6.m6.2.3.2.2.cmml" xref="S2.SS2.p2.6.m6.2.3.2.2">𝑎</ci><ci id="S2.SS2.p2.6.m6.2.3.2.3.cmml" xref="S2.SS2.p2.6.m6.2.3.2.3">𝑡</ci></apply><interval closure="open" id="S2.SS2.p2.6.m6.2.3.3.1.cmml" xref="S2.SS2.p2.6.m6.2.3.3.2"><cn id="S2.SS2.p2.6.m6.1.1.cmml" type="integer" xref="S2.SS2.p2.6.m6.1.1">0</cn><cn id="S2.SS2.p2.6.m6.2.2.cmml" type="integer" xref="S2.SS2.p2.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m6.2c">a_{t}\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m6.2d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ ( 0 , 1 )</annotation></semantics></math>, and <math alttext="(\widetilde{\bm{x}}_{t},\bm{b}_{t})" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m7.2"><semantics id="S2.SS2.p2.7.m7.2a"><mrow id="S2.SS2.p2.7.m7.2.2.2" xref="S2.SS2.p2.7.m7.2.2.3.cmml"><mo id="S2.SS2.p2.7.m7.2.2.2.3" stretchy="false" xref="S2.SS2.p2.7.m7.2.2.3.cmml">(</mo><msub id="S2.SS2.p2.7.m7.1.1.1.1" xref="S2.SS2.p2.7.m7.1.1.1.1.cmml"><mover accent="true" id="S2.SS2.p2.7.m7.1.1.1.1.2" xref="S2.SS2.p2.7.m7.1.1.1.1.2.cmml"><mi id="S2.SS2.p2.7.m7.1.1.1.1.2.2" xref="S2.SS2.p2.7.m7.1.1.1.1.2.2.cmml">𝒙</mi><mo id="S2.SS2.p2.7.m7.1.1.1.1.2.1" xref="S2.SS2.p2.7.m7.1.1.1.1.2.1.cmml">~</mo></mover><mi id="S2.SS2.p2.7.m7.1.1.1.1.3" xref="S2.SS2.p2.7.m7.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.SS2.p2.7.m7.2.2.2.4" xref="S2.SS2.p2.7.m7.2.2.3.cmml">,</mo><msub id="S2.SS2.p2.7.m7.2.2.2.2" xref="S2.SS2.p2.7.m7.2.2.2.2.cmml"><mi id="S2.SS2.p2.7.m7.2.2.2.2.2" xref="S2.SS2.p2.7.m7.2.2.2.2.2.cmml">𝒃</mi><mi id="S2.SS2.p2.7.m7.2.2.2.2.3" xref="S2.SS2.p2.7.m7.2.2.2.2.3.cmml">t</mi></msub><mo id="S2.SS2.p2.7.m7.2.2.2.5" stretchy="false" xref="S2.SS2.p2.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m7.2b"><interval closure="open" id="S2.SS2.p2.7.m7.2.2.3.cmml" xref="S2.SS2.p2.7.m7.2.2.2"><apply id="S2.SS2.p2.7.m7.1.1.1.1.cmml" xref="S2.SS2.p2.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.7.m7.1.1.1.1.1.cmml" xref="S2.SS2.p2.7.m7.1.1.1.1">subscript</csymbol><apply id="S2.SS2.p2.7.m7.1.1.1.1.2.cmml" xref="S2.SS2.p2.7.m7.1.1.1.1.2"><ci id="S2.SS2.p2.7.m7.1.1.1.1.2.1.cmml" xref="S2.SS2.p2.7.m7.1.1.1.1.2.1">~</ci><ci id="S2.SS2.p2.7.m7.1.1.1.1.2.2.cmml" xref="S2.SS2.p2.7.m7.1.1.1.1.2.2">𝒙</ci></apply><ci id="S2.SS2.p2.7.m7.1.1.1.1.3.cmml" xref="S2.SS2.p2.7.m7.1.1.1.1.3">𝑡</ci></apply><apply id="S2.SS2.p2.7.m7.2.2.2.2.cmml" xref="S2.SS2.p2.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.7.m7.2.2.2.2.1.cmml" xref="S2.SS2.p2.7.m7.2.2.2.2">subscript</csymbol><ci id="S2.SS2.p2.7.m7.2.2.2.2.2.cmml" xref="S2.SS2.p2.7.m7.2.2.2.2.2">𝒃</ci><ci id="S2.SS2.p2.7.m7.2.2.2.2.3.cmml" xref="S2.SS2.p2.7.m7.2.2.2.2.3">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m7.2c">(\widetilde{\bm{x}}_{t},\bm{b}_{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m7.2d">( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>. Subsequently, we compute the state projection <math alttext="\bm{y}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m8.1"><semantics id="S2.SS2.p2.8.m8.1a"><msub id="S2.SS2.p2.8.m8.1.1" xref="S2.SS2.p2.8.m8.1.1.cmml"><mi id="S2.SS2.p2.8.m8.1.1.2" xref="S2.SS2.p2.8.m8.1.1.2.cmml">𝒚</mi><mi id="S2.SS2.p2.8.m8.1.1.3" xref="S2.SS2.p2.8.m8.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m8.1b"><apply id="S2.SS2.p2.8.m8.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.8.m8.1.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1">subscript</csymbol><ci id="S2.SS2.p2.8.m8.1.1.2.cmml" xref="S2.SS2.p2.8.m8.1.1.2">𝒚</ci><ci id="S2.SS2.p2.8.m8.1.1.3.cmml" xref="S2.SS2.p2.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m8.1c">\bm{y}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m8.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, modulate it with an input-selective ReLU term to yield <math alttext="\bm{z}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m9.1"><semantics id="S2.SS2.p2.9.m9.1a"><msub id="S2.SS2.p2.9.m9.1.1" xref="S2.SS2.p2.9.m9.1.1.cmml"><mi id="S2.SS2.p2.9.m9.1.1.2" xref="S2.SS2.p2.9.m9.1.1.2.cmml">𝒛</mi><mi id="S2.SS2.p2.9.m9.1.1.3" xref="S2.SS2.p2.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m9.1b"><apply id="S2.SS2.p2.9.m9.1.1.cmml" xref="S2.SS2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.9.m9.1.1.1.cmml" xref="S2.SS2.p2.9.m9.1.1">subscript</csymbol><ci id="S2.SS2.p2.9.m9.1.1.2.cmml" xref="S2.SS2.p2.9.m9.1.1.2">𝒛</ci><ci id="S2.SS2.p2.9.m9.1.1.3.cmml" xref="S2.SS2.p2.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m9.1c">\bm{z}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m9.1d">bold_italic_z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, and finally project it down to get the <math alttext="d" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m10.1"><semantics id="S2.SS2.p2.10.m10.1a"><mi id="S2.SS2.p2.10.m10.1.1" xref="S2.SS2.p2.10.m10.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.10.m10.1b"><ci id="S2.SS2.p2.10.m10.1.1.cmml" xref="S2.SS2.p2.10.m10.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m10.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m10.1d">italic_d</annotation></semantics></math>-dimensional output embedding <math alttext="\bm{o}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.11.m11.1"><semantics id="S2.SS2.p2.11.m11.1a"><msub id="S2.SS2.p2.11.m11.1.1" xref="S2.SS2.p2.11.m11.1.1.cmml"><mi id="S2.SS2.p2.11.m11.1.1.2" xref="S2.SS2.p2.11.m11.1.1.2.cmml">𝒐</mi><mi id="S2.SS2.p2.11.m11.1.1.3" xref="S2.SS2.p2.11.m11.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.11.m11.1b"><apply id="S2.SS2.p2.11.m11.1.1.cmml" xref="S2.SS2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.11.m11.1.1.1.cmml" xref="S2.SS2.p2.11.m11.1.1">subscript</csymbol><ci id="S2.SS2.p2.11.m11.1.1.2.cmml" xref="S2.SS2.p2.11.m11.1.1.2">𝒐</ci><ci id="S2.SS2.p2.11.m11.1.1.3.cmml" xref="S2.SS2.p2.11.m11.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.11.m11.1c">\bm{o}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.11.m11.1d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. Thus the output <math alttext="\bm{o}_{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.12.m12.1"><semantics id="S2.SS2.p2.12.m12.1a"><msub id="S2.SS2.p2.12.m12.1.1" xref="S2.SS2.p2.12.m12.1.1.cmml"><mi id="S2.SS2.p2.12.m12.1.1.2" xref="S2.SS2.p2.12.m12.1.1.2.cmml">𝒐</mi><mi id="S2.SS2.p2.12.m12.1.1.3" xref="S2.SS2.p2.12.m12.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.12.m12.1b"><apply id="S2.SS2.p2.12.m12.1.1.cmml" xref="S2.SS2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m12.1.1.1.cmml" xref="S2.SS2.p2.12.m12.1.1">subscript</csymbol><ci id="S2.SS2.p2.12.m12.1.1.2.cmml" xref="S2.SS2.p2.12.m12.1.1.2">𝒐</ci><ci id="S2.SS2.p2.12.m12.1.1.3.cmml" xref="S2.SS2.p2.12.m12.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.12.m12.1c">\bm{o}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.12.m12.1d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p2.13.m13.1"><semantics id="S2.SS2.p2.13.m13.1a"><mi id="S2.SS2.p2.13.m13.1.1" xref="S2.SS2.p2.13.m13.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.13.m13.1b"><ci id="S2.SS2.p2.13.m13.1.1.cmml" xref="S2.SS2.p2.13.m13.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.13.m13.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.13.m13.1d">italic_t</annotation></semantics></math> is a function of the entire input sequence till then, <math alttext="\bm{x}_{1}^{t}" class="ltx_Math" display="inline" id="S2.SS2.p2.14.m14.1"><semantics id="S2.SS2.p2.14.m14.1a"><msubsup id="S2.SS2.p2.14.m14.1.1" xref="S2.SS2.p2.14.m14.1.1.cmml"><mi id="S2.SS2.p2.14.m14.1.1.2.2" xref="S2.SS2.p2.14.m14.1.1.2.2.cmml">𝒙</mi><mn id="S2.SS2.p2.14.m14.1.1.2.3" xref="S2.SS2.p2.14.m14.1.1.2.3.cmml">1</mn><mi id="S2.SS2.p2.14.m14.1.1.3" xref="S2.SS2.p2.14.m14.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.14.m14.1b"><apply id="S2.SS2.p2.14.m14.1.1.cmml" xref="S2.SS2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.14.m14.1.1.1.cmml" xref="S2.SS2.p2.14.m14.1.1">superscript</csymbol><apply id="S2.SS2.p2.14.m14.1.1.2.cmml" xref="S2.SS2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.14.m14.1.1.2.1.cmml" xref="S2.SS2.p2.14.m14.1.1">subscript</csymbol><ci id="S2.SS2.p2.14.m14.1.1.2.2.cmml" xref="S2.SS2.p2.14.m14.1.1.2.2">𝒙</ci><cn id="S2.SS2.p2.14.m14.1.1.2.3.cmml" type="integer" xref="S2.SS2.p2.14.m14.1.1.2.3">1</cn></apply><ci id="S2.SS2.p2.14.m14.1.1.3.cmml" xref="S2.SS2.p2.14.m14.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.14.m14.1c">\bm{x}_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.14.m14.1d">bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, yielding <math alttext="\bm{o}_{t}=\mathsf{Mamba}(\bm{x}_{1}^{t})" class="ltx_Math" display="inline" id="S2.SS2.p2.15.m15.1"><semantics id="S2.SS2.p2.15.m15.1a"><mrow id="S2.SS2.p2.15.m15.1.1" xref="S2.SS2.p2.15.m15.1.1.cmml"><msub id="S2.SS2.p2.15.m15.1.1.3" xref="S2.SS2.p2.15.m15.1.1.3.cmml"><mi id="S2.SS2.p2.15.m15.1.1.3.2" xref="S2.SS2.p2.15.m15.1.1.3.2.cmml">𝒐</mi><mi id="S2.SS2.p2.15.m15.1.1.3.3" xref="S2.SS2.p2.15.m15.1.1.3.3.cmml">t</mi></msub><mo id="S2.SS2.p2.15.m15.1.1.2" xref="S2.SS2.p2.15.m15.1.1.2.cmml">=</mo><mrow id="S2.SS2.p2.15.m15.1.1.1" xref="S2.SS2.p2.15.m15.1.1.1.cmml"><mi id="S2.SS2.p2.15.m15.1.1.1.3" xref="S2.SS2.p2.15.m15.1.1.1.3.cmml">𝖬𝖺𝗆𝖻𝖺</mi><mo id="S2.SS2.p2.15.m15.1.1.1.2" xref="S2.SS2.p2.15.m15.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.15.m15.1.1.1.1.1" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.15.m15.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.cmml">(</mo><msubsup id="S2.SS2.p2.15.m15.1.1.1.1.1.1" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.2" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mn id="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.3" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S2.SS2.p2.15.m15.1.1.1.1.1.1.3" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S2.SS2.p2.15.m15.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.15.m15.1b"><apply id="S2.SS2.p2.15.m15.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1"><eq id="S2.SS2.p2.15.m15.1.1.2.cmml" xref="S2.SS2.p2.15.m15.1.1.2"></eq><apply id="S2.SS2.p2.15.m15.1.1.3.cmml" xref="S2.SS2.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.15.m15.1.1.3.1.cmml" xref="S2.SS2.p2.15.m15.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.15.m15.1.1.3.2.cmml" xref="S2.SS2.p2.15.m15.1.1.3.2">𝒐</ci><ci id="S2.SS2.p2.15.m15.1.1.3.3.cmml" xref="S2.SS2.p2.15.m15.1.1.3.3">𝑡</ci></apply><apply id="S2.SS2.p2.15.m15.1.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1.1"><times id="S2.SS2.p2.15.m15.1.1.1.2.cmml" xref="S2.SS2.p2.15.m15.1.1.1.2"></times><ci id="S2.SS2.p2.15.m15.1.1.1.3.cmml" xref="S2.SS2.p2.15.m15.1.1.1.3">𝖬𝖺𝗆𝖻𝖺</ci><apply id="S2.SS2.p2.15.m15.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.15.m15.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1.1.1.1">superscript</csymbol><apply id="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p2.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.2">𝒙</ci><cn id="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S2.SS2.p2.15.m15.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.15.m15.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.15.m15.1c">\bm{o}_{t}=\mathsf{Mamba}(\bm{x}_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.15.m15.1d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = sansserif_Mamba ( bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S2.F2"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S2.F2.1" style="width:181.9pt;height:193.6pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-22.7pt,24.2pt) scale(0.8,0.8) ;"><svg class="ltx_picture" height="334.86" id="S2.F2.1.pic1" overflow="visible" version="1.1" width="277.77"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,334.86) matrix(1 0 0 -1 0 0) translate(57.53,0) translate(0,11.53)"><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -5.89 -1.73)"><foreignobject height="8.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="11.78"><math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">x</mi><mn id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑥</ci><cn id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 35.35 -3.46)"><foreignobject height="12.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.38"><math alttext="\dots" class="ltx_Math" display="inline" id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1" mathvariant="normal" xref="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">…</mi><annotation-xml encoding="MathML-Content" id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1.1">…</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1c">\dots</annotation><annotation encoding="application/x-llamapun" id="S2.F2.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.m1.1d">…</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 75.19 -1.79)"><foreignobject height="8.34" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.71"><math alttext="x_{t}" class="ltx_Math" display="inline" id="S2.F2.1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.F2.1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S2.F2.1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S2.F2.1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S2.F2.1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S2.F2.1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">x</mi><mi 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id="S2.F2.1.pic1.34.34.34.34.34.34.34.34.34.34.34.34.1.1.1.1.1.1.1.1.1.m1.1c">\dots</annotation><annotation encoding="application/x-llamapun" id="S2.F2.1.pic1.34.34.34.34.34.34.34.34.34.34.34.34.1.1.1.1.1.1.1.1.1.m1.1d">…</annotation></semantics></math></foreignobject></g></g></svg> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F2.7.3.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S2.F2.5.2" style="font-size:90%;">Mamba-based language model with binary input data: for each <math alttext="t\in[T]" class="ltx_Math" display="inline" id="S2.F2.4.1.m1.1"><semantics id="S2.F2.4.1.m1.1b"><mrow id="S2.F2.4.1.m1.1.2" xref="S2.F2.4.1.m1.1.2.cmml"><mi id="S2.F2.4.1.m1.1.2.2" xref="S2.F2.4.1.m1.1.2.2.cmml">t</mi><mo id="S2.F2.4.1.m1.1.2.1" xref="S2.F2.4.1.m1.1.2.1.cmml">∈</mo><mrow id="S2.F2.4.1.m1.1.2.3.2" xref="S2.F2.4.1.m1.1.2.3.1.cmml"><mo id="S2.F2.4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.F2.4.1.m1.1.2.3.1.1.cmml">[</mo><mi id="S2.F2.4.1.m1.1.1" xref="S2.F2.4.1.m1.1.1.cmml">T</mi><mo id="S2.F2.4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.F2.4.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.4.1.m1.1c"><apply id="S2.F2.4.1.m1.1.2.cmml" xref="S2.F2.4.1.m1.1.2"><in id="S2.F2.4.1.m1.1.2.1.cmml" xref="S2.F2.4.1.m1.1.2.1"></in><ci id="S2.F2.4.1.m1.1.2.2.cmml" xref="S2.F2.4.1.m1.1.2.2">𝑡</ci><apply id="S2.F2.4.1.m1.1.2.3.1.cmml" xref="S2.F2.4.1.m1.1.2.3.2"><csymbol cd="latexml" id="S2.F2.4.1.m1.1.2.3.1.1.cmml" xref="S2.F2.4.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.F2.4.1.m1.1.1.cmml" xref="S2.F2.4.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.4.1.m1.1d">t\in[T]</annotation><annotation encoding="application/x-llamapun" id="S2.F2.4.1.m1.1e">italic_t ∈ [ italic_T ]</annotation></semantics></math>, the next-token prediction probability is <math alttext="f_{\bm{\theta}}(x_{1}^{t})=\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=\cdot\mid x_{% 1}^{t}\right)" class="ltx_math_unparsed" display="inline" id="S2.F2.5.2.m2.1"><semantics id="S2.F2.5.2.m2.1b"><mrow id="S2.F2.5.2.m2.1c"><msub id="S2.F2.5.2.m2.1.1"><mi id="S2.F2.5.2.m2.1.1.2">f</mi><mi id="S2.F2.5.2.m2.1.1.3">𝜽</mi></msub><mrow id="S2.F2.5.2.m2.1.2"><mo id="S2.F2.5.2.m2.1.2.1" stretchy="false">(</mo><msubsup id="S2.F2.5.2.m2.1.2.2"><mi id="S2.F2.5.2.m2.1.2.2.2.2">x</mi><mn id="S2.F2.5.2.m2.1.2.2.2.3">1</mn><mi id="S2.F2.5.2.m2.1.2.2.3">t</mi></msubsup><mo id="S2.F2.5.2.m2.1.2.3" stretchy="false">)</mo></mrow><mo id="S2.F2.5.2.m2.1.3">=</mo><msub id="S2.F2.5.2.m2.1.4"><mi id="S2.F2.5.2.m2.1.4.2">ℙ</mi><mi id="S2.F2.5.2.m2.1.4.3">𝜽</mi></msub><mrow id="S2.F2.5.2.m2.1.5"><mo id="S2.F2.5.2.m2.1.5.1">(</mo><msub id="S2.F2.5.2.m2.1.5.2"><mi id="S2.F2.5.2.m2.1.5.2.2">x</mi><mrow id="S2.F2.5.2.m2.1.5.2.3"><mi id="S2.F2.5.2.m2.1.5.2.3.2">t</mi><mo id="S2.F2.5.2.m2.1.5.2.3.1">+</mo><mn id="S2.F2.5.2.m2.1.5.2.3.3">1</mn></mrow></msub><mo id="S2.F2.5.2.m2.1.5.3" rspace="0em">=</mo><mo id="S2.F2.5.2.m2.1.5.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.F2.5.2.m2.1.5.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S2.F2.5.2.m2.1.5.6"><mi id="S2.F2.5.2.m2.1.5.6.2.2">x</mi><mn id="S2.F2.5.2.m2.1.5.6.2.3">1</mn><mi id="S2.F2.5.2.m2.1.5.6.3">t</mi></msubsup><mo id="S2.F2.5.2.m2.1.5.7">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.F2.5.2.m2.1d">f_{\bm{\theta}}(x_{1}^{t})=\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=\cdot\mid x_{% 1}^{t}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.F2.5.2.m2.1e">italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>.</span></figcaption> </figure> <div class="ltx_para ltx_noindent" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.5"><span class="ltx_text ltx_font_bold" id="S2.SS2.p3.5.1">Mamba-based language model.</span> <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex1" title="Equation Mamba ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Mamba</span></a> block is then incorporated into a full-fledged language model as follows: let <math alttext="x=(x_{1},x_{2},\cdots,x_{T})\in\mathcal{X}^{T}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.4"><semantics id="S2.SS2.p3.1.m1.4a"><mrow id="S2.SS2.p3.1.m1.4.4" xref="S2.SS2.p3.1.m1.4.4.cmml"><mi id="S2.SS2.p3.1.m1.4.4.5" xref="S2.SS2.p3.1.m1.4.4.5.cmml">x</mi><mo id="S2.SS2.p3.1.m1.4.4.6" xref="S2.SS2.p3.1.m1.4.4.6.cmml">=</mo><mrow id="S2.SS2.p3.1.m1.4.4.3.3" xref="S2.SS2.p3.1.m1.4.4.3.4.cmml"><mo id="S2.SS2.p3.1.m1.4.4.3.3.4" stretchy="false" xref="S2.SS2.p3.1.m1.4.4.3.4.cmml">(</mo><msub id="S2.SS2.p3.1.m1.2.2.1.1.1" xref="S2.SS2.p3.1.m1.2.2.1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.2.2.1.1.1.2" xref="S2.SS2.p3.1.m1.2.2.1.1.1.2.cmml">x</mi><mn id="S2.SS2.p3.1.m1.2.2.1.1.1.3" xref="S2.SS2.p3.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p3.1.m1.4.4.3.3.5" xref="S2.SS2.p3.1.m1.4.4.3.4.cmml">,</mo><msub id="S2.SS2.p3.1.m1.3.3.2.2.2" xref="S2.SS2.p3.1.m1.3.3.2.2.2.cmml"><mi id="S2.SS2.p3.1.m1.3.3.2.2.2.2" xref="S2.SS2.p3.1.m1.3.3.2.2.2.2.cmml">x</mi><mn id="S2.SS2.p3.1.m1.3.3.2.2.2.3" xref="S2.SS2.p3.1.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS2.p3.1.m1.4.4.3.3.6" xref="S2.SS2.p3.1.m1.4.4.3.4.cmml">,</mo><mi id="S2.SS2.p3.1.m1.1.1" mathvariant="normal" xref="S2.SS2.p3.1.m1.1.1.cmml">⋯</mi><mo id="S2.SS2.p3.1.m1.4.4.3.3.7" xref="S2.SS2.p3.1.m1.4.4.3.4.cmml">,</mo><msub id="S2.SS2.p3.1.m1.4.4.3.3.3" xref="S2.SS2.p3.1.m1.4.4.3.3.3.cmml"><mi id="S2.SS2.p3.1.m1.4.4.3.3.3.2" xref="S2.SS2.p3.1.m1.4.4.3.3.3.2.cmml">x</mi><mi id="S2.SS2.p3.1.m1.4.4.3.3.3.3" xref="S2.SS2.p3.1.m1.4.4.3.3.3.3.cmml">T</mi></msub><mo id="S2.SS2.p3.1.m1.4.4.3.3.8" stretchy="false" xref="S2.SS2.p3.1.m1.4.4.3.4.cmml">)</mo></mrow><mo id="S2.SS2.p3.1.m1.4.4.7" xref="S2.SS2.p3.1.m1.4.4.7.cmml">∈</mo><msup id="S2.SS2.p3.1.m1.4.4.8" xref="S2.SS2.p3.1.m1.4.4.8.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.4.4.8.2" xref="S2.SS2.p3.1.m1.4.4.8.2.cmml">𝒳</mi><mi id="S2.SS2.p3.1.m1.4.4.8.3" xref="S2.SS2.p3.1.m1.4.4.8.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.4b"><apply id="S2.SS2.p3.1.m1.4.4.cmml" xref="S2.SS2.p3.1.m1.4.4"><and id="S2.SS2.p3.1.m1.4.4a.cmml" xref="S2.SS2.p3.1.m1.4.4"></and><apply id="S2.SS2.p3.1.m1.4.4b.cmml" xref="S2.SS2.p3.1.m1.4.4"><eq id="S2.SS2.p3.1.m1.4.4.6.cmml" xref="S2.SS2.p3.1.m1.4.4.6"></eq><ci id="S2.SS2.p3.1.m1.4.4.5.cmml" xref="S2.SS2.p3.1.m1.4.4.5">𝑥</ci><vector id="S2.SS2.p3.1.m1.4.4.3.4.cmml" xref="S2.SS2.p3.1.m1.4.4.3.3"><apply id="S2.SS2.p3.1.m1.2.2.1.1.1.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2.p3.1.m1.2.2.1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.2">𝑥</ci><cn id="S2.SS2.p3.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS2.p3.1.m1.2.2.1.1.1.3">1</cn></apply><apply id="S2.SS2.p3.1.m1.3.3.2.2.2.cmml" xref="S2.SS2.p3.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.3.3.2.2.2.1.cmml" xref="S2.SS2.p3.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S2.SS2.p3.1.m1.3.3.2.2.2.2.cmml" xref="S2.SS2.p3.1.m1.3.3.2.2.2.2">𝑥</ci><cn id="S2.SS2.p3.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S2.SS2.p3.1.m1.3.3.2.2.2.3">2</cn></apply><ci id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">⋯</ci><apply id="S2.SS2.p3.1.m1.4.4.3.3.3.cmml" xref="S2.SS2.p3.1.m1.4.4.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.4.4.3.3.3.1.cmml" xref="S2.SS2.p3.1.m1.4.4.3.3.3">subscript</csymbol><ci id="S2.SS2.p3.1.m1.4.4.3.3.3.2.cmml" xref="S2.SS2.p3.1.m1.4.4.3.3.3.2">𝑥</ci><ci id="S2.SS2.p3.1.m1.4.4.3.3.3.3.cmml" xref="S2.SS2.p3.1.m1.4.4.3.3.3.3">𝑇</ci></apply></vector></apply><apply id="S2.SS2.p3.1.m1.4.4c.cmml" xref="S2.SS2.p3.1.m1.4.4"><in id="S2.SS2.p3.1.m1.4.4.7.cmml" xref="S2.SS2.p3.1.m1.4.4.7"></in><share href="https://arxiv.org/html/2502.10178v1#S2.SS2.p3.1.m1.4.4.3.cmml" id="S2.SS2.p3.1.m1.4.4d.cmml" xref="S2.SS2.p3.1.m1.4.4"></share><apply id="S2.SS2.p3.1.m1.4.4.8.cmml" xref="S2.SS2.p3.1.m1.4.4.8"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.4.4.8.1.cmml" xref="S2.SS2.p3.1.m1.4.4.8">superscript</csymbol><ci id="S2.SS2.p3.1.m1.4.4.8.2.cmml" xref="S2.SS2.p3.1.m1.4.4.8.2">𝒳</ci><ci id="S2.SS2.p3.1.m1.4.4.8.3.cmml" xref="S2.SS2.p3.1.m1.4.4.8.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.4c">x=(x_{1},x_{2},\cdots,x_{T})\in\mathcal{X}^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.4d">italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) ∈ caligraphic_X start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> be an input token-sequence over the alphabet <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><ci id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">caligraphic_X</annotation></semantics></math>; here <math alttext="\mathcal{X}=\{0,1\}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.2"><semantics id="S2.SS2.p3.3.m3.2a"><mrow id="S2.SS2.p3.3.m3.2.3" xref="S2.SS2.p3.3.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.3.m3.2.3.2" xref="S2.SS2.p3.3.m3.2.3.2.cmml">𝒳</mi><mo id="S2.SS2.p3.3.m3.2.3.1" xref="S2.SS2.p3.3.m3.2.3.1.cmml">=</mo><mrow id="S2.SS2.p3.3.m3.2.3.3.2" xref="S2.SS2.p3.3.m3.2.3.3.1.cmml"><mo id="S2.SS2.p3.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS2.p3.3.m3.2.3.3.1.cmml">{</mo><mn id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml">0</mn><mo id="S2.SS2.p3.3.m3.2.3.3.2.2" xref="S2.SS2.p3.3.m3.2.3.3.1.cmml">,</mo><mn id="S2.SS2.p3.3.m3.2.2" xref="S2.SS2.p3.3.m3.2.2.cmml">1</mn><mo id="S2.SS2.p3.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS2.p3.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m3.2b"><apply id="S2.SS2.p3.3.m3.2.3.cmml" xref="S2.SS2.p3.3.m3.2.3"><eq id="S2.SS2.p3.3.m3.2.3.1.cmml" xref="S2.SS2.p3.3.m3.2.3.1"></eq><ci id="S2.SS2.p3.3.m3.2.3.2.cmml" xref="S2.SS2.p3.3.m3.2.3.2">𝒳</ci><set id="S2.SS2.p3.3.m3.2.3.3.1.cmml" xref="S2.SS2.p3.3.m3.2.3.3.2"><cn id="S2.SS2.p3.3.m3.1.1.cmml" type="integer" xref="S2.SS2.p3.3.m3.1.1">0</cn><cn id="S2.SS2.p3.3.m3.2.2.cmml" type="integer" xref="S2.SS2.p3.3.m3.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m3.2c">\mathcal{X}=\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m3.2d">caligraphic_X = { 0 , 1 }</annotation></semantics></math> as explained in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="2.1 Input data: Random Markov chains ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.1</span></a>. Then, at every <math alttext="t\in[T]" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m4.1"><semantics id="S2.SS2.p3.4.m4.1a"><mrow id="S2.SS2.p3.4.m4.1.2" xref="S2.SS2.p3.4.m4.1.2.cmml"><mi id="S2.SS2.p3.4.m4.1.2.2" xref="S2.SS2.p3.4.m4.1.2.2.cmml">t</mi><mo id="S2.SS2.p3.4.m4.1.2.1" xref="S2.SS2.p3.4.m4.1.2.1.cmml">∈</mo><mrow id="S2.SS2.p3.4.m4.1.2.3.2" xref="S2.SS2.p3.4.m4.1.2.3.1.cmml"><mo id="S2.SS2.p3.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS2.p3.4.m4.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.p3.4.m4.1.1" xref="S2.SS2.p3.4.m4.1.1.cmml">T</mi><mo id="S2.SS2.p3.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS2.p3.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m4.1b"><apply id="S2.SS2.p3.4.m4.1.2.cmml" xref="S2.SS2.p3.4.m4.1.2"><in id="S2.SS2.p3.4.m4.1.2.1.cmml" xref="S2.SS2.p3.4.m4.1.2.1"></in><ci id="S2.SS2.p3.4.m4.1.2.2.cmml" xref="S2.SS2.p3.4.m4.1.2.2">𝑡</ci><apply id="S2.SS2.p3.4.m4.1.2.3.1.cmml" 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id="S2.SS2.p3.5.m5.1d">bold_italic_θ</annotation></semantics></math> is given by the following sequence of equations <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">dao2024transformers</span>]</cite>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx2"> <tbody id="S2.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{x}_{t}" class="ltx_Math" display="inline" id="S2.Ex3.m1.1"><semantics id="S2.Ex3.m1.1a"><msub id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml"><mi id="S2.Ex3.m1.1.1.2" xref="S2.Ex3.m1.1.1.2.cmml">𝒙</mi><mi id="S2.Ex3.m1.1.1.3" xref="S2.Ex3.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.1b"><apply id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex3.m1.1.1.1.cmml" xref="S2.Ex3.m1.1.1">subscript</csymbol><ci id="S2.Ex3.m1.1.1.2.cmml" xref="S2.Ex3.m1.1.1.2">𝒙</ci><ci id="S2.Ex3.m1.1.1.3.cmml" xref="S2.Ex3.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.1c">\displaystyle\bm{x}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.1d">bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\bm{e}_{x_{t}}=x_{t}\,\bm{e}_{1}+(1-x_{t})\,\bm{e}_{0}\in\mathbb% {R}^{d}," class="ltx_Math" display="inline" id="S2.Ex3.m2.1"><semantics id="S2.Ex3.m2.1a"><mrow id="S2.Ex3.m2.1.1.1" xref="S2.Ex3.m2.1.1.1.1.cmml"><mrow id="S2.Ex3.m2.1.1.1.1" xref="S2.Ex3.m2.1.1.1.1.cmml"><mi id="S2.Ex3.m2.1.1.1.1.3" xref="S2.Ex3.m2.1.1.1.1.3.cmml"></mi><mo id="S2.Ex3.m2.1.1.1.1.4" xref="S2.Ex3.m2.1.1.1.1.4.cmml">=</mo><msub id="S2.Ex3.m2.1.1.1.1.5" xref="S2.Ex3.m2.1.1.1.1.5.cmml"><mi 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id="S2.Ex3.m2.1c">\displaystyle=\bm{e}_{x_{t}}=x_{t}\,\bm{e}_{1}+(1-x_{t})\,\bm{e}_{0}\in\mathbb% {R}^{d},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m2.1d">= bold_italic_e start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ( 1 - italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) bold_italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Embedding)</span></td> </tr></tbody> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{u}_{t}" class="ltx_Math" display="inline" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><msub id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.2" xref="S2.Ex4.m1.1.1.2.cmml">𝒖</mi><mi id="S2.Ex4.m1.1.1.3" xref="S2.Ex4.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><apply id="S2.Ex4.m1.1.1.cmml" xref="S2.Ex4.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.1.cmml" xref="S2.Ex4.m1.1.1">subscript</csymbol><ci id="S2.Ex4.m1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.2">𝒖</ci><ci id="S2.Ex4.m1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\displaystyle\bm{u}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">bold_italic_u start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\bm{x}_{t}+\mathsf{Mamba}(\bm{x}_{1}^{t})\in\mathbb{R}^{d}," class="ltx_Math" display="inline" id="S2.Ex4.m2.1"><semantics id="S2.Ex4.m2.1a"><mrow id="S2.Ex4.m2.1.1.1" xref="S2.Ex4.m2.1.1.1.1.cmml"><mrow id="S2.Ex4.m2.1.1.1.1" xref="S2.Ex4.m2.1.1.1.1.cmml"><mi id="S2.Ex4.m2.1.1.1.1.3" xref="S2.Ex4.m2.1.1.1.1.3.cmml"></mi><mo id="S2.Ex4.m2.1.1.1.1.4" xref="S2.Ex4.m2.1.1.1.1.4.cmml">=</mo><mrow id="S2.Ex4.m2.1.1.1.1.1" xref="S2.Ex4.m2.1.1.1.1.1.cmml"><msub id="S2.Ex4.m2.1.1.1.1.1.3" xref="S2.Ex4.m2.1.1.1.1.1.3.cmml"><mi id="S2.Ex4.m2.1.1.1.1.1.3.2" xref="S2.Ex4.m2.1.1.1.1.1.3.2.cmml">𝒙</mi><mi id="S2.Ex4.m2.1.1.1.1.1.3.3" xref="S2.Ex4.m2.1.1.1.1.1.3.3.cmml">t</mi></msub><mo id="S2.Ex4.m2.1.1.1.1.1.2" xref="S2.Ex4.m2.1.1.1.1.1.2.cmml">+</mo><mrow id="S2.Ex4.m2.1.1.1.1.1.1" xref="S2.Ex4.m2.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m2.1.1.1.1.1.1.3" xref="S2.Ex4.m2.1.1.1.1.1.1.3.cmml">𝖬𝖺𝗆𝖻𝖺</mi><mo id="S2.Ex4.m2.1.1.1.1.1.1.2" xref="S2.Ex4.m2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m2.1.1.1.1.1.1.1.1" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S2.Ex4.m2.1.1.1.1.1.1.1.1.1" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.2.2" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mn id="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.2.3" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S2.Ex4.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex4.m2.1.1.1.1.5" xref="S2.Ex4.m2.1.1.1.1.5.cmml">∈</mo><msup id="S2.Ex4.m2.1.1.1.1.6" xref="S2.Ex4.m2.1.1.1.1.6.cmml"><mi id="S2.Ex4.m2.1.1.1.1.6.2" xref="S2.Ex4.m2.1.1.1.1.6.2.cmml">ℝ</mi><mi id="S2.Ex4.m2.1.1.1.1.6.3" xref="S2.Ex4.m2.1.1.1.1.6.3.cmml">d</mi></msup></mrow><mo id="S2.Ex4.m2.1.1.1.2" 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xref="S2.Ex4.m2.1.1.1.1.5"></in><share href="https://arxiv.org/html/2502.10178v1#S2.Ex4.m2.1.1.1.1.1.cmml" id="S2.Ex4.m2.1.1.1.1d.cmml" xref="S2.Ex4.m2.1.1.1"></share><apply id="S2.Ex4.m2.1.1.1.1.6.cmml" xref="S2.Ex4.m2.1.1.1.1.6"><csymbol cd="ambiguous" id="S2.Ex4.m2.1.1.1.1.6.1.cmml" xref="S2.Ex4.m2.1.1.1.1.6">superscript</csymbol><ci id="S2.Ex4.m2.1.1.1.1.6.2.cmml" xref="S2.Ex4.m2.1.1.1.1.6.2">ℝ</ci><ci id="S2.Ex4.m2.1.1.1.1.6.3.cmml" xref="S2.Ex4.m2.1.1.1.1.6.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.1c">\displaystyle=\bm{x}_{t}+\mathsf{Mamba}(\bm{x}_{1}^{t})\in\mathbb{R}^{d},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.1d">= bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + sansserif_Mamba ( bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Mamba)</span></td> </tr></tbody> <tbody id="S2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{v}_{t}" class="ltx_Math" display="inline" id="S2.Ex5.m1.1"><semantics id="S2.Ex5.m1.1a"><msub id="S2.Ex5.m1.1.1" xref="S2.Ex5.m1.1.1.cmml"><mi id="S2.Ex5.m1.1.1.2" xref="S2.Ex5.m1.1.1.2.cmml">𝒗</mi><mi id="S2.Ex5.m1.1.1.3" xref="S2.Ex5.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.1b"><apply id="S2.Ex5.m1.1.1.cmml" xref="S2.Ex5.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.1.1.1.cmml" xref="S2.Ex5.m1.1.1">subscript</csymbol><ci id="S2.Ex5.m1.1.1.2.cmml" xref="S2.Ex5.m1.1.1.2">𝒗</ci><ci id="S2.Ex5.m1.1.1.3.cmml" xref="S2.Ex5.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.1c">\displaystyle\bm{v}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.1d">bold_italic_v start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\bm{u}_{t}+W_{2}[\mathrm{ReLU}(W_{1}\bm{u}_{t})+\odot W_{3}\bm{u% }_{t}]\in\mathbb{R}^{d}," class="ltx_math_unparsed" display="inline" id="S2.Ex5.m2.1"><semantics id="S2.Ex5.m2.1a"><mrow id="S2.Ex5.m2.1b"><mo id="S2.Ex5.m2.1.1">=</mo><msub id="S2.Ex5.m2.1.2"><mi id="S2.Ex5.m2.1.2.2">𝒖</mi><mi id="S2.Ex5.m2.1.2.3">t</mi></msub><mo id="S2.Ex5.m2.1.3">+</mo><msub id="S2.Ex5.m2.1.4"><mi id="S2.Ex5.m2.1.4.2">W</mi><mn id="S2.Ex5.m2.1.4.3">2</mn></msub><mrow id="S2.Ex5.m2.1.5"><mo id="S2.Ex5.m2.1.5.1" stretchy="false">[</mo><mi id="S2.Ex5.m2.1.5.2">ReLU</mi><mrow id="S2.Ex5.m2.1.5.3"><mo id="S2.Ex5.m2.1.5.3.1" stretchy="false">(</mo><msub id="S2.Ex5.m2.1.5.3.2"><mi id="S2.Ex5.m2.1.5.3.2.2">W</mi><mn id="S2.Ex5.m2.1.5.3.2.3">1</mn></msub><msub id="S2.Ex5.m2.1.5.3.3"><mi id="S2.Ex5.m2.1.5.3.3.2">𝒖</mi><mi id="S2.Ex5.m2.1.5.3.3.3">t</mi></msub><mo id="S2.Ex5.m2.1.5.3.4" stretchy="false">)</mo></mrow><mo id="S2.Ex5.m2.1.5.4" rspace="0em">+</mo><mo id="S2.Ex5.m2.1.5.5" lspace="0em" rspace="0.222em">⊙</mo><msub id="S2.Ex5.m2.1.5.6"><mi id="S2.Ex5.m2.1.5.6.2">W</mi><mn id="S2.Ex5.m2.1.5.6.3">3</mn></msub><msub id="S2.Ex5.m2.1.5.7"><mi id="S2.Ex5.m2.1.5.7.2">𝒖</mi><mi id="S2.Ex5.m2.1.5.7.3">t</mi></msub><mo id="S2.Ex5.m2.1.5.8" stretchy="false">]</mo></mrow><mo id="S2.Ex5.m2.1.6">∈</mo><msup id="S2.Ex5.m2.1.7"><mi id="S2.Ex5.m2.1.7.2">ℝ</mi><mi id="S2.Ex5.m2.1.7.3">d</mi></msup><mo id="S2.Ex5.m2.1.8">,</mo></mrow><annotation encoding="application/x-tex" id="S2.Ex5.m2.1c">\displaystyle=\bm{u}_{t}+W_{2}[\mathrm{ReLU}(W_{1}\bm{u}_{t})+\odot W_{3}\bm{u% }_{t}]\in\mathbb{R}^{d},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m2.1d">= bold_italic_u start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ roman_ReLU ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_italic_u start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) + ⊙ italic_W start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT bold_italic_u start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ] ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(<span class="ltx_text" id="S2.Ex5.2.1.1" style="font-size:90%;">MLP</span>)</span></td> </tr></tbody> <tbody id="S2.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathrm{logit}_{t}" class="ltx_Math" display="inline" id="S2.Ex6.m1.1"><semantics id="S2.Ex6.m1.1a"><msub id="S2.Ex6.m1.1.1" xref="S2.Ex6.m1.1.1.cmml"><mi id="S2.Ex6.m1.1.1.2" xref="S2.Ex6.m1.1.1.2.cmml">logit</mi><mi id="S2.Ex6.m1.1.1.3" xref="S2.Ex6.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Ex6.m1.1b"><apply id="S2.Ex6.m1.1.1.cmml" xref="S2.Ex6.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex6.m1.1.1.1.cmml" xref="S2.Ex6.m1.1.1">subscript</csymbol><ci id="S2.Ex6.m1.1.1.2.cmml" xref="S2.Ex6.m1.1.1.2">logit</ci><ci id="S2.Ex6.m1.1.1.3.cmml" xref="S2.Ex6.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m1.1c">\displaystyle\mathrm{logit}_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.1d">roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=W_{\ell}\,\bm{v}_{t}\in\mathbb{R}^{2}," class="ltx_Math" display="inline" id="S2.Ex6.m2.1"><semantics id="S2.Ex6.m2.1a"><mrow id="S2.Ex6.m2.1.1.1" xref="S2.Ex6.m2.1.1.1.1.cmml"><mrow id="S2.Ex6.m2.1.1.1.1" xref="S2.Ex6.m2.1.1.1.1.cmml"><mi id="S2.Ex6.m2.1.1.1.1.2" xref="S2.Ex6.m2.1.1.1.1.2.cmml"></mi><mo id="S2.Ex6.m2.1.1.1.1.3" xref="S2.Ex6.m2.1.1.1.1.3.cmml">=</mo><mrow id="S2.Ex6.m2.1.1.1.1.4" xref="S2.Ex6.m2.1.1.1.1.4.cmml"><msub id="S2.Ex6.m2.1.1.1.1.4.2" xref="S2.Ex6.m2.1.1.1.1.4.2.cmml"><mi id="S2.Ex6.m2.1.1.1.1.4.2.2" xref="S2.Ex6.m2.1.1.1.1.4.2.2.cmml">W</mi><mi id="S2.Ex6.m2.1.1.1.1.4.2.3" mathvariant="normal" xref="S2.Ex6.m2.1.1.1.1.4.2.3.cmml">ℓ</mi></msub><mo id="S2.Ex6.m2.1.1.1.1.4.1" xref="S2.Ex6.m2.1.1.1.1.4.1.cmml">⁢</mo><msub id="S2.Ex6.m2.1.1.1.1.4.3" xref="S2.Ex6.m2.1.1.1.1.4.3.cmml"><mi id="S2.Ex6.m2.1.1.1.1.4.3.2" xref="S2.Ex6.m2.1.1.1.1.4.3.2.cmml">𝒗</mi><mi id="S2.Ex6.m2.1.1.1.1.4.3.3" xref="S2.Ex6.m2.1.1.1.1.4.3.3.cmml">t</mi></msub></mrow><mo id="S2.Ex6.m2.1.1.1.1.5" xref="S2.Ex6.m2.1.1.1.1.5.cmml">∈</mo><msup id="S2.Ex6.m2.1.1.1.1.6" xref="S2.Ex6.m2.1.1.1.1.6.cmml"><mi id="S2.Ex6.m2.1.1.1.1.6.2" xref="S2.Ex6.m2.1.1.1.1.6.2.cmml">ℝ</mi><mn id="S2.Ex6.m2.1.1.1.1.6.3" xref="S2.Ex6.m2.1.1.1.1.6.3.cmml">2</mn></msup></mrow><mo id="S2.Ex6.m2.1.1.1.2" xref="S2.Ex6.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex6.m2.1b"><apply id="S2.Ex6.m2.1.1.1.1.cmml" xref="S2.Ex6.m2.1.1.1"><and id="S2.Ex6.m2.1.1.1.1a.cmml" xref="S2.Ex6.m2.1.1.1"></and><apply id="S2.Ex6.m2.1.1.1.1b.cmml" xref="S2.Ex6.m2.1.1.1"><eq id="S2.Ex6.m2.1.1.1.1.3.cmml" xref="S2.Ex6.m2.1.1.1.1.3"></eq><csymbol cd="latexml" id="S2.Ex6.m2.1.1.1.1.2.cmml" xref="S2.Ex6.m2.1.1.1.1.2">absent</csymbol><apply id="S2.Ex6.m2.1.1.1.1.4.cmml" xref="S2.Ex6.m2.1.1.1.1.4"><times id="S2.Ex6.m2.1.1.1.1.4.1.cmml" xref="S2.Ex6.m2.1.1.1.1.4.1"></times><apply id="S2.Ex6.m2.1.1.1.1.4.2.cmml" xref="S2.Ex6.m2.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S2.Ex6.m2.1.1.1.1.4.2.1.cmml" xref="S2.Ex6.m2.1.1.1.1.4.2">subscript</csymbol><ci id="S2.Ex6.m2.1.1.1.1.4.2.2.cmml" xref="S2.Ex6.m2.1.1.1.1.4.2.2">𝑊</ci><ci id="S2.Ex6.m2.1.1.1.1.4.2.3.cmml" xref="S2.Ex6.m2.1.1.1.1.4.2.3">ℓ</ci></apply><apply id="S2.Ex6.m2.1.1.1.1.4.3.cmml" xref="S2.Ex6.m2.1.1.1.1.4.3"><csymbol cd="ambiguous" id="S2.Ex6.m2.1.1.1.1.4.3.1.cmml" xref="S2.Ex6.m2.1.1.1.1.4.3">subscript</csymbol><ci id="S2.Ex6.m2.1.1.1.1.4.3.2.cmml" xref="S2.Ex6.m2.1.1.1.1.4.3.2">𝒗</ci><ci id="S2.Ex6.m2.1.1.1.1.4.3.3.cmml" xref="S2.Ex6.m2.1.1.1.1.4.3.3">𝑡</ci></apply></apply></apply><apply id="S2.Ex6.m2.1.1.1.1c.cmml" xref="S2.Ex6.m2.1.1.1"><in id="S2.Ex6.m2.1.1.1.1.5.cmml" xref="S2.Ex6.m2.1.1.1.1.5"></in><share href="https://arxiv.org/html/2502.10178v1#S2.Ex6.m2.1.1.1.1.4.cmml" id="S2.Ex6.m2.1.1.1.1d.cmml" xref="S2.Ex6.m2.1.1.1"></share><apply id="S2.Ex6.m2.1.1.1.1.6.cmml" xref="S2.Ex6.m2.1.1.1.1.6"><csymbol cd="ambiguous" id="S2.Ex6.m2.1.1.1.1.6.1.cmml" xref="S2.Ex6.m2.1.1.1.1.6">superscript</csymbol><ci id="S2.Ex6.m2.1.1.1.1.6.2.cmml" xref="S2.Ex6.m2.1.1.1.1.6.2">ℝ</ci><cn id="S2.Ex6.m2.1.1.1.1.6.3.cmml" type="integer" xref="S2.Ex6.m2.1.1.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m2.1c">\displaystyle=W_{\ell}\,\bm{v}_{t}\in\mathbb{R}^{2},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m2.1d">= italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT bold_italic_v start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Linear)</span></td> </tr></tbody> <tbody id="S2.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{\bm{\theta}}(x_{1}^{t})" class="ltx_Math" display="inline" id="S2.Ex7.m1.1"><semantics id="S2.Ex7.m1.1a"><mrow id="S2.Ex7.m1.1.1" xref="S2.Ex7.m1.1.1.cmml"><msub id="S2.Ex7.m1.1.1.3" xref="S2.Ex7.m1.1.1.3.cmml"><mi id="S2.Ex7.m1.1.1.3.2" xref="S2.Ex7.m1.1.1.3.2.cmml">f</mi><mi id="S2.Ex7.m1.1.1.3.3" xref="S2.Ex7.m1.1.1.3.3.cmml">𝜽</mi></msub><mo id="S2.Ex7.m1.1.1.2" xref="S2.Ex7.m1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex7.m1.1.1.1.1" xref="S2.Ex7.m1.1.1.1.1.1.cmml"><mo id="S2.Ex7.m1.1.1.1.1.2" stretchy="false" xref="S2.Ex7.m1.1.1.1.1.1.cmml">(</mo><msubsup id="S2.Ex7.m1.1.1.1.1.1" xref="S2.Ex7.m1.1.1.1.1.1.cmml"><mi id="S2.Ex7.m1.1.1.1.1.1.2.2" xref="S2.Ex7.m1.1.1.1.1.1.2.2.cmml">x</mi><mn id="S2.Ex7.m1.1.1.1.1.1.2.3" xref="S2.Ex7.m1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S2.Ex7.m1.1.1.1.1.1.3" xref="S2.Ex7.m1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S2.Ex7.m1.1.1.1.1.3" stretchy="false" xref="S2.Ex7.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex7.m1.1b"><apply id="S2.Ex7.m1.1.1.cmml" xref="S2.Ex7.m1.1.1"><times id="S2.Ex7.m1.1.1.2.cmml" xref="S2.Ex7.m1.1.1.2"></times><apply id="S2.Ex7.m1.1.1.3.cmml" xref="S2.Ex7.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex7.m1.1.1.3.1.cmml" xref="S2.Ex7.m1.1.1.3">subscript</csymbol><ci id="S2.Ex7.m1.1.1.3.2.cmml" xref="S2.Ex7.m1.1.1.3.2">𝑓</ci><ci id="S2.Ex7.m1.1.1.3.3.cmml" xref="S2.Ex7.m1.1.1.3.3">𝜽</ci></apply><apply id="S2.Ex7.m1.1.1.1.1.1.cmml" xref="S2.Ex7.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex7.m1.1.1.1.1.1.1.cmml" xref="S2.Ex7.m1.1.1.1.1">superscript</csymbol><apply id="S2.Ex7.m1.1.1.1.1.1.2.cmml" xref="S2.Ex7.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex7.m1.1.1.1.1.1.2.1.cmml" xref="S2.Ex7.m1.1.1.1.1">subscript</csymbol><ci id="S2.Ex7.m1.1.1.1.1.1.2.2.cmml" xref="S2.Ex7.m1.1.1.1.1.1.2.2">𝑥</ci><cn id="S2.Ex7.m1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.Ex7.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="S2.Ex7.m1.1.1.1.1.1.3.cmml" xref="S2.Ex7.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex7.m1.1c">\displaystyle f_{\bm{\theta}}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.1d">italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\triangleq\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=\cdot\mid x_{1}^{% t}\right)=\mathrm{softmax}(\mathrm{logit}_{t})\in[0,1]^{2}," class="ltx_math_unparsed" display="inline" id="S2.Ex7.m2.1"><semantics id="S2.Ex7.m2.1a"><mrow id="S2.Ex7.m2.1b"><mo id="S2.Ex7.m2.1.1">≜</mo><msub id="S2.Ex7.m2.1.2"><mi id="S2.Ex7.m2.1.2.2">ℙ</mi><mi id="S2.Ex7.m2.1.2.3">𝜽</mi></msub><mrow id="S2.Ex7.m2.1.3"><mo id="S2.Ex7.m2.1.3.1">(</mo><msub id="S2.Ex7.m2.1.3.2"><mi id="S2.Ex7.m2.1.3.2.2">x</mi><mrow id="S2.Ex7.m2.1.3.2.3"><mi id="S2.Ex7.m2.1.3.2.3.2">t</mi><mo id="S2.Ex7.m2.1.3.2.3.1">+</mo><mn id="S2.Ex7.m2.1.3.2.3.3">1</mn></mrow></msub><mo id="S2.Ex7.m2.1.3.3" rspace="0em">=</mo><mo id="S2.Ex7.m2.1.3.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.Ex7.m2.1.3.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S2.Ex7.m2.1.3.6"><mi id="S2.Ex7.m2.1.3.6.2.2">x</mi><mn id="S2.Ex7.m2.1.3.6.2.3">1</mn><mi id="S2.Ex7.m2.1.3.6.3">t</mi></msubsup><mo id="S2.Ex7.m2.1.3.7">)</mo></mrow><mo id="S2.Ex7.m2.1.4">=</mo><mi id="S2.Ex7.m2.1.5">softmax</mi><mrow id="S2.Ex7.m2.1.6"><mo id="S2.Ex7.m2.1.6.1" stretchy="false">(</mo><msub id="S2.Ex7.m2.1.6.2"><mi id="S2.Ex7.m2.1.6.2.2">logit</mi><mi id="S2.Ex7.m2.1.6.2.3">t</mi></msub><mo id="S2.Ex7.m2.1.6.3" stretchy="false">)</mo></mrow><mo id="S2.Ex7.m2.1.7">∈</mo><msup id="S2.Ex7.m2.1.8"><mrow id="S2.Ex7.m2.1.8.2"><mo id="S2.Ex7.m2.1.8.2.1" stretchy="false">[</mo><mn id="S2.Ex7.m2.1.8.2.2">0</mn><mo id="S2.Ex7.m2.1.8.2.3">,</mo><mn id="S2.Ex7.m2.1.8.2.4">1</mn><mo id="S2.Ex7.m2.1.8.2.5" stretchy="false">]</mo></mrow><mn id="S2.Ex7.m2.1.8.3">2</mn></msup><mo id="S2.Ex7.m2.1.9">,</mo></mrow><annotation encoding="application/x-tex" id="S2.Ex7.m2.1c">\displaystyle\triangleq\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=\cdot\mid x_{1}^{% t}\right)=\mathrm{softmax}(\mathrm{logit}_{t})\in[0,1]^{2},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m2.1d">≜ blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = roman_softmax ( roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Prediction)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.12">where the parameters <math alttext="\bm{e}_{0},\bm{e}_{1}\in\mathbb{R}^{d},W_{1}\in\mathbb{R}^{4d\times d},W_{2}% \in\mathbb{R}^{d\times 4d}" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m1.2"><semantics id="S2.SS2.p3.6.m1.2a"><mrow id="S2.SS2.p3.6.m1.2.2.2" xref="S2.SS2.p3.6.m1.2.2.3.cmml"><mrow id="S2.SS2.p3.6.m1.1.1.1.1" xref="S2.SS2.p3.6.m1.1.1.1.1.cmml"><mrow id="S2.SS2.p3.6.m1.1.1.1.1.2.2" xref="S2.SS2.p3.6.m1.1.1.1.1.2.3.cmml"><msub id="S2.SS2.p3.6.m1.1.1.1.1.1.1.1" xref="S2.SS2.p3.6.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p3.6.m1.1.1.1.1.1.1.1.2" xref="S2.SS2.p3.6.m1.1.1.1.1.1.1.1.2.cmml">𝒆</mi><mn id="S2.SS2.p3.6.m1.1.1.1.1.1.1.1.3" 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start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 4 italic_d × italic_d end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d × 4 italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="W_{\ell}\in\mathbb{R}^{2\times d}" class="ltx_Math" display="inline" id="S2.SS2.p3.7.m2.1"><semantics id="S2.SS2.p3.7.m2.1a"><mrow id="S2.SS2.p3.7.m2.1.1" xref="S2.SS2.p3.7.m2.1.1.cmml"><msub id="S2.SS2.p3.7.m2.1.1.2" xref="S2.SS2.p3.7.m2.1.1.2.cmml"><mi id="S2.SS2.p3.7.m2.1.1.2.2" xref="S2.SS2.p3.7.m2.1.1.2.2.cmml">W</mi><mi id="S2.SS2.p3.7.m2.1.1.2.3" mathvariant="normal" xref="S2.SS2.p3.7.m2.1.1.2.3.cmml">ℓ</mi></msub><mo id="S2.SS2.p3.7.m2.1.1.1" xref="S2.SS2.p3.7.m2.1.1.1.cmml">∈</mo><msup id="S2.SS2.p3.7.m2.1.1.3" xref="S2.SS2.p3.7.m2.1.1.3.cmml"><mi id="S2.SS2.p3.7.m2.1.1.3.2" xref="S2.SS2.p3.7.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS2.p3.7.m2.1.1.3.3" xref="S2.SS2.p3.7.m2.1.1.3.3.cmml"><mn id="S2.SS2.p3.7.m2.1.1.3.3.2" xref="S2.SS2.p3.7.m2.1.1.3.3.2.cmml">2</mn><mo id="S2.SS2.p3.7.m2.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p3.7.m2.1.1.3.3.1.cmml">×</mo><mi id="S2.SS2.p3.7.m2.1.1.3.3.3" xref="S2.SS2.p3.7.m2.1.1.3.3.3.cmml">d</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.7.m2.1b"><apply id="S2.SS2.p3.7.m2.1.1.cmml" xref="S2.SS2.p3.7.m2.1.1"><in id="S2.SS2.p3.7.m2.1.1.1.cmml" xref="S2.SS2.p3.7.m2.1.1.1"></in><apply id="S2.SS2.p3.7.m2.1.1.2.cmml" xref="S2.SS2.p3.7.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m2.1.1.2.1.cmml" xref="S2.SS2.p3.7.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.p3.7.m2.1.1.2.2.cmml" xref="S2.SS2.p3.7.m2.1.1.2.2">𝑊</ci><ci id="S2.SS2.p3.7.m2.1.1.2.3.cmml" xref="S2.SS2.p3.7.m2.1.1.2.3">ℓ</ci></apply><apply id="S2.SS2.p3.7.m2.1.1.3.cmml" xref="S2.SS2.p3.7.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m2.1.1.3.1.cmml" xref="S2.SS2.p3.7.m2.1.1.3">superscript</csymbol><ci id="S2.SS2.p3.7.m2.1.1.3.2.cmml" xref="S2.SS2.p3.7.m2.1.1.3.2">ℝ</ci><apply id="S2.SS2.p3.7.m2.1.1.3.3.cmml" xref="S2.SS2.p3.7.m2.1.1.3.3"><times id="S2.SS2.p3.7.m2.1.1.3.3.1.cmml" xref="S2.SS2.p3.7.m2.1.1.3.3.1"></times><cn id="S2.SS2.p3.7.m2.1.1.3.3.2.cmml" type="integer" xref="S2.SS2.p3.7.m2.1.1.3.3.2">2</cn><ci id="S2.SS2.p3.7.m2.1.1.3.3.3.cmml" xref="S2.SS2.p3.7.m2.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.7.m2.1c">W_{\ell}\in\mathbb{R}^{2\times d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.7.m2.1d">italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 × italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> are learnable, and <math alttext="f_{\bm{\theta}}(x_{1}^{t})" class="ltx_Math" display="inline" id="S2.SS2.p3.8.m3.1"><semantics id="S2.SS2.p3.8.m3.1a"><mrow id="S2.SS2.p3.8.m3.1.1" xref="S2.SS2.p3.8.m3.1.1.cmml"><msub id="S2.SS2.p3.8.m3.1.1.3" xref="S2.SS2.p3.8.m3.1.1.3.cmml"><mi id="S2.SS2.p3.8.m3.1.1.3.2" xref="S2.SS2.p3.8.m3.1.1.3.2.cmml">f</mi><mi id="S2.SS2.p3.8.m3.1.1.3.3" xref="S2.SS2.p3.8.m3.1.1.3.3.cmml">𝜽</mi></msub><mo id="S2.SS2.p3.8.m3.1.1.2" xref="S2.SS2.p3.8.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p3.8.m3.1.1.1.1" xref="S2.SS2.p3.8.m3.1.1.1.1.1.cmml"><mo id="S2.SS2.p3.8.m3.1.1.1.1.2" stretchy="false" xref="S2.SS2.p3.8.m3.1.1.1.1.1.cmml">(</mo><msubsup id="S2.SS2.p3.8.m3.1.1.1.1.1" xref="S2.SS2.p3.8.m3.1.1.1.1.1.cmml"><mi id="S2.SS2.p3.8.m3.1.1.1.1.1.2.2" xref="S2.SS2.p3.8.m3.1.1.1.1.1.2.2.cmml">x</mi><mn id="S2.SS2.p3.8.m3.1.1.1.1.1.2.3" xref="S2.SS2.p3.8.m3.1.1.1.1.1.2.3.cmml">1</mn><mi id="S2.SS2.p3.8.m3.1.1.1.1.1.3" xref="S2.SS2.p3.8.m3.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S2.SS2.p3.8.m3.1.1.1.1.3" stretchy="false" xref="S2.SS2.p3.8.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.8.m3.1b"><apply id="S2.SS2.p3.8.m3.1.1.cmml" xref="S2.SS2.p3.8.m3.1.1"><times id="S2.SS2.p3.8.m3.1.1.2.cmml" xref="S2.SS2.p3.8.m3.1.1.2"></times><apply id="S2.SS2.p3.8.m3.1.1.3.cmml" xref="S2.SS2.p3.8.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.8.m3.1.1.3.1.cmml" xref="S2.SS2.p3.8.m3.1.1.3">subscript</csymbol><ci id="S2.SS2.p3.8.m3.1.1.3.2.cmml" xref="S2.SS2.p3.8.m3.1.1.3.2">𝑓</ci><ci id="S2.SS2.p3.8.m3.1.1.3.3.cmml" xref="S2.SS2.p3.8.m3.1.1.3.3">𝜽</ci></apply><apply id="S2.SS2.p3.8.m3.1.1.1.1.1.cmml" xref="S2.SS2.p3.8.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.8.m3.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.8.m3.1.1.1.1">superscript</csymbol><apply id="S2.SS2.p3.8.m3.1.1.1.1.1.2.cmml" xref="S2.SS2.p3.8.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.8.m3.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p3.8.m3.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p3.8.m3.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p3.8.m3.1.1.1.1.1.2.2">𝑥</ci><cn id="S2.SS2.p3.8.m3.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS2.p3.8.m3.1.1.1.1.1.2.3">1</cn></apply><ci id="S2.SS2.p3.8.m3.1.1.1.1.1.3.cmml" xref="S2.SS2.p3.8.m3.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.8.m3.1c">f_{\bm{\theta}}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.8.m3.1d">italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> is the probability law for the next symbol <math alttext="x_{t+1}" class="ltx_Math" display="inline" id="S2.SS2.p3.9.m4.1"><semantics id="S2.SS2.p3.9.m4.1a"><msub id="S2.SS2.p3.9.m4.1.1" xref="S2.SS2.p3.9.m4.1.1.cmml"><mi id="S2.SS2.p3.9.m4.1.1.2" xref="S2.SS2.p3.9.m4.1.1.2.cmml">x</mi><mrow id="S2.SS2.p3.9.m4.1.1.3" xref="S2.SS2.p3.9.m4.1.1.3.cmml"><mi id="S2.SS2.p3.9.m4.1.1.3.2" xref="S2.SS2.p3.9.m4.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p3.9.m4.1.1.3.1" xref="S2.SS2.p3.9.m4.1.1.3.1.cmml">+</mo><mn id="S2.SS2.p3.9.m4.1.1.3.3" xref="S2.SS2.p3.9.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.9.m4.1b"><apply id="S2.SS2.p3.9.m4.1.1.cmml" xref="S2.SS2.p3.9.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.9.m4.1.1.1.cmml" xref="S2.SS2.p3.9.m4.1.1">subscript</csymbol><ci id="S2.SS2.p3.9.m4.1.1.2.cmml" xref="S2.SS2.p3.9.m4.1.1.2">𝑥</ci><apply id="S2.SS2.p3.9.m4.1.1.3.cmml" xref="S2.SS2.p3.9.m4.1.1.3"><plus id="S2.SS2.p3.9.m4.1.1.3.1.cmml" xref="S2.SS2.p3.9.m4.1.1.3.1"></plus><ci id="S2.SS2.p3.9.m4.1.1.3.2.cmml" xref="S2.SS2.p3.9.m4.1.1.3.2">𝑡</ci><cn id="S2.SS2.p3.9.m4.1.1.3.3.cmml" type="integer" xref="S2.SS2.p3.9.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.9.m4.1c">x_{t+1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.9.m4.1d">italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> conditioned on the past <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="S2.SS2.p3.10.m5.1"><semantics id="S2.SS2.p3.10.m5.1a"><msubsup id="S2.SS2.p3.10.m5.1.1" xref="S2.SS2.p3.10.m5.1.1.cmml"><mi id="S2.SS2.p3.10.m5.1.1.2.2" xref="S2.SS2.p3.10.m5.1.1.2.2.cmml">x</mi><mn id="S2.SS2.p3.10.m5.1.1.2.3" xref="S2.SS2.p3.10.m5.1.1.2.3.cmml">1</mn><mi id="S2.SS2.p3.10.m5.1.1.3" xref="S2.SS2.p3.10.m5.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.10.m5.1b"><apply id="S2.SS2.p3.10.m5.1.1.cmml" xref="S2.SS2.p3.10.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.10.m5.1.1.1.cmml" xref="S2.SS2.p3.10.m5.1.1">superscript</csymbol><apply id="S2.SS2.p3.10.m5.1.1.2.cmml" xref="S2.SS2.p3.10.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.10.m5.1.1.2.1.cmml" xref="S2.SS2.p3.10.m5.1.1">subscript</csymbol><ci id="S2.SS2.p3.10.m5.1.1.2.2.cmml" xref="S2.SS2.p3.10.m5.1.1.2.2">𝑥</ci><cn id="S2.SS2.p3.10.m5.1.1.2.3.cmml" type="integer" xref="S2.SS2.p3.10.m5.1.1.2.3">1</cn></apply><ci id="S2.SS2.p3.10.m5.1.1.3.cmml" xref="S2.SS2.p3.10.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.10.m5.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.10.m5.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. We omit the layer norm here for simplicity. We compactly denote the set of all model parameters as <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S2.SS2.p3.11.m6.1"><semantics id="S2.SS2.p3.11.m6.1a"><mi id="S2.SS2.p3.11.m6.1.1" xref="S2.SS2.p3.11.m6.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.11.m6.1b"><ci id="S2.SS2.p3.11.m6.1.1.cmml" xref="S2.SS2.p3.11.m6.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.11.m6.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.11.m6.1d">bold_italic_θ</annotation></semantics></math>, i.e. <math alttext="\bm{\theta}=(\bm{e}_{0},\bm{e}_{1},\bm{\theta}_{\mathsf{Mamba}},W_{1,2,3},W_{% \ell})\in\mathbb{R}^{D}" class="ltx_Math" display="inline" id="S2.SS2.p3.12.m7.8"><semantics id="S2.SS2.p3.12.m7.8a"><mrow id="S2.SS2.p3.12.m7.8.8" xref="S2.SS2.p3.12.m7.8.8.cmml"><mi id="S2.SS2.p3.12.m7.8.8.7" xref="S2.SS2.p3.12.m7.8.8.7.cmml">𝜽</mi><mo id="S2.SS2.p3.12.m7.8.8.8" xref="S2.SS2.p3.12.m7.8.8.8.cmml">=</mo><mrow id="S2.SS2.p3.12.m7.8.8.5.5" xref="S2.SS2.p3.12.m7.8.8.5.6.cmml"><mo id="S2.SS2.p3.12.m7.8.8.5.5.6" stretchy="false" xref="S2.SS2.p3.12.m7.8.8.5.6.cmml">(</mo><msub id="S2.SS2.p3.12.m7.4.4.1.1.1" xref="S2.SS2.p3.12.m7.4.4.1.1.1.cmml"><mi id="S2.SS2.p3.12.m7.4.4.1.1.1.2" xref="S2.SS2.p3.12.m7.4.4.1.1.1.2.cmml">𝒆</mi><mn id="S2.SS2.p3.12.m7.4.4.1.1.1.3" xref="S2.SS2.p3.12.m7.4.4.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS2.p3.12.m7.8.8.5.5.7" xref="S2.SS2.p3.12.m7.8.8.5.6.cmml">,</mo><msub id="S2.SS2.p3.12.m7.5.5.2.2.2" xref="S2.SS2.p3.12.m7.5.5.2.2.2.cmml"><mi id="S2.SS2.p3.12.m7.5.5.2.2.2.2" xref="S2.SS2.p3.12.m7.5.5.2.2.2.2.cmml">𝒆</mi><mn id="S2.SS2.p3.12.m7.5.5.2.2.2.3" xref="S2.SS2.p3.12.m7.5.5.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS2.p3.12.m7.8.8.5.5.8" xref="S2.SS2.p3.12.m7.8.8.5.6.cmml">,</mo><msub id="S2.SS2.p3.12.m7.6.6.3.3.3" xref="S2.SS2.p3.12.m7.6.6.3.3.3.cmml"><mi id="S2.SS2.p3.12.m7.6.6.3.3.3.2" 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id="S2.SS2.p3.12.m7.8c">\bm{\theta}=(\bm{e}_{0},\bm{e}_{1},\bm{\theta}_{\mathsf{Mamba}},W_{1,2,3},W_{% \ell})\in\mathbb{R}^{D}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.12.m7.8d">bold_italic_θ = ( bold_italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT sansserif_Mamba end_POSTSUBSCRIPT , italic_W start_POSTSUBSCRIPT 1 , 2 , 3 end_POSTSUBSCRIPT , italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Learning task: next-token prediction</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.4">With the objective of auto-regressively estimating the next token, we train the model parameters <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mi id="S2.SS3.p1.1.m1.1.1" xref="S2.SS3.p1.1.m1.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><ci id="S2.SS3.p1.1.m1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">bold_italic_θ</annotation></semantics></math> to minimize the cross-entropy loss between the next-token predicted probability <math alttext="f_{\bm{\theta}}(x_{1}^{t})" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.1"><semantics id="S2.SS3.p1.2.m2.1a"><mrow id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml"><msub id="S2.SS3.p1.2.m2.1.1.3" xref="S2.SS3.p1.2.m2.1.1.3.cmml"><mi id="S2.SS3.p1.2.m2.1.1.3.2" xref="S2.SS3.p1.2.m2.1.1.3.2.cmml">f</mi><mi id="S2.SS3.p1.2.m2.1.1.3.3" 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encoding="application/x-tex" id="S2.SS3.p1.2.m2.1c">f_{\bm{\theta}}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.1d">italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> and the corresponding ground-truth symbol <math alttext="x_{t+1}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m3.1"><semantics id="S2.SS3.p1.3.m3.1a"><msub id="S2.SS3.p1.3.m3.1.1" xref="S2.SS3.p1.3.m3.1.1.cmml"><mi id="S2.SS3.p1.3.m3.1.1.2" xref="S2.SS3.p1.3.m3.1.1.2.cmml">x</mi><mrow id="S2.SS3.p1.3.m3.1.1.3" xref="S2.SS3.p1.3.m3.1.1.3.cmml"><mi id="S2.SS3.p1.3.m3.1.1.3.2" xref="S2.SS3.p1.3.m3.1.1.3.2.cmml">t</mi><mo id="S2.SS3.p1.3.m3.1.1.3.1" xref="S2.SS3.p1.3.m3.1.1.3.1.cmml">+</mo><mn id="S2.SS3.p1.3.m3.1.1.3.3" xref="S2.SS3.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.3.m3.1b"><apply id="S2.SS3.p1.3.m3.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.3.m3.1.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS3.p1.3.m3.1.1.2.cmml" xref="S2.SS3.p1.3.m3.1.1.2">𝑥</ci><apply id="S2.SS3.p1.3.m3.1.1.3.cmml" xref="S2.SS3.p1.3.m3.1.1.3"><plus id="S2.SS3.p1.3.m3.1.1.3.1.cmml" xref="S2.SS3.p1.3.m3.1.1.3.1"></plus><ci id="S2.SS3.p1.3.m3.1.1.3.2.cmml" xref="S2.SS3.p1.3.m3.1.1.3.2">𝑡</ci><cn id="S2.SS3.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS3.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m3.1c">x_{t+1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT</annotation></semantics></math> across all the positions <math alttext="t\in[T]" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m4.1"><semantics id="S2.SS3.p1.4.m4.1a"><mrow id="S2.SS3.p1.4.m4.1.2" xref="S2.SS3.p1.4.m4.1.2.cmml"><mi id="S2.SS3.p1.4.m4.1.2.2" xref="S2.SS3.p1.4.m4.1.2.2.cmml">t</mi><mo id="S2.SS3.p1.4.m4.1.2.1" xref="S2.SS3.p1.4.m4.1.2.1.cmml">∈</mo><mrow id="S2.SS3.p1.4.m4.1.2.3.2" xref="S2.SS3.p1.4.m4.1.2.3.1.cmml"><mo id="S2.SS3.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS3.p1.4.m4.1.2.3.1.1.cmml">[</mo><mi id="S2.SS3.p1.4.m4.1.1" xref="S2.SS3.p1.4.m4.1.1.cmml">T</mi><mo id="S2.SS3.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS3.p1.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m4.1b"><apply id="S2.SS3.p1.4.m4.1.2.cmml" xref="S2.SS3.p1.4.m4.1.2"><in id="S2.SS3.p1.4.m4.1.2.1.cmml" xref="S2.SS3.p1.4.m4.1.2.1"></in><ci id="S2.SS3.p1.4.m4.1.2.2.cmml" xref="S2.SS3.p1.4.m4.1.2.2">𝑡</ci><apply id="S2.SS3.p1.4.m4.1.2.3.1.cmml" xref="S2.SS3.p1.4.m4.1.2.3.2"><csymbol cd="latexml" id="S2.SS3.p1.4.m4.1.2.3.1.1.cmml" xref="S2.SS3.p1.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS3.p1.4.m4.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m4.1c">t\in[T]</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m4.1d">italic_t ∈ [ italic_T ]</annotation></semantics></math>:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="L(\bm{\theta})\triangleq-\frac{1}{T}\sum_{t\in[T]}\mathbb{E}_{P}\mathbb{E}_{x_% {1}^{t+1}\sim P}\big{[}x_{t+1}\cdot\log f_{\bm{\theta}}^{(1)}(x_{1}^{t})+(1-x_% {t+1})\cdot\log f_{\bm{\theta}}^{(0)}(x_{1}^{t})\big{]}," class="ltx_Math" display="block" id="S2.E2.m1.5"><semantics id="S2.E2.m1.5a"><mrow id="S2.E2.m1.5.5.1" xref="S2.E2.m1.5.5.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1" xref="S2.E2.m1.5.5.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1.3" xref="S2.E2.m1.5.5.1.1.3.cmml"><mi id="S2.E2.m1.5.5.1.1.3.2" xref="S2.E2.m1.5.5.1.1.3.2.cmml">L</mi><mo id="S2.E2.m1.5.5.1.1.3.1" xref="S2.E2.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.3.3.2" xref="S2.E2.m1.5.5.1.1.3.cmml"><mo id="S2.E2.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.E2.m1.5.5.1.1.3.cmml">(</mo><mi id="S2.E2.m1.4.4" xref="S2.E2.m1.4.4.cmml">𝜽</mi><mo id="S2.E2.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S2.E2.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.5.5.1.1.2" xref="S2.E2.m1.5.5.1.1.2.cmml">≜</mo><mrow id="S2.E2.m1.5.5.1.1.1" xref="S2.E2.m1.5.5.1.1.1.cmml"><mo id="S2.E2.m1.5.5.1.1.1a" xref="S2.E2.m1.5.5.1.1.1.cmml">−</mo><mrow id="S2.E2.m1.5.5.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.cmml"><mfrac id="S2.E2.m1.5.5.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.1.1.3.cmml"><mn id="S2.E2.m1.5.5.1.1.1.1.3.2" xref="S2.E2.m1.5.5.1.1.1.1.3.2.cmml">1</mn><mi id="S2.E2.m1.5.5.1.1.1.1.3.3" xref="S2.E2.m1.5.5.1.1.1.1.3.3.cmml">T</mi></mfrac><mo id="S2.E2.m1.5.5.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.1.cmml"><munder id="S2.E2.m1.5.5.1.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.1.1.1.2.cmml"><mo id="S2.E2.m1.5.5.1.1.1.1.1.2.2" movablelimits="false" xref="S2.E2.m1.5.5.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.1.1.1.3.cmml">t</mi><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.2.cmml">∈</mo><mrow id="S2.E2.m1.1.1.1.4.2" xref="S2.E2.m1.1.1.1.4.1.cmml"><mo id="S2.E2.m1.1.1.1.4.2.1" stretchy="false" xref="S2.E2.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">T</mi><mo id="S2.E2.m1.1.1.1.4.2.2" stretchy="false" xref="S2.E2.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.1.1.cmml"><msub id="S2.E2.m1.5.5.1.1.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.3.cmml"><mi id="S2.E2.m1.5.5.1.1.1.1.1.1.3.2" xref="S2.E2.m1.5.5.1.1.1.1.1.1.3.2.cmml">𝔼</mi><mi id="S2.E2.m1.5.5.1.1.1.1.1.1.3.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.3.3.cmml">P</mi></msub><mo id="S2.E2.m1.5.5.1.1.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.E2.m1.5.5.1.1.1.1.1.1.4" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.cmml"><mi id="S2.E2.m1.5.5.1.1.1.1.1.1.4.2" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.2.cmml">𝔼</mi><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.cmml"><msubsup id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.cmml"><mi id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.2.2" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.2.2.cmml">x</mi><mn id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.2.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.2.3.cmml">1</mn><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.cmml"><mi id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.2" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.2.cmml">t</mi><mo id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.1" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.1.cmml">+</mo><mn id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.1" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.1.cmml">∼</mo><mi id="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.4.3.3.cmml">P</mi></mrow></msub><mo id="S2.E2.m1.5.5.1.1.1.1.1.1.2a" xref="S2.E2.m1.5.5.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S2.E2.m1.5.5.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.cmml"><msub id="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.2" 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xref="S2.E2.m1.5.5.1.1.1.1.1.1.1.1.1.3.2.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.5c">L(\bm{\theta})\triangleq-\frac{1}{T}\sum_{t\in[T]}\mathbb{E}_{P}\mathbb{E}_{x_% {1}^{t+1}\sim P}\big{[}x_{t+1}\cdot\log f_{\bm{\theta}}^{(1)}(x_{1}^{t})+(1-x_% {t+1})\cdot\log f_{\bm{\theta}}^{(0)}(x_{1}^{t})\big{]},</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.5d">italic_L ( bold_italic_θ ) ≜ - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t ∈ [ italic_T ] end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ∼ italic_P end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ⋅ roman_log italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) + ( 1 - italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ) ⋅ roman_log italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.9">where <math alttext="f_{\bm{\theta}}^{(j)}(x_{1}^{t})\triangleq\mathbb{P}_{\bm{\theta}}\left(x_{t+1% }=j\mid x_{1}^{t}\right)" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m1.3"><semantics id="S2.SS3.p1.5.m1.3a"><mrow 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end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ≜ blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = italic_j ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> for <math alttext="j\in\{0,1\}" class="ltx_Math" display="inline" id="S2.SS3.p1.6.m2.2"><semantics id="S2.SS3.p1.6.m2.2a"><mrow id="S2.SS3.p1.6.m2.2.3" xref="S2.SS3.p1.6.m2.2.3.cmml"><mi id="S2.SS3.p1.6.m2.2.3.2" xref="S2.SS3.p1.6.m2.2.3.2.cmml">j</mi><mo id="S2.SS3.p1.6.m2.2.3.1" xref="S2.SS3.p1.6.m2.2.3.1.cmml">∈</mo><mrow id="S2.SS3.p1.6.m2.2.3.3.2" xref="S2.SS3.p1.6.m2.2.3.3.1.cmml"><mo id="S2.SS3.p1.6.m2.2.3.3.2.1" stretchy="false" xref="S2.SS3.p1.6.m2.2.3.3.1.cmml">{</mo><mn id="S2.SS3.p1.6.m2.1.1" xref="S2.SS3.p1.6.m2.1.1.cmml">0</mn><mo id="S2.SS3.p1.6.m2.2.3.3.2.2" 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kernels <math alttext="P" class="ltx_Math" display="inline" id="S2.SS3.p1.7.m3.1"><semantics id="S2.SS3.p1.7.m3.1a"><mi id="S2.SS3.p1.7.m3.1.1" xref="S2.SS3.p1.7.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.7.m3.1b"><ci id="S2.SS3.p1.7.m3.1.1.cmml" xref="S2.SS3.p1.7.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.7.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.7.m3.1d">italic_P</annotation></semantics></math> and the Markov sequences <math alttext="x=(x_{t})_{t=1}^{T}" class="ltx_Math" display="inline" id="S2.SS3.p1.8.m4.1"><semantics id="S2.SS3.p1.8.m4.1a"><mrow id="S2.SS3.p1.8.m4.1.1" xref="S2.SS3.p1.8.m4.1.1.cmml"><mi id="S2.SS3.p1.8.m4.1.1.3" xref="S2.SS3.p1.8.m4.1.1.3.cmml">x</mi><mo id="S2.SS3.p1.8.m4.1.1.2" xref="S2.SS3.p1.8.m4.1.1.2.cmml">=</mo><msubsup id="S2.SS3.p1.8.m4.1.1.1" xref="S2.SS3.p1.8.m4.1.1.1.cmml"><mrow id="S2.SS3.p1.8.m4.1.1.1.1.1.1" 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id="S2.SS3.p1.8.m4.1b"><apply id="S2.SS3.p1.8.m4.1.1.cmml" xref="S2.SS3.p1.8.m4.1.1"><eq id="S2.SS3.p1.8.m4.1.1.2.cmml" xref="S2.SS3.p1.8.m4.1.1.2"></eq><ci id="S2.SS3.p1.8.m4.1.1.3.cmml" xref="S2.SS3.p1.8.m4.1.1.3">𝑥</ci><apply id="S2.SS3.p1.8.m4.1.1.1.cmml" xref="S2.SS3.p1.8.m4.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m4.1.1.1.2.cmml" xref="S2.SS3.p1.8.m4.1.1.1">superscript</csymbol><apply id="S2.SS3.p1.8.m4.1.1.1.1.cmml" xref="S2.SS3.p1.8.m4.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m4.1.1.1.1.2.cmml" xref="S2.SS3.p1.8.m4.1.1.1">subscript</csymbol><apply id="S2.SS3.p1.8.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m4.1.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p1.8.m4.1.1.1.1.1.1.1.2.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S2.SS3.p1.8.m4.1.1.1.1.1.1.1.3.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S2.SS3.p1.8.m4.1.1.1.1.3.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.3"><eq id="S2.SS3.p1.8.m4.1.1.1.1.3.1.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.3.1"></eq><ci id="S2.SS3.p1.8.m4.1.1.1.1.3.2.cmml" xref="S2.SS3.p1.8.m4.1.1.1.1.3.2">𝑡</ci><cn id="S2.SS3.p1.8.m4.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.p1.8.m4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S2.SS3.p1.8.m4.1.1.1.3.cmml" xref="S2.SS3.p1.8.m4.1.1.1.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.8.m4.1c">x=(x_{t})_{t=1}^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.8.m4.1d">italic_x = ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> sampled from <math alttext="P" class="ltx_Math" display="inline" id="S2.SS3.p1.9.m5.1"><semantics id="S2.SS3.p1.9.m5.1a"><mi id="S2.SS3.p1.9.m5.1.1" xref="S2.SS3.p1.9.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.9.m5.1b"><ci id="S2.SS3.p1.9.m5.1.1.cmml" xref="S2.SS3.p1.9.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.9.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.9.m5.1d">italic_P</annotation></semantics></math>. In practice, it is replaced by empirical average across a finite set of batches, sampled according to the random Markov distribution in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="2.1 Input data: Random Markov chains ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.1</span></a>. For our experiments we use the AdamW optimizer <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">Kingma2017</span>]</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.4 </span>Optimal estimator: Laplacian smoothing</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.6">Given the Bayesian prediction loss in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.E2" title="Equation 2 ‣ 2.3 Learning task: next-token prediction ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2</span></a>), it is natural to ask: <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.1.1">what’s the optimal <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S2.SS4.p1.1.1.m1.1"><semantics id="S2.SS4.p1.1.1.m1.1a"><mi id="S2.SS4.p1.1.1.m1.1.1" xref="S2.SS4.p1.1.1.m1.1.1.cmml">𝛉</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.1.m1.1b"><ci id="S2.SS4.p1.1.1.m1.1.1.cmml" xref="S2.SS4.p1.1.1.m1.1.1">𝛉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.1.m1.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.1.m1.1d">bold_italic_θ</annotation></semantics></math> minimizing it?</em> It follows from a classical result in statistics (<cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">rissanen1984</span>]</cite>, § <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A1" title="Appendix A Preliminaries on Laplacian smoothing ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">A</span></a>) that this minimum is achieved when the corresponding model prediction matches the (averaged) ground-truth predictive distribution, i.e. <math alttext="\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=1\mid x_{1}^{t}\right)=\mathbb{E}_{P|x_{% 1}^{t}}\left[\mathbb{P}\left(x_{t+1}=1\mid x_{1}^{t}\right)\right]" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m1.2"><semantics id="S2.SS4.p1.2.m1.2a"><mrow id="S2.SS4.p1.2.m1.2.2" xref="S2.SS4.p1.2.m1.2.2.cmml"><mrow id="S2.SS4.p1.2.m1.1.1.1" xref="S2.SS4.p1.2.m1.1.1.1.cmml"><msub id="S2.SS4.p1.2.m1.1.1.1.3" xref="S2.SS4.p1.2.m1.1.1.1.3.cmml"><mi id="S2.SS4.p1.2.m1.1.1.1.3.2" 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id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.2">1</cn><apply id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.cmml" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.1.cmml" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3">superscript</csymbol><apply id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.2.cmml" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.2.1.cmml" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.2.2.cmml" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.2.2">𝑥</ci><cn id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.2.3">1</cn></apply><ci id="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.3.cmml" xref="S2.SS4.p1.2.m1.2.2.2.1.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m1.2c">\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=1\mid x_{1}^{t}\right)=\mathbb{E}_{P|x_{% 1}^{t}}\left[\mathbb{P}\left(x_{t+1}=1\mid x_{1}^{t}\right)\right]</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m1.2d">blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = blackboard_E start_POSTSUBSCRIPT italic_P | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ blackboard_P ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ]</annotation></semantics></math>, for all <math alttext="t" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m2.1"><semantics id="S2.SS4.p1.3.m2.1a"><mi id="S2.SS4.p1.3.m2.1.1" xref="S2.SS4.p1.3.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m2.1b"><ci id="S2.SS4.p1.3.m2.1.1.cmml" xref="S2.SS4.p1.3.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m2.1d">italic_t</annotation></semantics></math>. Given the joint distribution of the pair <math alttext="(P,x_{1}^{t+1})" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m3.2"><semantics id="S2.SS4.p1.4.m3.2a"><mrow id="S2.SS4.p1.4.m3.2.2.1" xref="S2.SS4.p1.4.m3.2.2.2.cmml"><mo id="S2.SS4.p1.4.m3.2.2.1.2" stretchy="false" xref="S2.SS4.p1.4.m3.2.2.2.cmml">(</mo><mi id="S2.SS4.p1.4.m3.1.1" xref="S2.SS4.p1.4.m3.1.1.cmml">P</mi><mo id="S2.SS4.p1.4.m3.2.2.1.3" xref="S2.SS4.p1.4.m3.2.2.2.cmml">,</mo><msubsup id="S2.SS4.p1.4.m3.2.2.1.1" xref="S2.SS4.p1.4.m3.2.2.1.1.cmml"><mi id="S2.SS4.p1.4.m3.2.2.1.1.2.2" xref="S2.SS4.p1.4.m3.2.2.1.1.2.2.cmml">x</mi><mn id="S2.SS4.p1.4.m3.2.2.1.1.2.3" xref="S2.SS4.p1.4.m3.2.2.1.1.2.3.cmml">1</mn><mrow id="S2.SS4.p1.4.m3.2.2.1.1.3" xref="S2.SS4.p1.4.m3.2.2.1.1.3.cmml"><mi id="S2.SS4.p1.4.m3.2.2.1.1.3.2" xref="S2.SS4.p1.4.m3.2.2.1.1.3.2.cmml">t</mi><mo id="S2.SS4.p1.4.m3.2.2.1.1.3.1" xref="S2.SS4.p1.4.m3.2.2.1.1.3.1.cmml">+</mo><mn id="S2.SS4.p1.4.m3.2.2.1.1.3.3" xref="S2.SS4.p1.4.m3.2.2.1.1.3.3.cmml">1</mn></mrow></msubsup><mo id="S2.SS4.p1.4.m3.2.2.1.4" stretchy="false" xref="S2.SS4.p1.4.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m3.2b"><interval closure="open" id="S2.SS4.p1.4.m3.2.2.2.cmml" xref="S2.SS4.p1.4.m3.2.2.1"><ci id="S2.SS4.p1.4.m3.1.1.cmml" xref="S2.SS4.p1.4.m3.1.1">𝑃</ci><apply id="S2.SS4.p1.4.m3.2.2.1.1.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m3.2.2.1.1.1.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1">superscript</csymbol><apply id="S2.SS4.p1.4.m3.2.2.1.1.2.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m3.2.2.1.1.2.1.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1">subscript</csymbol><ci id="S2.SS4.p1.4.m3.2.2.1.1.2.2.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1.2.2">𝑥</ci><cn id="S2.SS4.p1.4.m3.2.2.1.1.2.3.cmml" type="integer" xref="S2.SS4.p1.4.m3.2.2.1.1.2.3">1</cn></apply><apply id="S2.SS4.p1.4.m3.2.2.1.1.3.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1.3"><plus id="S2.SS4.p1.4.m3.2.2.1.1.3.1.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1.3.1"></plus><ci id="S2.SS4.p1.4.m3.2.2.1.1.3.2.cmml" xref="S2.SS4.p1.4.m3.2.2.1.1.3.2">𝑡</ci><cn id="S2.SS4.p1.4.m3.2.2.1.1.3.3.cmml" type="integer" xref="S2.SS4.p1.4.m3.2.2.1.1.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m3.2c">(P,x_{1}^{t+1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m3.2d">( italic_P , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="2.1 Input data: Random Markov chains ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.1</span></a>, where the kernel <math alttext="P\sim\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="S2.SS4.p1.5.m4.1"><semantics id="S2.SS4.p1.5.m4.1a"><mrow id="S2.SS4.p1.5.m4.1.1" xref="S2.SS4.p1.5.m4.1.1.cmml"><mi id="S2.SS4.p1.5.m4.1.1.3" xref="S2.SS4.p1.5.m4.1.1.3.cmml">P</mi><mo id="S2.SS4.p1.5.m4.1.1.2" xref="S2.SS4.p1.5.m4.1.1.2.cmml">∼</mo><mrow id="S2.SS4.p1.5.m4.1.1.1" xref="S2.SS4.p1.5.m4.1.1.1.cmml"><mi id="S2.SS4.p1.5.m4.1.1.1.3" xref="S2.SS4.p1.5.m4.1.1.1.3.cmml">Dir</mi><mo id="S2.SS4.p1.5.m4.1.1.1.2" xref="S2.SS4.p1.5.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.p1.5.m4.1.1.1.1.1" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.cmml"><mo id="S2.SS4.p1.5.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.p1.5.m4.1.1.1.1.1.1" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p1.5.m4.1.1.1.1.1.1.2" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.2.cmml">β</mi><mo id="S2.SS4.p1.5.m4.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.1.cmml">⋅</mo><mn id="S2.SS4.p1.5.m4.1.1.1.1.1.1.3" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S2.SS4.p1.5.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.5.m4.1b"><apply id="S2.SS4.p1.5.m4.1.1.cmml" xref="S2.SS4.p1.5.m4.1.1"><csymbol cd="latexml" id="S2.SS4.p1.5.m4.1.1.2.cmml" xref="S2.SS4.p1.5.m4.1.1.2">similar-to</csymbol><ci id="S2.SS4.p1.5.m4.1.1.3.cmml" xref="S2.SS4.p1.5.m4.1.1.3">𝑃</ci><apply id="S2.SS4.p1.5.m4.1.1.1.cmml" xref="S2.SS4.p1.5.m4.1.1.1"><times id="S2.SS4.p1.5.m4.1.1.1.2.cmml" xref="S2.SS4.p1.5.m4.1.1.1.2"></times><ci id="S2.SS4.p1.5.m4.1.1.1.3.cmml" xref="S2.SS4.p1.5.m4.1.1.1.3">Dir</ci><apply id="S2.SS4.p1.5.m4.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.5.m4.1.1.1.1.1"><ci id="S2.SS4.p1.5.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.1">⋅</ci><ci id="S2.SS4.p1.5.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.2">𝛽</ci><cn id="S2.SS4.p1.5.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.p1.5.m4.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m4.1c">P\sim\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m4.1d">italic_P ∼ roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math>, it can be shown (§ <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A1" title="Appendix A Preliminaries on Laplacian smoothing ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">A</span></a>) that the conditional expectation above simplifies to the well-known <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.6.3">Laplacian smoothing</em>, also known as the <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.6.2">add-<math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS4.p1.6.2.m1.1"><semantics id="S2.SS4.p1.6.2.m1.1a"><mi id="S2.SS4.p1.6.2.m1.1.1" xref="S2.SS4.p1.6.2.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.6.2.m1.1b"><ci id="S2.SS4.p1.6.2.m1.1.1.cmml" xref="S2.SS4.p1.6.2.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.6.2.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.6.2.m1.1d">italic_β</annotation></semantics></math> estimator</em> (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">merhav1998</span>]</cite>):</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx3"> <tbody id="S2.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{P}_{\beta}^{(k)}\left(x_{t+1}=1\mid x_{1}^{t}\right)% \triangleq\mathbb{E}_{P|x_{1}^{t}}\left[\mathbb{P}\left(x_{t+1}=1\mid x_{1}^{t% }\right)\right]=\frac{n_{1}+\beta}{n+2\beta}," class="ltx_Math" display="inline" id="S2.Ex8.m1.2"><semantics id="S2.Ex8.m1.2a"><mrow id="S2.Ex8.m1.2.2.1" xref="S2.Ex8.m1.2.2.1.1.cmml"><mrow id="S2.Ex8.m1.2.2.1.1" xref="S2.Ex8.m1.2.2.1.1.cmml"><mrow id="S2.Ex8.m1.2.2.1.1.1" xref="S2.Ex8.m1.2.2.1.1.1.cmml"><msubsup id="S2.Ex8.m1.2.2.1.1.1.3" xref="S2.Ex8.m1.2.2.1.1.1.3.cmml"><mi id="S2.Ex8.m1.2.2.1.1.1.3.2.2" xref="S2.Ex8.m1.2.2.1.1.1.3.2.2.cmml">ℙ</mi><mi id="S2.Ex8.m1.2.2.1.1.1.3.2.3" xref="S2.Ex8.m1.2.2.1.1.1.3.2.3.cmml">β</mi><mrow id="S2.Ex8.m1.1.1.1.3" xref="S2.Ex8.m1.2.2.1.1.1.3.cmml"><mo id="S2.Ex8.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex8.m1.2.2.1.1.1.3.cmml">(</mo><mi id="S2.Ex8.m1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.cmml">k</mi><mo id="S2.Ex8.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex8.m1.2.2.1.1.1.3.cmml">)</mo></mrow></msubsup><mo id="S2.Ex8.m1.2.2.1.1.1.2" xref="S2.Ex8.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex8.m1.2.2.1.1.1.1.1" xref="S2.Ex8.m1.2.2.1.1.1.1.1.1.cmml"><mo 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}\right)\right]=\frac{n_{1}+\beta}{n+2\beta},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.2d">blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ≜ blackboard_E start_POSTSUBSCRIPT italic_P | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ blackboard_P ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ] = divide start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n + 2 italic_β end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Laplacian smoothing)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p1.18">where <math alttext="n_{1}" class="ltx_Math" display="inline" id="S2.SS4.p1.7.m1.1"><semantics id="S2.SS4.p1.7.m1.1a"><msub id="S2.SS4.p1.7.m1.1.1" xref="S2.SS4.p1.7.m1.1.1.cmml"><mi id="S2.SS4.p1.7.m1.1.1.2" xref="S2.SS4.p1.7.m1.1.1.2.cmml">n</mi><mn id="S2.SS4.p1.7.m1.1.1.3" xref="S2.SS4.p1.7.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.7.m1.1b"><apply id="S2.SS4.p1.7.m1.1.1.cmml" xref="S2.SS4.p1.7.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.7.m1.1.1.1.cmml" xref="S2.SS4.p1.7.m1.1.1">subscript</csymbol><ci id="S2.SS4.p1.7.m1.1.1.2.cmml" xref="S2.SS4.p1.7.m1.1.1.2">𝑛</ci><cn id="S2.SS4.p1.7.m1.1.1.3.cmml" type="integer" xref="S2.SS4.p1.7.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.7.m1.1c">n_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.7.m1.1d">italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is the number of times token <math alttext="1" class="ltx_Math" display="inline" id="S2.SS4.p1.8.m2.1"><semantics id="S2.SS4.p1.8.m2.1a"><mn id="S2.SS4.p1.8.m2.1.1" xref="S2.SS4.p1.8.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.8.m2.1b"><cn id="S2.SS4.p1.8.m2.1.1.cmml" type="integer" xref="S2.SS4.p1.8.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.8.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.8.m2.1d">1</annotation></semantics></math> follows the current <math alttext="k^{\text{th}}" class="ltx_Math" display="inline" id="S2.SS4.p1.9.m3.1"><semantics id="S2.SS4.p1.9.m3.1a"><msup id="S2.SS4.p1.9.m3.1.1" xref="S2.SS4.p1.9.m3.1.1.cmml"><mi id="S2.SS4.p1.9.m3.1.1.2" xref="S2.SS4.p1.9.m3.1.1.2.cmml">k</mi><mtext id="S2.SS4.p1.9.m3.1.1.3" xref="S2.SS4.p1.9.m3.1.1.3a.cmml">th</mtext></msup><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.9.m3.1b"><apply id="S2.SS4.p1.9.m3.1.1.cmml" xref="S2.SS4.p1.9.m3.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.9.m3.1.1.1.cmml" xref="S2.SS4.p1.9.m3.1.1">superscript</csymbol><ci id="S2.SS4.p1.9.m3.1.1.2.cmml" xref="S2.SS4.p1.9.m3.1.1.2">𝑘</ci><ci id="S2.SS4.p1.9.m3.1.1.3a.cmml" xref="S2.SS4.p1.9.m3.1.1.3"><mtext id="S2.SS4.p1.9.m3.1.1.3.cmml" mathsize="70%" xref="S2.SS4.p1.9.m3.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.9.m3.1c">k^{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.9.m3.1d">italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT</annotation></semantics></math>-order context <math alttext="x_{t-k+1}^{t}" class="ltx_Math" display="inline" id="S2.SS4.p1.10.m4.1"><semantics id="S2.SS4.p1.10.m4.1a"><msubsup id="S2.SS4.p1.10.m4.1.1" xref="S2.SS4.p1.10.m4.1.1.cmml"><mi id="S2.SS4.p1.10.m4.1.1.2.2" xref="S2.SS4.p1.10.m4.1.1.2.2.cmml">x</mi><mrow id="S2.SS4.p1.10.m4.1.1.2.3" xref="S2.SS4.p1.10.m4.1.1.2.3.cmml"><mrow id="S2.SS4.p1.10.m4.1.1.2.3.2" xref="S2.SS4.p1.10.m4.1.1.2.3.2.cmml"><mi id="S2.SS4.p1.10.m4.1.1.2.3.2.2" xref="S2.SS4.p1.10.m4.1.1.2.3.2.2.cmml">t</mi><mo id="S2.SS4.p1.10.m4.1.1.2.3.2.1" xref="S2.SS4.p1.10.m4.1.1.2.3.2.1.cmml">−</mo><mi id="S2.SS4.p1.10.m4.1.1.2.3.2.3" xref="S2.SS4.p1.10.m4.1.1.2.3.2.3.cmml">k</mi></mrow><mo id="S2.SS4.p1.10.m4.1.1.2.3.1" xref="S2.SS4.p1.10.m4.1.1.2.3.1.cmml">+</mo><mn id="S2.SS4.p1.10.m4.1.1.2.3.3" xref="S2.SS4.p1.10.m4.1.1.2.3.3.cmml">1</mn></mrow><mi id="S2.SS4.p1.10.m4.1.1.3" xref="S2.SS4.p1.10.m4.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.10.m4.1b"><apply id="S2.SS4.p1.10.m4.1.1.cmml" xref="S2.SS4.p1.10.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.10.m4.1.1.1.cmml" xref="S2.SS4.p1.10.m4.1.1">superscript</csymbol><apply id="S2.SS4.p1.10.m4.1.1.2.cmml" xref="S2.SS4.p1.10.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.10.m4.1.1.2.1.cmml" xref="S2.SS4.p1.10.m4.1.1">subscript</csymbol><ci id="S2.SS4.p1.10.m4.1.1.2.2.cmml" xref="S2.SS4.p1.10.m4.1.1.2.2">𝑥</ci><apply id="S2.SS4.p1.10.m4.1.1.2.3.cmml" xref="S2.SS4.p1.10.m4.1.1.2.3"><plus id="S2.SS4.p1.10.m4.1.1.2.3.1.cmml" xref="S2.SS4.p1.10.m4.1.1.2.3.1"></plus><apply id="S2.SS4.p1.10.m4.1.1.2.3.2.cmml" xref="S2.SS4.p1.10.m4.1.1.2.3.2"><minus id="S2.SS4.p1.10.m4.1.1.2.3.2.1.cmml" xref="S2.SS4.p1.10.m4.1.1.2.3.2.1"></minus><ci id="S2.SS4.p1.10.m4.1.1.2.3.2.2.cmml" xref="S2.SS4.p1.10.m4.1.1.2.3.2.2">𝑡</ci><ci id="S2.SS4.p1.10.m4.1.1.2.3.2.3.cmml" xref="S2.SS4.p1.10.m4.1.1.2.3.2.3">𝑘</ci></apply><cn id="S2.SS4.p1.10.m4.1.1.2.3.3.cmml" type="integer" xref="S2.SS4.p1.10.m4.1.1.2.3.3">1</cn></apply></apply><ci id="S2.SS4.p1.10.m4.1.1.3.cmml" xref="S2.SS4.p1.10.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.10.m4.1c">x_{t-k+1}^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.10.m4.1d">italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> in the sequence <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="S2.SS4.p1.11.m5.1"><semantics id="S2.SS4.p1.11.m5.1a"><msubsup id="S2.SS4.p1.11.m5.1.1" xref="S2.SS4.p1.11.m5.1.1.cmml"><mi id="S2.SS4.p1.11.m5.1.1.2.2" xref="S2.SS4.p1.11.m5.1.1.2.2.cmml">x</mi><mn id="S2.SS4.p1.11.m5.1.1.2.3" xref="S2.SS4.p1.11.m5.1.1.2.3.cmml">1</mn><mi id="S2.SS4.p1.11.m5.1.1.3" xref="S2.SS4.p1.11.m5.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.11.m5.1b"><apply id="S2.SS4.p1.11.m5.1.1.cmml" xref="S2.SS4.p1.11.m5.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.11.m5.1.1.1.cmml" 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xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.2.3.cmml"><mo id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.2.2.3" stretchy="false" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.2.3.cmml">(</mo><msubsup id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.2" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mrow id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.2" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.1" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.1.cmml">−</mo><mi id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.3" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.2.3.3.cmml">k</mi></mrow><mrow id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.3" xref="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.SS4.p1.12.m6.3.3.1.1.1.1.1.1.1.1.3.2" 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id="S2.SS4.p1.12.m6.3d">italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = | { italic_i : ( italic_x start_POSTSUBSCRIPT italic_i - italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i - 1 end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = ( italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , 1 ) } |</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S2.SS4.p1.13.m7.1"><semantics id="S2.SS4.p1.13.m7.1a"><mi id="S2.SS4.p1.13.m7.1.1" xref="S2.SS4.p1.13.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.13.m7.1b"><ci id="S2.SS4.p1.13.m7.1.1.cmml" xref="S2.SS4.p1.13.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.13.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.13.m7.1d">italic_n</annotation></semantics></math> is the frequency of this context, 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Adjusting these counts by <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS4.p1.15.m9.1"><semantics id="S2.SS4.p1.15.m9.1a"><mi id="S2.SS4.p1.15.m9.1.1" xref="S2.SS4.p1.15.m9.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.15.m9.1b"><ci id="S2.SS4.p1.15.m9.1.1.cmml" xref="S2.SS4.p1.15.m9.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.15.m9.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.15.m9.1d">italic_β</annotation></semantics></math> plays the role of additive smoothing, which avoids assigning zero probabilities to unseen events, an idea dating back to Laplace <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">laplace1814essaiprob</span>]</cite>. It is also known that the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS4.p1.16.m10.1"><semantics id="S2.SS4.p1.16.m10.1a"><mi id="S2.SS4.p1.16.m10.1.1" xref="S2.SS4.p1.16.m10.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.16.m10.1b"><ci id="S2.SS4.p1.16.m10.1.1.cmml" xref="S2.SS4.p1.16.m10.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.16.m10.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.16.m10.1d">italic_β</annotation></semantics></math> estimator is asymptotically minimax optimal, as <math alttext="T\to\infty" class="ltx_Math" display="inline" id="S2.SS4.p1.17.m11.1"><semantics id="S2.SS4.p1.17.m11.1a"><mrow id="S2.SS4.p1.17.m11.1.1" xref="S2.SS4.p1.17.m11.1.1.cmml"><mi id="S2.SS4.p1.17.m11.1.1.2" xref="S2.SS4.p1.17.m11.1.1.2.cmml">T</mi><mo id="S2.SS4.p1.17.m11.1.1.1" stretchy="false" xref="S2.SS4.p1.17.m11.1.1.1.cmml">→</mo><mi id="S2.SS4.p1.17.m11.1.1.3" mathvariant="normal" xref="S2.SS4.p1.17.m11.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.17.m11.1b"><apply id="S2.SS4.p1.17.m11.1.1.cmml" xref="S2.SS4.p1.17.m11.1.1"><ci id="S2.SS4.p1.17.m11.1.1.1.cmml" xref="S2.SS4.p1.17.m11.1.1.1">→</ci><ci id="S2.SS4.p1.17.m11.1.1.2.cmml" xref="S2.SS4.p1.17.m11.1.1.2">𝑇</ci><infinity id="S2.SS4.p1.17.m11.1.1.3.cmml" xref="S2.SS4.p1.17.m11.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.17.m11.1c">T\to\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.17.m11.1d">italic_T → ∞</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">xie-barron-1997</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">orlitsky2018mc</span>]</cite>. 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(<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.E2" title="Equation 2 ‣ 2.3 Learning task: next-token prediction ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2</span></a>) on random Markov chains. We repeat the same procedure for both single and two layer transformers and report the best performance for all these models (we refer to § <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A4" title="Appendix D Model architectures and hyper-parameters ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">D</span></a> for more experimental details). These experiments reveal interesting and rather surprising insights about Mamba:</p> </div> <div class="ltx_para" id="S3.p2"> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1">Mamba learns the optimal Laplacian smoothing estimator on the Markov prediction task, even with a single layer (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1.F1.sf1" title="Figure 1(a) ‣ Figure 1 ‣ 1 Introduction ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1(a)</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.1">Convolution mechanism plays a fundamental role in Mamba, aiding in its learning abilities (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3.F3.sf1" title="Figure 3(a) ‣ Figure 3 ‣ 3 Does Mamba learn in-context estimators? ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3(a)</span></a>)</p> </div> </li> </ol> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">In the sequel, we expand upon these observations in detail.</p> </div> <div class="ltx_para ltx_noindent" id="S3.p4"> <p class="ltx_p" id="S3.p4.1"><span class="ltx_text ltx_font_bold" id="S3.p4.1.1">1) Mamba learns the Laplacian smoothing.</span> After training, we evaluate the Mamba and transformer models on the same test sequence fixed beforehand and compare their performance to that of the optimal <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex8" title="Equation Laplacian smoothing ‣ 2.4 Optimal estimator: Laplacian smoothing ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Laplacian smoothing</span></a> estimator. Specifically, we compare their next-token prediction probabilities with those of the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_β</annotation></semantics></math> estimator. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1</span></a> illustrates these results, which uncovers a surprising phenomenon: <em class="ltx_emph ltx_font_italic" id="S3.p4.1.2">even a single-layer Mamba sharply matches the optimal estimator on the whole sequence</em>. In fact, this approximation error is small even for various Markov orders. On the other hand, for transformers we observe that a two-layer model also matches the predictor, albeit less sharply and more variance, whereas a single layer transformers fails to solve the task. This is not fully surprising, given a recent theoretical result <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">sanford2024onelayer</span>]</cite> that while an induction head (realizing the counting estimator) can be efficiently represented by a two-layer transformer, its single-layer variant needs to be exponentially large.</p> </div> <div class="ltx_para ltx_noindent" id="S3.p5"> <p class="ltx_p" id="S3.p5.6"><span class="ltx_text ltx_font_bold" id="S3.p5.6.1">2) Convolution is the key.</span> To decipher the key architectural component behind Mamba’s success in Markov prediction task, we do an ablation study on its three main features: (i) convolution in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a>, (ii) ReLU non-linearity in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a>, and (iii) the gating mechanism in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex1" title="Equation Mamba ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Mamba</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex5" title="Equation MLP ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag"><span class="ltx_text" style="font-size:90%;">MLP</span></span></a>. Amongst them, surprisingly, <em class="ltx_emph ltx_font_italic" id="S3.p5.6.2">convolution</em> plays a fundamental role in the model’s performance, as illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3.F3.sf1" title="Figure 3(a) ‣ Figure 3 ‣ 3 Does Mamba learn in-context estimators? ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3(a)</span></a>. Here we compare the full Mamba architecture from Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS2" title="2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.2</span></a>, Mamba with just the convolution in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a> removed, and a simplified Mamba architecture with only convolution (<math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mi id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><ci id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">sansserif_MambaZero</annotation></semantics></math> in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS1" title="4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">4.1</span></a>). As a metric of comparison, we use the closeness of each of these models <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mi id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><ci id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">bold_italic_θ</annotation></semantics></math> to the optimal add-<math alttext="\beta" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mi id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_β</annotation></semantics></math> estimator, i.e. i.e. <math alttext="|L(\bm{\theta})-L(\beta)|" class="ltx_Math" display="inline" id="S3.p5.4.m4.3"><semantics id="S3.p5.4.m4.3a"><mrow id="S3.p5.4.m4.3.3.1" xref="S3.p5.4.m4.3.3.2.cmml"><mo id="S3.p5.4.m4.3.3.1.2" stretchy="false" xref="S3.p5.4.m4.3.3.2.1.cmml">|</mo><mrow id="S3.p5.4.m4.3.3.1.1" xref="S3.p5.4.m4.3.3.1.1.cmml"><mrow id="S3.p5.4.m4.3.3.1.1.2" xref="S3.p5.4.m4.3.3.1.1.2.cmml"><mi id="S3.p5.4.m4.3.3.1.1.2.2" xref="S3.p5.4.m4.3.3.1.1.2.2.cmml">L</mi><mo id="S3.p5.4.m4.3.3.1.1.2.1" xref="S3.p5.4.m4.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.p5.4.m4.3.3.1.1.2.3.2" xref="S3.p5.4.m4.3.3.1.1.2.cmml"><mo id="S3.p5.4.m4.3.3.1.1.2.3.2.1" stretchy="false" xref="S3.p5.4.m4.3.3.1.1.2.cmml">(</mo><mi id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml">𝜽</mi><mo id="S3.p5.4.m4.3.3.1.1.2.3.2.2" stretchy="false" xref="S3.p5.4.m4.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.4.m4.3.3.1.1.1" xref="S3.p5.4.m4.3.3.1.1.1.cmml">−</mo><mrow id="S3.p5.4.m4.3.3.1.1.3" xref="S3.p5.4.m4.3.3.1.1.3.cmml"><mi id="S3.p5.4.m4.3.3.1.1.3.2" xref="S3.p5.4.m4.3.3.1.1.3.2.cmml">L</mi><mo id="S3.p5.4.m4.3.3.1.1.3.1" xref="S3.p5.4.m4.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.p5.4.m4.3.3.1.1.3.3.2" xref="S3.p5.4.m4.3.3.1.1.3.cmml"><mo id="S3.p5.4.m4.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.p5.4.m4.3.3.1.1.3.cmml">(</mo><mi id="S3.p5.4.m4.2.2" xref="S3.p5.4.m4.2.2.cmml">β</mi><mo id="S3.p5.4.m4.3.3.1.1.3.3.2.2" stretchy="false" xref="S3.p5.4.m4.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p5.4.m4.3.3.1.3" stretchy="false" xref="S3.p5.4.m4.3.3.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.3b"><apply id="S3.p5.4.m4.3.3.2.cmml" xref="S3.p5.4.m4.3.3.1"><abs id="S3.p5.4.m4.3.3.2.1.cmml" xref="S3.p5.4.m4.3.3.1.2"></abs><apply id="S3.p5.4.m4.3.3.1.1.cmml" xref="S3.p5.4.m4.3.3.1.1"><minus id="S3.p5.4.m4.3.3.1.1.1.cmml" xref="S3.p5.4.m4.3.3.1.1.1"></minus><apply id="S3.p5.4.m4.3.3.1.1.2.cmml" xref="S3.p5.4.m4.3.3.1.1.2"><times id="S3.p5.4.m4.3.3.1.1.2.1.cmml" xref="S3.p5.4.m4.3.3.1.1.2.1"></times><ci id="S3.p5.4.m4.3.3.1.1.2.2.cmml" xref="S3.p5.4.m4.3.3.1.1.2.2">𝐿</ci><ci id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1">𝜽</ci></apply><apply id="S3.p5.4.m4.3.3.1.1.3.cmml" xref="S3.p5.4.m4.3.3.1.1.3"><times id="S3.p5.4.m4.3.3.1.1.3.1.cmml" xref="S3.p5.4.m4.3.3.1.1.3.1"></times><ci id="S3.p5.4.m4.3.3.1.1.3.2.cmml" xref="S3.p5.4.m4.3.3.1.1.3.2">𝐿</ci><ci id="S3.p5.4.m4.2.2.cmml" xref="S3.p5.4.m4.2.2">𝛽</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.3c">|L(\bm{\theta})-L(\beta)|</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.3d">| italic_L ( bold_italic_θ ) - italic_L ( italic_β ) |</annotation></semantics></math>, where <math alttext="L(\beta)" class="ltx_Math" display="inline" id="S3.p5.5.m5.1"><semantics id="S3.p5.5.m5.1a"><mrow id="S3.p5.5.m5.1.2" xref="S3.p5.5.m5.1.2.cmml"><mi id="S3.p5.5.m5.1.2.2" xref="S3.p5.5.m5.1.2.2.cmml">L</mi><mo id="S3.p5.5.m5.1.2.1" xref="S3.p5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.p5.5.m5.1.2.3.2" xref="S3.p5.5.m5.1.2.cmml"><mo id="S3.p5.5.m5.1.2.3.2.1" stretchy="false" xref="S3.p5.5.m5.1.2.cmml">(</mo><mi id="S3.p5.5.m5.1.1" xref="S3.p5.5.m5.1.1.cmml">β</mi><mo id="S3.p5.5.m5.1.2.3.2.2" stretchy="false" xref="S3.p5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.1b"><apply id="S3.p5.5.m5.1.2.cmml" xref="S3.p5.5.m5.1.2"><times id="S3.p5.5.m5.1.2.1.cmml" xref="S3.p5.5.m5.1.2.1"></times><ci id="S3.p5.5.m5.1.2.2.cmml" xref="S3.p5.5.m5.1.2.2">𝐿</ci><ci id="S3.p5.5.m5.1.1.cmml" xref="S3.p5.5.m5.1.1">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.1c">L(\beta)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.1d">italic_L ( italic_β )</annotation></semantics></math> is the optimal loss incurred by the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S3.p5.6.m6.1"><semantics id="S3.p5.6.m6.1a"><mi id="S3.p5.6.m6.1.1" xref="S3.p5.6.m6.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.p5.6.m6.1b"><ci id="S3.p5.6.m6.1.1.cmml" xref="S3.p5.6.m6.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m6.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m6.1d">italic_β</annotation></semantics></math> estimator. Closer this metric is to zero, the better the model’s performance is. Remarkably, the simplified Mamba with just the convolution succeeds on the Markov prediction task, while the full model without convolution fails, highlighting its fundamental importance. This raises a natural question: <em class="ltx_emph ltx_font_italic" id="S3.p5.6.3">how does convolution help Mamba to implement the optimal Laplacian estimator?</em></p> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.1">In the next section, we investigate this theoretically.</p> </div> <figure class="ltx_figure" id="S3.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S3.F3.sf1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="624" id="S3.F3.sf1.g1" src="x3.png" width="830"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S3.F3.sf1.1"><svg height="65.19" overflow="visible" version="1.1" width="400.97"><g transform="translate(0,65.19) scale(1,-1)"><g transform="translate(-188.18,-9.69)"><text fill="black" transform="scale(1, -1)" x="0" y="0">Iteration (<g transform="translate(0,1.383700013837) scale(1, -1)"><foreignobject height="1.383700013837" overflow="visible" width="1.383700013837"><math alttext="\times 200" class="ltx_Math" display="inline" id="S3.F3.sf1.1.pic1.1.1.m1.1"><semantics id="S3.F3.sf1.1.pic1.1.1.m1.1a"><mrow id="S3.F3.sf1.1.pic1.1.1.m1.1.1" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.cmml"><mi id="S3.F3.sf1.1.pic1.1.1.m1.1.1.2" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.2.cmml"></mi><mo id="S3.F3.sf1.1.pic1.1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.1.cmml">×</mo><mn id="S3.F3.sf1.1.pic1.1.1.m1.1.1.3" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.3.cmml">200</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.sf1.1.pic1.1.1.m1.1b"><apply id="S3.F3.sf1.1.pic1.1.1.m1.1.1.cmml" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1"><times id="S3.F3.sf1.1.pic1.1.1.m1.1.1.1.cmml" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.1"></times><csymbol cd="latexml" id="S3.F3.sf1.1.pic1.1.1.m1.1.1.2.cmml" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.2">absent</csymbol><cn id="S3.F3.sf1.1.pic1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.F3.sf1.1.pic1.1.1.m1.1.1.3">200</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.sf1.1.pic1.1.1.m1.1c">\times 200</annotation><annotation encoding="application/x-llamapun" id="S3.F3.sf1.1.pic1.1.1.m1.1d">× 200</annotation></semantics></math></foreignobject></g>)</text></g><g transform="translate(-329.32,119)"><g transform="translate(0,52.1654905216549) scale(1, -1)"><foreignobject height="61.7130206171302" overflow="visible" width="9.5475300954753"> <div class="ltx_inline-block ltx_transformed_outer" id="S3.F3.sf1.1.pic1.5.1" style="width:6.9pt;height:37.7pt;vertical-align:6.9pt;"><span class="ltx_transformed_inner" style="width:37.7pt;transform:translate(-15.36pt,-22.31pt) rotate(-90deg) ;"> <p class="ltx_p" id="S3.F3.sf1.1.pic1.5.1.1">Test loss</p> </span></div></foreignobject></g></g><g transform="translate(-156.36,212.4)"><g transform="translate(0,9.5475300954753) scale(1, -1)"><foreignobject height="9.5475300954753" overflow="visible" width="45.3853604538536"><math alttext="\mathsf{Mamba}" class="ltx_Math" display="inline" id="S3.F3.sf1.1.pic1.2.m1.1"><semantics id="S3.F3.sf1.1.pic1.2.m1.1a"><mi id="S3.F3.sf1.1.pic1.2.m1.1.1" xref="S3.F3.sf1.1.pic1.2.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺</mi><annotation-xml encoding="MathML-Content" id="S3.F3.sf1.1.pic1.2.m1.1b"><ci id="S3.F3.sf1.1.pic1.2.m1.1.1.cmml" xref="S3.F3.sf1.1.pic1.2.m1.1.1">𝖬𝖺𝗆𝖻𝖺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.sf1.1.pic1.2.m1.1c">\mathsf{Mamba}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.sf1.1.pic1.2.m1.1d">sansserif_Mamba</annotation></semantics></math></foreignobject></g></g><g transform="translate(-156.36,197.87)"><g transform="translate(0,9.5475300954753) scale(1, -1)"><foreignobject height="9.5475300954753" overflow="visible" width="72.3675107236751"><math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S3.F3.sf1.1.pic1.3.m1.1"><semantics id="S3.F3.sf1.1.pic1.3.m1.1a"><mi id="S3.F3.sf1.1.pic1.3.m1.1.1" xref="S3.F3.sf1.1.pic1.3.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S3.F3.sf1.1.pic1.3.m1.1b"><ci id="S3.F3.sf1.1.pic1.3.m1.1.1.cmml" xref="S3.F3.sf1.1.pic1.3.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.sf1.1.pic1.3.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.sf1.1.pic1.3.m1.1d">sansserif_MambaZero</annotation></semantics></math></foreignobject></g></g><g transform="translate(-156.36,182.65)"><g transform="translate(0,9.5475300954753) scale(1, -1)"><foreignobject height="9.5475300954753" overflow="visible" width="171.025321710253"><math alttext="\mathsf{Mamba}" class="ltx_Math" display="inline" id="S3.F3.sf1.1.pic1.4.m1.1"><semantics id="S3.F3.sf1.1.pic1.4.m1.1a"><mi id="S3.F3.sf1.1.pic1.4.m1.1.1" xref="S3.F3.sf1.1.pic1.4.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺</mi><annotation-xml encoding="MathML-Content" id="S3.F3.sf1.1.pic1.4.m1.1b"><ci id="S3.F3.sf1.1.pic1.4.m1.1.1.cmml" xref="S3.F3.sf1.1.pic1.4.m1.1.1">𝖬𝖺𝗆𝖻𝖺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.sf1.1.pic1.4.m1.1c">\mathsf{Mamba}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.sf1.1.pic1.4.m1.1d">sansserif_Mamba</annotation></semantics></math></foreignobject></g><text fill="black" transform="scale(1, -1)" x="0" y="0">without convolution</text></g></g></svg></p> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F3.sf1.3.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S3.F3.sf1.4.2" style="font-size:90%;">Importance of convolution</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S3.F3.sf2"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="624" id="S3.F3.sf2.g1" src="x4.png" width="830"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S3.F3.sf2.1"><svg height="61.73" overflow="visible" version="1.1" width="386.98"><g transform="translate(0,61.73) scale(1,-1)"><g transform="translate(-235.23,-9.69)"><text fill="black" transform="scale(1, -1)" x="0" y="0">Convolution window of Mamba</text></g><g transform="translate(-329.32,117.61)"><g transform="translate(0,52.1654905216549) scale(1, -1)"><foreignobject height="61.7130206171302" overflow="visible" width="9.5475300954753"> <div class="ltx_inline-block ltx_transformed_outer" 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transform="translate(-53.96,166.04)"><text fill="black" transform="scale(1, -1)" x="0" y="0">Order <g transform="translate(0,1.383700013837) scale(1, -1)"><foreignobject height="1.383700013837" overflow="visible" width="1.383700013837"><math alttext="4" class="ltx_Math" display="inline" id="S3.F3.sf2.1.pic1.4.1.m1.1"><semantics id="S3.F3.sf2.1.pic1.4.1.m1.1a"><mn id="S3.F3.sf2.1.pic1.4.1.m1.1.1" xref="S3.F3.sf2.1.pic1.4.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S3.F3.sf2.1.pic1.4.1.m1.1b"><cn id="S3.F3.sf2.1.pic1.4.1.m1.1.1.cmml" type="integer" xref="S3.F3.sf2.1.pic1.4.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.sf2.1.pic1.4.1.m1.1c">4</annotation><annotation encoding="application/x-llamapun" id="S3.F3.sf2.1.pic1.4.1.m1.1d">4</annotation></semantics></math></foreignobject></g></text></g></g></svg></p> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F3.sf2.3.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S3.F3.sf2.4.2" style="font-size:90%;">Relation between window size and Markov order</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F3.14.7.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S3.F3.12.6" style="font-size:90%;">(a) illustrates the fundamental role of convolution, without which the model fails to learn the task. In contrast, a simplified variant with just the convolution (<math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S3.F3.7.1.m1.1"><semantics id="S3.F3.7.1.m1.1b"><mi id="S3.F3.7.1.m1.1.1" xref="S3.F3.7.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S3.F3.7.1.m1.1c"><ci id="S3.F3.7.1.m1.1.1.cmml" xref="S3.F3.7.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.7.1.m1.1d">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.7.1.m1.1e">sansserif_MambaZero</annotation></semantics></math>) matches the performance of that of the full model. (b) highlights the relation between the Markov order <math alttext="k" class="ltx_Math" display="inline" id="S3.F3.8.2.m2.1"><semantics id="S3.F3.8.2.m2.1b"><mi id="S3.F3.8.2.m2.1.1" xref="S3.F3.8.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.F3.8.2.m2.1c"><ci id="S3.F3.8.2.m2.1.1.cmml" xref="S3.F3.8.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.8.2.m2.1d">k</annotation><annotation encoding="application/x-llamapun" id="S3.F3.8.2.m2.1e">italic_k</annotation></semantics></math> and the window size <math alttext="w" class="ltx_Math" display="inline" id="S3.F3.9.3.m3.1"><semantics id="S3.F3.9.3.m3.1b"><mi id="S3.F3.9.3.m3.1.1" xref="S3.F3.9.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.F3.9.3.m3.1c"><ci id="S3.F3.9.3.m3.1.1.cmml" xref="S3.F3.9.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.9.3.m3.1d">w</annotation><annotation encoding="application/x-llamapun" id="S3.F3.9.3.m3.1e">italic_w</annotation></semantics></math> of <math alttext="\mathsf{Mamba}" class="ltx_Math" display="inline" id="S3.F3.10.4.m4.1"><semantics id="S3.F3.10.4.m4.1b"><mi id="S3.F3.10.4.m4.1.1" xref="S3.F3.10.4.m4.1.1.cmml">𝖬𝖺𝗆𝖻𝖺</mi><annotation-xml encoding="MathML-Content" id="S3.F3.10.4.m4.1c"><ci id="S3.F3.10.4.m4.1.1.cmml" xref="S3.F3.10.4.m4.1.1">𝖬𝖺𝗆𝖻𝖺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.10.4.m4.1d">\mathsf{Mamba}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.10.4.m4.1e">sansserif_Mamba</annotation></semantics></math>. It is required that <math alttext="w\geq k+1" class="ltx_Math" display="inline" id="S3.F3.11.5.m5.1"><semantics id="S3.F3.11.5.m5.1b"><mrow id="S3.F3.11.5.m5.1.1" xref="S3.F3.11.5.m5.1.1.cmml"><mi id="S3.F3.11.5.m5.1.1.2" xref="S3.F3.11.5.m5.1.1.2.cmml">w</mi><mo id="S3.F3.11.5.m5.1.1.1" xref="S3.F3.11.5.m5.1.1.1.cmml">≥</mo><mrow id="S3.F3.11.5.m5.1.1.3" xref="S3.F3.11.5.m5.1.1.3.cmml"><mi id="S3.F3.11.5.m5.1.1.3.2" xref="S3.F3.11.5.m5.1.1.3.2.cmml">k</mi><mo id="S3.F3.11.5.m5.1.1.3.1" xref="S3.F3.11.5.m5.1.1.3.1.cmml">+</mo><mn id="S3.F3.11.5.m5.1.1.3.3" xref="S3.F3.11.5.m5.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.11.5.m5.1c"><apply id="S3.F3.11.5.m5.1.1.cmml" xref="S3.F3.11.5.m5.1.1"><geq id="S3.F3.11.5.m5.1.1.1.cmml" xref="S3.F3.11.5.m5.1.1.1"></geq><ci id="S3.F3.11.5.m5.1.1.2.cmml" xref="S3.F3.11.5.m5.1.1.2">𝑤</ci><apply id="S3.F3.11.5.m5.1.1.3.cmml" xref="S3.F3.11.5.m5.1.1.3"><plus id="S3.F3.11.5.m5.1.1.3.1.cmml" xref="S3.F3.11.5.m5.1.1.3.1"></plus><ci id="S3.F3.11.5.m5.1.1.3.2.cmml" xref="S3.F3.11.5.m5.1.1.3.2">𝑘</ci><cn id="S3.F3.11.5.m5.1.1.3.3.cmml" type="integer" xref="S3.F3.11.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.11.5.m5.1d">w\geq k+1</annotation><annotation encoding="application/x-llamapun" id="S3.F3.11.5.m5.1e">italic_w ≥ italic_k + 1</annotation></semantics></math> for the model to learn the order-<math alttext="k" class="ltx_Math" display="inline" id="S3.F3.12.6.m6.1"><semantics id="S3.F3.12.6.m6.1b"><mi id="S3.F3.12.6.m6.1.1" xref="S3.F3.12.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.F3.12.6.m6.1c"><ci id="S3.F3.12.6.m6.1.1.cmml" xref="S3.F3.12.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.12.6.m6.1d">k</annotation><annotation encoding="application/x-llamapun" id="S3.F3.12.6.m6.1e">italic_k</annotation></semantics></math> prediction task.</span></figcaption> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Theoretical results</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">Motivated by Mamba’s success in learning the optimal smoothing estimator, and convolution’s pivotal role in it, here we study how it can represent Laplacian smoothing.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>MambaZero: Simplified model</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.2">Building upon the insight that Mamba with just the convolution achieves the same performance as that of the full model (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3.F3.sf1" title="Figure 3(a) ‣ Figure 3 ‣ 3 Does Mamba learn in-context estimators? ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3(a)</span></a>), we consider its simplified version: <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">sansserif_MambaZero</annotation></semantics></math>. <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><mi id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><ci id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">sansserif_MambaZero</annotation></semantics></math> retains only the essential elements of the full model in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS2" title="2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.2</span></a>: the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex3" title="Equation Embedding ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Embedding</span></a> layer, the convolution inside the Mamba block in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a>, and the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex6" title="Equation Linear ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Linear</span></a> layer. More formally, it’s given by:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx4"> <tbody id="S4.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{x}_{t}" class="ltx_Math" display="inline" id="S4.Ex9.m1.1"><semantics id="S4.Ex9.m1.1a"><msub id="S4.Ex9.m1.1.1" xref="S4.Ex9.m1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.2" xref="S4.Ex9.m1.1.1.2.cmml">𝒙</mi><mi id="S4.Ex9.m1.1.1.3" xref="S4.Ex9.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Ex9.m1.1b"><apply id="S4.Ex9.m1.1.1.cmml" xref="S4.Ex9.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex9.m1.1.1.1.cmml" xref="S4.Ex9.m1.1.1">subscript</csymbol><ci id="S4.Ex9.m1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.2">𝒙</ci><ci id="S4.Ex9.m1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.1c">\displaystyle\bm{x}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.1d">bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=x_{t}\,\bm{e}_{1}+(1-x_{t})\,\bm{e}_{0}\in\mathbb{R}^{d}," class="ltx_Math" display="inline" id="S4.Ex9.m2.1"><semantics id="S4.Ex9.m2.1a"><mrow id="S4.Ex9.m2.1.1.1" xref="S4.Ex9.m2.1.1.1.1.cmml"><mrow id="S4.Ex9.m2.1.1.1.1" xref="S4.Ex9.m2.1.1.1.1.cmml"><mi id="S4.Ex9.m2.1.1.1.1.3" xref="S4.Ex9.m2.1.1.1.1.3.cmml"></mi><mo id="S4.Ex9.m2.1.1.1.1.4" xref="S4.Ex9.m2.1.1.1.1.4.cmml">=</mo><mrow id="S4.Ex9.m2.1.1.1.1.1" xref="S4.Ex9.m2.1.1.1.1.1.cmml"><mrow id="S4.Ex9.m2.1.1.1.1.1.3" xref="S4.Ex9.m2.1.1.1.1.1.3.cmml"><msub id="S4.Ex9.m2.1.1.1.1.1.3.2" xref="S4.Ex9.m2.1.1.1.1.1.3.2.cmml"><mi id="S4.Ex9.m2.1.1.1.1.1.3.2.2" xref="S4.Ex9.m2.1.1.1.1.1.3.2.2.cmml">x</mi><mi id="S4.Ex9.m2.1.1.1.1.1.3.2.3" 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id="S4.Ex10.m2.1c">\displaystyle=\bm{x}_{t}+\mathsf{MambaZero}(\bm{x}_{1}^{t})\in\mathbb{R}^{d},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m2.1d">= bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + sansserif_MambaZero ( bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(MambaZero)</span></td> </tr></tbody> <tbody id="S4.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathrm{logit}_{t}" class="ltx_Math" display="inline" id="S4.Ex11.m1.1"><semantics id="S4.Ex11.m1.1a"><msub id="S4.Ex11.m1.1.1" xref="S4.Ex11.m1.1.1.cmml"><mi id="S4.Ex11.m1.1.1.2" xref="S4.Ex11.m1.1.1.2.cmml">logit</mi><mi id="S4.Ex11.m1.1.1.3" xref="S4.Ex11.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Ex11.m1.1b"><apply id="S4.Ex11.m1.1.1.cmml" xref="S4.Ex11.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.1.1.1.cmml" xref="S4.Ex11.m1.1.1">subscript</csymbol><ci id="S4.Ex11.m1.1.1.2.cmml" xref="S4.Ex11.m1.1.1.2">logit</ci><ci id="S4.Ex11.m1.1.1.3.cmml" xref="S4.Ex11.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.1c">\displaystyle\mathrm{logit}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.1d">roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=W_{\ell}\,{\bm{u}_{t}}\in\mathbb{R}^{2}," class="ltx_Math" display="inline" id="S4.Ex11.m2.1"><semantics id="S4.Ex11.m2.1a"><mrow id="S4.Ex11.m2.1.1.1" xref="S4.Ex11.m2.1.1.1.1.cmml"><mrow id="S4.Ex11.m2.1.1.1.1" xref="S4.Ex11.m2.1.1.1.1.cmml"><mi id="S4.Ex11.m2.1.1.1.1.2" xref="S4.Ex11.m2.1.1.1.1.2.cmml"></mi><mo id="S4.Ex11.m2.1.1.1.1.3" xref="S4.Ex11.m2.1.1.1.1.3.cmml">=</mo><mrow id="S4.Ex11.m2.1.1.1.1.4" xref="S4.Ex11.m2.1.1.1.1.4.cmml"><msub id="S4.Ex11.m2.1.1.1.1.4.2" xref="S4.Ex11.m2.1.1.1.1.4.2.cmml"><mi id="S4.Ex11.m2.1.1.1.1.4.2.2" xref="S4.Ex11.m2.1.1.1.1.4.2.2.cmml">W</mi><mi id="S4.Ex11.m2.1.1.1.1.4.2.3" mathvariant="normal" xref="S4.Ex11.m2.1.1.1.1.4.2.3.cmml">ℓ</mi></msub><mo id="S4.Ex11.m2.1.1.1.1.4.1" xref="S4.Ex11.m2.1.1.1.1.4.1.cmml">⁢</mo><msub id="S4.Ex11.m2.1.1.1.1.4.3" xref="S4.Ex11.m2.1.1.1.1.4.3.cmml"><mi id="S4.Ex11.m2.1.1.1.1.4.3.2" xref="S4.Ex11.m2.1.1.1.1.4.3.2.cmml">𝒖</mi><mi id="S4.Ex11.m2.1.1.1.1.4.3.3" xref="S4.Ex11.m2.1.1.1.1.4.3.3.cmml">t</mi></msub></mrow><mo id="S4.Ex11.m2.1.1.1.1.5" xref="S4.Ex11.m2.1.1.1.1.5.cmml">∈</mo><msup id="S4.Ex11.m2.1.1.1.1.6" xref="S4.Ex11.m2.1.1.1.1.6.cmml"><mi id="S4.Ex11.m2.1.1.1.1.6.2" xref="S4.Ex11.m2.1.1.1.1.6.2.cmml">ℝ</mi><mn id="S4.Ex11.m2.1.1.1.1.6.3" xref="S4.Ex11.m2.1.1.1.1.6.3.cmml">2</mn></msup></mrow><mo id="S4.Ex11.m2.1.1.1.2" xref="S4.Ex11.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex11.m2.1b"><apply id="S4.Ex11.m2.1.1.1.1.cmml" xref="S4.Ex11.m2.1.1.1"><and id="S4.Ex11.m2.1.1.1.1a.cmml" xref="S4.Ex11.m2.1.1.1"></and><apply id="S4.Ex11.m2.1.1.1.1b.cmml" xref="S4.Ex11.m2.1.1.1"><eq id="S4.Ex11.m2.1.1.1.1.3.cmml" xref="S4.Ex11.m2.1.1.1.1.3"></eq><csymbol cd="latexml" id="S4.Ex11.m2.1.1.1.1.2.cmml" xref="S4.Ex11.m2.1.1.1.1.2">absent</csymbol><apply id="S4.Ex11.m2.1.1.1.1.4.cmml" xref="S4.Ex11.m2.1.1.1.1.4"><times id="S4.Ex11.m2.1.1.1.1.4.1.cmml" xref="S4.Ex11.m2.1.1.1.1.4.1"></times><apply id="S4.Ex11.m2.1.1.1.1.4.2.cmml" xref="S4.Ex11.m2.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S4.Ex11.m2.1.1.1.1.4.2.1.cmml" xref="S4.Ex11.m2.1.1.1.1.4.2">subscript</csymbol><ci id="S4.Ex11.m2.1.1.1.1.4.2.2.cmml" xref="S4.Ex11.m2.1.1.1.1.4.2.2">𝑊</ci><ci id="S4.Ex11.m2.1.1.1.1.4.2.3.cmml" xref="S4.Ex11.m2.1.1.1.1.4.2.3">ℓ</ci></apply><apply id="S4.Ex11.m2.1.1.1.1.4.3.cmml" xref="S4.Ex11.m2.1.1.1.1.4.3"><csymbol cd="ambiguous" id="S4.Ex11.m2.1.1.1.1.4.3.1.cmml" xref="S4.Ex11.m2.1.1.1.1.4.3">subscript</csymbol><ci id="S4.Ex11.m2.1.1.1.1.4.3.2.cmml" xref="S4.Ex11.m2.1.1.1.1.4.3.2">𝒖</ci><ci id="S4.Ex11.m2.1.1.1.1.4.3.3.cmml" xref="S4.Ex11.m2.1.1.1.1.4.3.3">𝑡</ci></apply></apply></apply><apply id="S4.Ex11.m2.1.1.1.1c.cmml" xref="S4.Ex11.m2.1.1.1"><in id="S4.Ex11.m2.1.1.1.1.5.cmml" xref="S4.Ex11.m2.1.1.1.1.5"></in><share href="https://arxiv.org/html/2502.10178v1#S4.Ex11.m2.1.1.1.1.4.cmml" id="S4.Ex11.m2.1.1.1.1d.cmml" xref="S4.Ex11.m2.1.1.1"></share><apply id="S4.Ex11.m2.1.1.1.1.6.cmml" xref="S4.Ex11.m2.1.1.1.1.6"><csymbol cd="ambiguous" id="S4.Ex11.m2.1.1.1.1.6.1.cmml" xref="S4.Ex11.m2.1.1.1.1.6">superscript</csymbol><ci id="S4.Ex11.m2.1.1.1.1.6.2.cmml" xref="S4.Ex11.m2.1.1.1.1.6.2">ℝ</ci><cn id="S4.Ex11.m2.1.1.1.1.6.3.cmml" type="integer" xref="S4.Ex11.m2.1.1.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m2.1c">\displaystyle=W_{\ell}\,{\bm{u}_{t}}\in\mathbb{R}^{2},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m2.1d">= italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT bold_italic_u start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Linear)</span></td> </tr></tbody> <tbody id="S4.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{\bm{\theta}}(x_{1}^{t})" class="ltx_Math" display="inline" id="S4.Ex12.m1.1"><semantics id="S4.Ex12.m1.1a"><mrow id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml"><msub id="S4.Ex12.m1.1.1.3" xref="S4.Ex12.m1.1.1.3.cmml"><mi id="S4.Ex12.m1.1.1.3.2" xref="S4.Ex12.m1.1.1.3.2.cmml">f</mi><mi id="S4.Ex12.m1.1.1.3.3" xref="S4.Ex12.m1.1.1.3.3.cmml">𝜽</mi></msub><mo id="S4.Ex12.m1.1.1.2" xref="S4.Ex12.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex12.m1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.1.cmml"><mo id="S4.Ex12.m1.1.1.1.1.2" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.1.cmml">(</mo><msubsup id="S4.Ex12.m1.1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.1.1.1.1.1.2.2" xref="S4.Ex12.m1.1.1.1.1.1.2.2.cmml">x</mi><mn id="S4.Ex12.m1.1.1.1.1.1.2.3" xref="S4.Ex12.m1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S4.Ex12.m1.1.1.1.1.1.3" xref="S4.Ex12.m1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S4.Ex12.m1.1.1.1.1.3" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.1b"><apply id="S4.Ex12.m1.1.1.cmml" xref="S4.Ex12.m1.1.1"><times id="S4.Ex12.m1.1.1.2.cmml" xref="S4.Ex12.m1.1.1.2"></times><apply id="S4.Ex12.m1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.3.1.cmml" xref="S4.Ex12.m1.1.1.3">subscript</csymbol><ci id="S4.Ex12.m1.1.1.3.2.cmml" xref="S4.Ex12.m1.1.1.3.2">𝑓</ci><ci id="S4.Ex12.m1.1.1.3.3.cmml" xref="S4.Ex12.m1.1.1.3.3">𝜽</ci></apply><apply id="S4.Ex12.m1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1">superscript</csymbol><apply id="S4.Ex12.m1.1.1.1.1.1.2.cmml" xref="S4.Ex12.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.1.1.2.1.cmml" xref="S4.Ex12.m1.1.1.1.1">subscript</csymbol><ci id="S4.Ex12.m1.1.1.1.1.1.2.2.cmml" xref="S4.Ex12.m1.1.1.1.1.1.2.2">𝑥</ci><cn id="S4.Ex12.m1.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.Ex12.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="S4.Ex12.m1.1.1.1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.1c">\displaystyle f_{\bm{\theta}}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.1d">italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\triangleq\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=\cdot\mid x_{1}^{% t}\right)=\left(\mathrm{logit}_{t}/\|\mathrm{logit}_{t}\|_{1}\right)\in[0,1]^{% 2}," class="ltx_math_unparsed" display="inline" id="S4.Ex12.m2.1"><semantics id="S4.Ex12.m2.1a"><mrow id="S4.Ex12.m2.1b"><mo id="S4.Ex12.m2.1.1">≜</mo><msub id="S4.Ex12.m2.1.2"><mi id="S4.Ex12.m2.1.2.2">ℙ</mi><mi id="S4.Ex12.m2.1.2.3">𝜽</mi></msub><mrow id="S4.Ex12.m2.1.3"><mo id="S4.Ex12.m2.1.3.1">(</mo><msub id="S4.Ex12.m2.1.3.2"><mi id="S4.Ex12.m2.1.3.2.2">x</mi><mrow id="S4.Ex12.m2.1.3.2.3"><mi id="S4.Ex12.m2.1.3.2.3.2">t</mi><mo id="S4.Ex12.m2.1.3.2.3.1">+</mo><mn id="S4.Ex12.m2.1.3.2.3.3">1</mn></mrow></msub><mo id="S4.Ex12.m2.1.3.3" rspace="0em">=</mo><mo id="S4.Ex12.m2.1.3.4" lspace="0em" rspace="0em">⋅</mo><mo id="S4.Ex12.m2.1.3.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S4.Ex12.m2.1.3.6"><mi id="S4.Ex12.m2.1.3.6.2.2">x</mi><mn id="S4.Ex12.m2.1.3.6.2.3">1</mn><mi id="S4.Ex12.m2.1.3.6.3">t</mi></msubsup><mo id="S4.Ex12.m2.1.3.7">)</mo></mrow><mo id="S4.Ex12.m2.1.4">=</mo><mrow id="S4.Ex12.m2.1.5"><mo id="S4.Ex12.m2.1.5.1">(</mo><msub id="S4.Ex12.m2.1.5.2"><mi id="S4.Ex12.m2.1.5.2.2">logit</mi><mi id="S4.Ex12.m2.1.5.2.3">t</mi></msub><mo id="S4.Ex12.m2.1.5.3" rspace="0em">/</mo><mo id="S4.Ex12.m2.1.5.4" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.Ex12.m2.1.5.5"><mi id="S4.Ex12.m2.1.5.5.2">logit</mi><mi id="S4.Ex12.m2.1.5.5.3">t</mi></msub><msub id="S4.Ex12.m2.1.5.6"><mo id="S4.Ex12.m2.1.5.6.2" lspace="0em" rspace="0.167em">∥</mo><mn id="S4.Ex12.m2.1.5.6.3">1</mn></msub><mo id="S4.Ex12.m2.1.5.7">)</mo></mrow><mo id="S4.Ex12.m2.1.6">∈</mo><msup id="S4.Ex12.m2.1.7"><mrow id="S4.Ex12.m2.1.7.2"><mo id="S4.Ex12.m2.1.7.2.1" stretchy="false">[</mo><mn id="S4.Ex12.m2.1.7.2.2">0</mn><mo id="S4.Ex12.m2.1.7.2.3">,</mo><mn id="S4.Ex12.m2.1.7.2.4">1</mn><mo id="S4.Ex12.m2.1.7.2.5" stretchy="false">]</mo></mrow><mn id="S4.Ex12.m2.1.7.3">2</mn></msup><mo id="S4.Ex12.m2.1.8">,</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex12.m2.1c">\displaystyle\triangleq\mathbb{P}_{\bm{\theta}}\left(x_{t+1}=\cdot\mid x_{1}^{% t}\right)=\left(\mathrm{logit}_{t}/\|\mathrm{logit}_{t}\|_{1}\right)\in[0,1]^{% 2},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m2.1d">≜ blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = ( roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT / ∥ roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Prediction)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.3">where the <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m1.1"><semantics id="S4.SS1.p1.3.m1.1a"><mi id="S4.SS1.p1.3.m1.1.1" xref="S4.SS1.p1.3.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m1.1b"><ci id="S4.SS1.p1.3.m1.1.1.cmml" xref="S4.SS1.p1.3.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m1.1d">sansserif_MambaZero</annotation></semantics></math> block is</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}H_{t}&=a_{t}H_{t-1}+\widetilde{\bm{x}}_{t}\,\bm{b}_{t}^{\top}\in% \mathbb{R}^{ed\times N},\\ \bm{y}_{t}&=H_{t}\,\bm{c}_{t}\in\mathbb{R}^{ed},\\ \bm{o}_{t}&=W_{o}\,\bm{y}_{t}\in\mathbb{R}^{d},\end{split}" class="ltx_Math" display="block" id="S4.Ex13.m1.43"><semantics id="S4.Ex13.m1.43a"><mtable columnspacing="0pt" displaystyle="true" id="S4.Ex13.m1.43.43.4" rowspacing="0pt"><mtr id="S4.Ex13.m1.43.43.4a"><mtd class="ltx_align_right" columnalign="right" id="S4.Ex13.m1.43.43.4b"><msub id="S4.Ex13.m1.2.2.2.2.2"><mi id="S4.Ex13.m1.1.1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.1.1.cmml">H</mi><mi id="S4.Ex13.m1.2.2.2.2.2.2.1" xref="S4.Ex13.m1.2.2.2.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex13.m1.43.43.4c"><mrow id="S4.Ex13.m1.41.41.2.40.18.16.16"><mrow id="S4.Ex13.m1.41.41.2.40.18.16.16.1"><mi id="S4.Ex13.m1.41.41.2.40.18.16.16.1.2"></mi><mo id="S4.Ex13.m1.3.3.3.3.1.1" xref="S4.Ex13.m1.3.3.3.3.1.1.cmml">=</mo><mrow id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3"><mrow id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.1"><msub id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.1.2"><mi id="S4.Ex13.m1.4.4.4.4.2.2" xref="S4.Ex13.m1.4.4.4.4.2.2.cmml">a</mi><mi id="S4.Ex13.m1.5.5.5.5.3.3.1" xref="S4.Ex13.m1.5.5.5.5.3.3.1.cmml">t</mi></msub><mo id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.1.1">⁢</mo><msub id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.1.3"><mi id="S4.Ex13.m1.6.6.6.6.4.4" xref="S4.Ex13.m1.6.6.6.6.4.4.cmml">H</mi><mrow id="S4.Ex13.m1.7.7.7.7.5.5.1" xref="S4.Ex13.m1.7.7.7.7.5.5.1.cmml"><mi id="S4.Ex13.m1.7.7.7.7.5.5.1.2" xref="S4.Ex13.m1.7.7.7.7.5.5.1.2.cmml">t</mi><mo id="S4.Ex13.m1.7.7.7.7.5.5.1.1" xref="S4.Ex13.m1.7.7.7.7.5.5.1.1.cmml">−</mo><mn id="S4.Ex13.m1.7.7.7.7.5.5.1.3" xref="S4.Ex13.m1.7.7.7.7.5.5.1.3.cmml">1</mn></mrow></msub></mrow><mo id="S4.Ex13.m1.8.8.8.8.6.6" xref="S4.Ex13.m1.8.8.8.8.6.6.cmml">+</mo><mrow id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.2"><msub id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.2.2"><mover accent="true" id="S4.Ex13.m1.9.9.9.9.7.7" xref="S4.Ex13.m1.9.9.9.9.7.7.cmml"><mi id="S4.Ex13.m1.9.9.9.9.7.7.2" xref="S4.Ex13.m1.9.9.9.9.7.7.2.cmml">𝒙</mi><mo id="S4.Ex13.m1.9.9.9.9.7.7.1" xref="S4.Ex13.m1.9.9.9.9.7.7.1.cmml">~</mo></mover><mi id="S4.Ex13.m1.10.10.10.10.8.8.1" xref="S4.Ex13.m1.10.10.10.10.8.8.1.cmml">t</mi></msub><mo id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.2.1">⁢</mo><msubsup id="S4.Ex13.m1.41.41.2.40.18.16.16.1.3.2.3"><mi id="S4.Ex13.m1.11.11.11.11.9.9" xref="S4.Ex13.m1.11.11.11.11.9.9.cmml">𝒃</mi><mi id="S4.Ex13.m1.12.12.12.12.10.10.1" xref="S4.Ex13.m1.12.12.12.12.10.10.1.cmml">t</mi><mo id="S4.Ex13.m1.13.13.13.13.11.11.1" xref="S4.Ex13.m1.13.13.13.13.11.11.1.cmml">⊤</mo></msubsup></mrow></mrow><mo id="S4.Ex13.m1.14.14.14.14.12.12" xref="S4.Ex13.m1.14.14.14.14.12.12.cmml">∈</mo><msup id="S4.Ex13.m1.41.41.2.40.18.16.16.1.4"><mi id="S4.Ex13.m1.15.15.15.15.13.13" xref="S4.Ex13.m1.15.15.15.15.13.13.cmml">ℝ</mi><mrow id="S4.Ex13.m1.16.16.16.16.14.14.1" xref="S4.Ex13.m1.16.16.16.16.14.14.1.cmml"><mrow id="S4.Ex13.m1.16.16.16.16.14.14.1.2" xref="S4.Ex13.m1.16.16.16.16.14.14.1.2.cmml"><mi id="S4.Ex13.m1.16.16.16.16.14.14.1.2.2" xref="S4.Ex13.m1.16.16.16.16.14.14.1.2.2.cmml">e</mi><mo id="S4.Ex13.m1.16.16.16.16.14.14.1.2.1" xref="S4.Ex13.m1.16.16.16.16.14.14.1.2.1.cmml">⁢</mo><mi id="S4.Ex13.m1.16.16.16.16.14.14.1.2.3" xref="S4.Ex13.m1.16.16.16.16.14.14.1.2.3.cmml">d</mi></mrow><mo id="S4.Ex13.m1.16.16.16.16.14.14.1.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex13.m1.16.16.16.16.14.14.1.1.cmml">×</mo><mi id="S4.Ex13.m1.16.16.16.16.14.14.1.3" xref="S4.Ex13.m1.16.16.16.16.14.14.1.3.cmml">N</mi></mrow></msup></mrow><mo id="S4.Ex13.m1.17.17.17.17.15.15">,</mo></mrow></mtd></mtr><mtr id="S4.Ex13.m1.43.43.4d"><mtd class="ltx_align_right" columnalign="right" id="S4.Ex13.m1.43.43.4e"><msub id="S4.Ex13.m1.19.19.19.2.2"><mi id="S4.Ex13.m1.18.18.18.1.1.1" xref="S4.Ex13.m1.18.18.18.1.1.1.cmml">𝒚</mi><mi id="S4.Ex13.m1.19.19.19.2.2.2.1" xref="S4.Ex13.m1.19.19.19.2.2.2.1.cmml">t</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex13.m1.43.43.4f"><mrow id="S4.Ex13.m1.42.42.3.41.12.10.10"><mrow id="S4.Ex13.m1.42.42.3.41.12.10.10.1"><mi id="S4.Ex13.m1.42.42.3.41.12.10.10.1.2"></mi><mo id="S4.Ex13.m1.20.20.20.3.1.1" xref="S4.Ex13.m1.20.20.20.3.1.1.cmml">=</mo><mrow id="S4.Ex13.m1.42.42.3.41.12.10.10.1.3"><msub id="S4.Ex13.m1.42.42.3.41.12.10.10.1.3.2"><mi id="S4.Ex13.m1.21.21.21.4.2.2" xref="S4.Ex13.m1.21.21.21.4.2.2.cmml">H</mi><mi id="S4.Ex13.m1.22.22.22.5.3.3.1" xref="S4.Ex13.m1.22.22.22.5.3.3.1.cmml">t</mi></msub><mo id="S4.Ex13.m1.42.42.3.41.12.10.10.1.3.1">⁢</mo><msub id="S4.Ex13.m1.42.42.3.41.12.10.10.1.3.3"><mi id="S4.Ex13.m1.23.23.23.6.4.4" xref="S4.Ex13.m1.23.23.23.6.4.4.cmml">𝒄</mi><mi id="S4.Ex13.m1.24.24.24.7.5.5.1" xref="S4.Ex13.m1.24.24.24.7.5.5.1.cmml">t</mi></msub></mrow><mo id="S4.Ex13.m1.25.25.25.8.6.6" xref="S4.Ex13.m1.25.25.25.8.6.6.cmml">∈</mo><msup id="S4.Ex13.m1.42.42.3.41.12.10.10.1.4"><mi id="S4.Ex13.m1.26.26.26.9.7.7" xref="S4.Ex13.m1.26.26.26.9.7.7.cmml">ℝ</mi><mrow id="S4.Ex13.m1.27.27.27.10.8.8.1" xref="S4.Ex13.m1.27.27.27.10.8.8.1.cmml"><mi id="S4.Ex13.m1.27.27.27.10.8.8.1.2" xref="S4.Ex13.m1.27.27.27.10.8.8.1.2.cmml">e</mi><mo id="S4.Ex13.m1.27.27.27.10.8.8.1.1" xref="S4.Ex13.m1.27.27.27.10.8.8.1.1.cmml">⁢</mo><mi id="S4.Ex13.m1.27.27.27.10.8.8.1.3" 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end_POSTSUBSCRIPT bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_e italic_d end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_CELL start_CELL = italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(MambaZero)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.9">and the input-selective terms <math alttext="a_{t},\widetilde{\bm{x}}_{t},\bm{b}_{t}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m1.3"><semantics id="S4.SS1.p1.4.m1.3a"><mrow id="S4.SS1.p1.4.m1.3.3.3" 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xref="S4.SS1.p1.4.m1.3.3.3.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m1.3b"><list id="S4.SS1.p1.4.m1.3.3.4.cmml" xref="S4.SS1.p1.4.m1.3.3.3"><apply id="S4.SS1.p1.4.m1.1.1.1.1.cmml" xref="S4.SS1.p1.4.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m1.1.1.1.1.1.cmml" xref="S4.SS1.p1.4.m1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m1.1.1.1.1.2.cmml" xref="S4.SS1.p1.4.m1.1.1.1.1.2">𝑎</ci><ci id="S4.SS1.p1.4.m1.1.1.1.1.3.cmml" xref="S4.SS1.p1.4.m1.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS1.p1.4.m1.2.2.2.2.cmml" xref="S4.SS1.p1.4.m1.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m1.2.2.2.2.1.cmml" xref="S4.SS1.p1.4.m1.2.2.2.2">subscript</csymbol><apply id="S4.SS1.p1.4.m1.2.2.2.2.2.cmml" xref="S4.SS1.p1.4.m1.2.2.2.2.2"><ci id="S4.SS1.p1.4.m1.2.2.2.2.2.1.cmml" xref="S4.SS1.p1.4.m1.2.2.2.2.2.1">~</ci><ci id="S4.SS1.p1.4.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.p1.4.m1.2.2.2.2.2.2">𝒙</ci></apply><ci id="S4.SS1.p1.4.m1.2.2.2.2.3.cmml" xref="S4.SS1.p1.4.m1.2.2.2.2.3">𝑡</ci></apply><apply id="S4.SS1.p1.4.m1.3.3.3.3.cmml" xref="S4.SS1.p1.4.m1.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m1.3.3.3.3.1.cmml" xref="S4.SS1.p1.4.m1.3.3.3.3">subscript</csymbol><ci id="S4.SS1.p1.4.m1.3.3.3.3.2.cmml" xref="S4.SS1.p1.4.m1.3.3.3.3.2">𝒃</ci><ci id="S4.SS1.p1.4.m1.3.3.3.3.3.cmml" xref="S4.SS1.p1.4.m1.3.3.3.3.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m1.3c">a_{t},\widetilde{\bm{x}}_{t},\bm{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m1.3d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m2.1"><semantics id="S4.SS1.p1.5.m2.1a"><msub id="S4.SS1.p1.5.m2.1.1" xref="S4.SS1.p1.5.m2.1.1.cmml"><mi id="S4.SS1.p1.5.m2.1.1.2" xref="S4.SS1.p1.5.m2.1.1.2.cmml">𝒄</mi><mi id="S4.SS1.p1.5.m2.1.1.3" xref="S4.SS1.p1.5.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m2.1b"><apply id="S4.SS1.p1.5.m2.1.1.cmml" xref="S4.SS1.p1.5.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.5.m2.1.1.1.cmml" xref="S4.SS1.p1.5.m2.1.1">subscript</csymbol><ci id="S4.SS1.p1.5.m2.1.1.2.cmml" xref="S4.SS1.p1.5.m2.1.1.2">𝒄</ci><ci id="S4.SS1.p1.5.m2.1.1.3.cmml" xref="S4.SS1.p1.5.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m2.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m2.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> are computed as in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a> without ReLU and just the convolution. Here we use the <math alttext="L_{1}" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m3.1"><semantics id="S4.SS1.p1.6.m3.1a"><msub id="S4.SS1.p1.6.m3.1.1" xref="S4.SS1.p1.6.m3.1.1.cmml"><mi id="S4.SS1.p1.6.m3.1.1.2" xref="S4.SS1.p1.6.m3.1.1.2.cmml">L</mi><mn id="S4.SS1.p1.6.m3.1.1.3" xref="S4.SS1.p1.6.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m3.1b"><apply id="S4.SS1.p1.6.m3.1.1.cmml" xref="S4.SS1.p1.6.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.6.m3.1.1.1.cmml" xref="S4.SS1.p1.6.m3.1.1">subscript</csymbol><ci id="S4.SS1.p1.6.m3.1.1.2.cmml" xref="S4.SS1.p1.6.m3.1.1.2">𝐿</ci><cn id="S4.SS1.p1.6.m3.1.1.3.cmml" type="integer" xref="S4.SS1.p1.6.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m3.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m3.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> normalization instead of the <math alttext="\mathrm{softmax}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m4.1"><semantics id="S4.SS1.p1.7.m4.1a"><mi id="S4.SS1.p1.7.m4.1.1" xref="S4.SS1.p1.7.m4.1.1.cmml">softmax</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m4.1b"><ci id="S4.SS1.p1.7.m4.1.1.cmml" xref="S4.SS1.p1.7.m4.1.1">softmax</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m4.1c">\mathrm{softmax}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m4.1d">roman_softmax</annotation></semantics></math> in the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex12" title="Equation Prediction ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Prediction</span></a> layer ease of theoretical analysis, similar to <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">nic2024trans</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">rajaraman2024transformersmarkovdataconstant</span>]</cite>. Let <math alttext="\bm{\theta}=(\bm{e}_{0},\bm{e}_{1},\bm{\theta}_{\mathsf{MambaZero}},W_{\ell})% \in\mathbb{R}^{D}" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m5.4"><semantics id="S4.SS1.p1.8.m5.4a"><mrow id="S4.SS1.p1.8.m5.4.4" xref="S4.SS1.p1.8.m5.4.4.cmml"><mi id="S4.SS1.p1.8.m5.4.4.6" xref="S4.SS1.p1.8.m5.4.4.6.cmml">𝜽</mi><mo id="S4.SS1.p1.8.m5.4.4.7" xref="S4.SS1.p1.8.m5.4.4.7.cmml">=</mo><mrow id="S4.SS1.p1.8.m5.4.4.4.4" xref="S4.SS1.p1.8.m5.4.4.4.5.cmml"><mo id="S4.SS1.p1.8.m5.4.4.4.4.5" stretchy="false" xref="S4.SS1.p1.8.m5.4.4.4.5.cmml">(</mo><msub id="S4.SS1.p1.8.m5.1.1.1.1.1" xref="S4.SS1.p1.8.m5.1.1.1.1.1.cmml"><mi id="S4.SS1.p1.8.m5.1.1.1.1.1.2" xref="S4.SS1.p1.8.m5.1.1.1.1.1.2.cmml">𝒆</mi><mn id="S4.SS1.p1.8.m5.1.1.1.1.1.3" xref="S4.SS1.p1.8.m5.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.SS1.p1.8.m5.4.4.4.4.6" xref="S4.SS1.p1.8.m5.4.4.4.5.cmml">,</mo><msub id="S4.SS1.p1.8.m5.2.2.2.2.2" xref="S4.SS1.p1.8.m5.2.2.2.2.2.cmml"><mi id="S4.SS1.p1.8.m5.2.2.2.2.2.2" xref="S4.SS1.p1.8.m5.2.2.2.2.2.2.cmml">𝒆</mi><mn id="S4.SS1.p1.8.m5.2.2.2.2.2.3" xref="S4.SS1.p1.8.m5.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.SS1.p1.8.m5.4.4.4.4.7" xref="S4.SS1.p1.8.m5.4.4.4.5.cmml">,</mo><msub id="S4.SS1.p1.8.m5.3.3.3.3.3" xref="S4.SS1.p1.8.m5.3.3.3.3.3.cmml"><mi id="S4.SS1.p1.8.m5.3.3.3.3.3.2" xref="S4.SS1.p1.8.m5.3.3.3.3.3.2.cmml">𝜽</mi><mi id="S4.SS1.p1.8.m5.3.3.3.3.3.3" xref="S4.SS1.p1.8.m5.3.3.3.3.3.3.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi></msub><mo id="S4.SS1.p1.8.m5.4.4.4.4.8" xref="S4.SS1.p1.8.m5.4.4.4.5.cmml">,</mo><msub id="S4.SS1.p1.8.m5.4.4.4.4.4" xref="S4.SS1.p1.8.m5.4.4.4.4.4.cmml"><mi id="S4.SS1.p1.8.m5.4.4.4.4.4.2" xref="S4.SS1.p1.8.m5.4.4.4.4.4.2.cmml">W</mi><mi id="S4.SS1.p1.8.m5.4.4.4.4.4.3" mathvariant="normal" xref="S4.SS1.p1.8.m5.4.4.4.4.4.3.cmml">ℓ</mi></msub><mo id="S4.SS1.p1.8.m5.4.4.4.4.9" stretchy="false" xref="S4.SS1.p1.8.m5.4.4.4.5.cmml">)</mo></mrow><mo id="S4.SS1.p1.8.m5.4.4.8" xref="S4.SS1.p1.8.m5.4.4.8.cmml">∈</mo><msup id="S4.SS1.p1.8.m5.4.4.9" 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id="S4.SS1.p1.8.m5.4.4.4.4.4.2.cmml" xref="S4.SS1.p1.8.m5.4.4.4.4.4.2">𝑊</ci><ci id="S4.SS1.p1.8.m5.4.4.4.4.4.3.cmml" xref="S4.SS1.p1.8.m5.4.4.4.4.4.3">ℓ</ci></apply></vector></apply><apply id="S4.SS1.p1.8.m5.4.4c.cmml" xref="S4.SS1.p1.8.m5.4.4"><in id="S4.SS1.p1.8.m5.4.4.8.cmml" xref="S4.SS1.p1.8.m5.4.4.8"></in><share href="https://arxiv.org/html/2502.10178v1#S4.SS1.p1.8.m5.4.4.4.cmml" id="S4.SS1.p1.8.m5.4.4d.cmml" xref="S4.SS1.p1.8.m5.4.4"></share><apply id="S4.SS1.p1.8.m5.4.4.9.cmml" xref="S4.SS1.p1.8.m5.4.4.9"><csymbol cd="ambiguous" id="S4.SS1.p1.8.m5.4.4.9.1.cmml" xref="S4.SS1.p1.8.m5.4.4.9">superscript</csymbol><ci id="S4.SS1.p1.8.m5.4.4.9.2.cmml" xref="S4.SS1.p1.8.m5.4.4.9.2">ℝ</ci><ci id="S4.SS1.p1.8.m5.4.4.9.3.cmml" xref="S4.SS1.p1.8.m5.4.4.9.3">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m5.4c">\bm{\theta}=(\bm{e}_{0},\bm{e}_{1},\bm{\theta}_{\mathsf{MambaZero}},W_{\ell})% \in\mathbb{R}^{D}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m5.4d">bold_italic_θ = ( bold_italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT sansserif_MambaZero end_POSTSUBSCRIPT , italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT</annotation></semantics></math> denote the full set of parameters for appropriate <math alttext="D\geq 1" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m6.1"><semantics id="S4.SS1.p1.9.m6.1a"><mrow id="S4.SS1.p1.9.m6.1.1" xref="S4.SS1.p1.9.m6.1.1.cmml"><mi id="S4.SS1.p1.9.m6.1.1.2" xref="S4.SS1.p1.9.m6.1.1.2.cmml">D</mi><mo id="S4.SS1.p1.9.m6.1.1.1" xref="S4.SS1.p1.9.m6.1.1.1.cmml">≥</mo><mn id="S4.SS1.p1.9.m6.1.1.3" xref="S4.SS1.p1.9.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m6.1b"><apply id="S4.SS1.p1.9.m6.1.1.cmml" xref="S4.SS1.p1.9.m6.1.1"><geq id="S4.SS1.p1.9.m6.1.1.1.cmml" xref="S4.SS1.p1.9.m6.1.1.1"></geq><ci id="S4.SS1.p1.9.m6.1.1.2.cmml" xref="S4.SS1.p1.9.m6.1.1.2">𝐷</ci><cn id="S4.SS1.p1.9.m6.1.1.3.cmml" type="integer" xref="S4.SS1.p1.9.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m6.1c">D\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m6.1d">italic_D ≥ 1</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Main theorem for first-order Markov</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">Towards establishing how <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">sansserif_MambaZero</annotation></semantics></math> can represent Laplacian smoothing, we start with the random first-order Markov sequences. Our main result is the following.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem1.3.1.1">Theorem 1</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem1.4.2"> </span>(<math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="Thmtheorem1.1.m1.1"><semantics id="Thmtheorem1.1.m1.1b"><mi id="Thmtheorem1.1.m1.1.1" xref="Thmtheorem1.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.1.m1.1c"><ci id="Thmtheorem1.1.m1.1.1.cmml" xref="Thmtheorem1.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.1.m1.1d">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.1.m1.1e">sansserif_MambaZero</annotation></semantics></math> represents order-<math alttext="1" class="ltx_Math" display="inline" id="Thmtheorem1.2.m2.1"><semantics id="Thmtheorem1.2.m2.1b"><mn id="Thmtheorem1.2.m2.1.1" xref="Thmtheorem1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="Thmtheorem1.2.m2.1c"><cn id="Thmtheorem1.2.m2.1.1.cmml" type="integer" xref="Thmtheorem1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.2.m2.1d">1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.2.m2.1e">1</annotation></semantics></math> Laplacian smoothing)<span class="ltx_text ltx_font_bold" id="Thmtheorem1.5.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem1.p1"> <p class="ltx_p" id="Thmtheorem1.p1.9"><span class="ltx_text ltx_font_italic" id="Thmtheorem1.p1.9.9">For the canonical <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="Thmtheorem1.p1.1.1.m1.1"><semantics id="Thmtheorem1.p1.1.1.m1.1a"><mi id="Thmtheorem1.p1.1.1.m1.1.1" xref="Thmtheorem1.p1.1.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.1.1.m1.1b"><ci id="Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="Thmtheorem1.p1.1.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.1.1.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.1.1.m1.1d">sansserif_MambaZero</annotation></semantics></math> model with dimensions <math alttext="d=N=2" class="ltx_Math" display="inline" id="Thmtheorem1.p1.2.2.m2.1"><semantics id="Thmtheorem1.p1.2.2.m2.1a"><mrow id="Thmtheorem1.p1.2.2.m2.1.1" xref="Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem1.p1.2.2.m2.1.1.2" xref="Thmtheorem1.p1.2.2.m2.1.1.2.cmml">d</mi><mo id="Thmtheorem1.p1.2.2.m2.1.1.3" xref="Thmtheorem1.p1.2.2.m2.1.1.3.cmml">=</mo><mi id="Thmtheorem1.p1.2.2.m2.1.1.4" xref="Thmtheorem1.p1.2.2.m2.1.1.4.cmml">N</mi><mo id="Thmtheorem1.p1.2.2.m2.1.1.5" xref="Thmtheorem1.p1.2.2.m2.1.1.5.cmml">=</mo><mn id="Thmtheorem1.p1.2.2.m2.1.1.6" xref="Thmtheorem1.p1.2.2.m2.1.1.6.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.2.2.m2.1b"><apply id="Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1"><and id="Thmtheorem1.p1.2.2.m2.1.1a.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1"></and><apply id="Thmtheorem1.p1.2.2.m2.1.1b.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1"><eq id="Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1.3"></eq><ci id="Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1.2">𝑑</ci><ci id="Thmtheorem1.p1.2.2.m2.1.1.4.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1.4">𝑁</ci></apply><apply id="Thmtheorem1.p1.2.2.m2.1.1c.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1"><eq id="Thmtheorem1.p1.2.2.m2.1.1.5.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1.5"></eq><share href="https://arxiv.org/html/2502.10178v1#Thmtheorem1.p1.2.2.m2.1.1.4.cmml" id="Thmtheorem1.p1.2.2.m2.1.1d.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1"></share><cn id="Thmtheorem1.p1.2.2.m2.1.1.6.cmml" type="integer" xref="Thmtheorem1.p1.2.2.m2.1.1.6">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.2.2.m2.1c">d=N=2</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.2.2.m2.1d">italic_d = italic_N = 2</annotation></semantics></math>, <math alttext="e=1" class="ltx_Math" display="inline" id="Thmtheorem1.p1.3.3.m3.1"><semantics id="Thmtheorem1.p1.3.3.m3.1a"><mrow id="Thmtheorem1.p1.3.3.m3.1.1" xref="Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="Thmtheorem1.p1.3.3.m3.1.1.2" xref="Thmtheorem1.p1.3.3.m3.1.1.2.cmml">e</mi><mo id="Thmtheorem1.p1.3.3.m3.1.1.1" xref="Thmtheorem1.p1.3.3.m3.1.1.1.cmml">=</mo><mn id="Thmtheorem1.p1.3.3.m3.1.1.3" xref="Thmtheorem1.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.3.3.m3.1b"><apply id="Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="Thmtheorem1.p1.3.3.m3.1.1"><eq id="Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem1.p1.3.3.m3.1.1.1"></eq><ci id="Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem1.p1.3.3.m3.1.1.2">𝑒</ci><cn id="Thmtheorem1.p1.3.3.m3.1.1.3.cmml" type="integer" xref="Thmtheorem1.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.3.3.m3.1c">e=1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.3.3.m3.1d">italic_e = 1</annotation></semantics></math>, and convolution window <math alttext="w=2" class="ltx_Math" display="inline" id="Thmtheorem1.p1.4.4.m4.1"><semantics id="Thmtheorem1.p1.4.4.m4.1a"><mrow id="Thmtheorem1.p1.4.4.m4.1.1" xref="Thmtheorem1.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem1.p1.4.4.m4.1.1.2" xref="Thmtheorem1.p1.4.4.m4.1.1.2.cmml">w</mi><mo id="Thmtheorem1.p1.4.4.m4.1.1.1" xref="Thmtheorem1.p1.4.4.m4.1.1.1.cmml">=</mo><mn id="Thmtheorem1.p1.4.4.m4.1.1.3" xref="Thmtheorem1.p1.4.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.4.4.m4.1b"><apply id="Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="Thmtheorem1.p1.4.4.m4.1.1"><eq id="Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem1.p1.4.4.m4.1.1.1"></eq><ci id="Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem1.p1.4.4.m4.1.1.2">𝑤</ci><cn id="Thmtheorem1.p1.4.4.m4.1.1.3.cmml" type="integer" xref="Thmtheorem1.p1.4.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.4.4.m4.1c">w=2</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.4.4.m4.1d">italic_w = 2</annotation></semantics></math>, there is a choice of parameters such that the model prediction is arbitrarily close to the Laplacian estimator for random first-order Markov chains. More formally, for any <math alttext="\beta>0" class="ltx_Math" display="inline" id="Thmtheorem1.p1.5.5.m5.1"><semantics id="Thmtheorem1.p1.5.5.m5.1a"><mrow id="Thmtheorem1.p1.5.5.m5.1.1" xref="Thmtheorem1.p1.5.5.m5.1.1.cmml"><mi id="Thmtheorem1.p1.5.5.m5.1.1.2" xref="Thmtheorem1.p1.5.5.m5.1.1.2.cmml">β</mi><mo id="Thmtheorem1.p1.5.5.m5.1.1.1" xref="Thmtheorem1.p1.5.5.m5.1.1.1.cmml">></mo><mn id="Thmtheorem1.p1.5.5.m5.1.1.3" xref="Thmtheorem1.p1.5.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.5.5.m5.1b"><apply id="Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="Thmtheorem1.p1.5.5.m5.1.1"><gt id="Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem1.p1.5.5.m5.1.1.1"></gt><ci id="Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem1.p1.5.5.m5.1.1.2">𝛽</ci><cn id="Thmtheorem1.p1.5.5.m5.1.1.3.cmml" type="integer" xref="Thmtheorem1.p1.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.5.5.m5.1c">\beta>0</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.5.5.m5.1d">italic_β > 0</annotation></semantics></math> and <math alttext="\epsilon\in(0,1)" class="ltx_Math" display="inline" id="Thmtheorem1.p1.6.6.m6.2"><semantics id="Thmtheorem1.p1.6.6.m6.2a"><mrow id="Thmtheorem1.p1.6.6.m6.2.3" xref="Thmtheorem1.p1.6.6.m6.2.3.cmml"><mi id="Thmtheorem1.p1.6.6.m6.2.3.2" xref="Thmtheorem1.p1.6.6.m6.2.3.2.cmml">ϵ</mi><mo id="Thmtheorem1.p1.6.6.m6.2.3.1" xref="Thmtheorem1.p1.6.6.m6.2.3.1.cmml">∈</mo><mrow id="Thmtheorem1.p1.6.6.m6.2.3.3.2" xref="Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml"><mo id="Thmtheorem1.p1.6.6.m6.2.3.3.2.1" stretchy="false" xref="Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">(</mo><mn id="Thmtheorem1.p1.6.6.m6.1.1" xref="Thmtheorem1.p1.6.6.m6.1.1.cmml">0</mn><mo id="Thmtheorem1.p1.6.6.m6.2.3.3.2.2" xref="Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">,</mo><mn id="Thmtheorem1.p1.6.6.m6.2.2" xref="Thmtheorem1.p1.6.6.m6.2.2.cmml">1</mn><mo id="Thmtheorem1.p1.6.6.m6.2.3.3.2.3" stretchy="false" xref="Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.6.6.m6.2b"><apply id="Thmtheorem1.p1.6.6.m6.2.3.cmml" xref="Thmtheorem1.p1.6.6.m6.2.3"><in id="Thmtheorem1.p1.6.6.m6.2.3.1.cmml" xref="Thmtheorem1.p1.6.6.m6.2.3.1"></in><ci id="Thmtheorem1.p1.6.6.m6.2.3.2.cmml" xref="Thmtheorem1.p1.6.6.m6.2.3.2">italic-ϵ</ci><interval closure="open" id="Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml" xref="Thmtheorem1.p1.6.6.m6.2.3.3.2"><cn id="Thmtheorem1.p1.6.6.m6.1.1.cmml" type="integer" xref="Thmtheorem1.p1.6.6.m6.1.1">0</cn><cn id="Thmtheorem1.p1.6.6.m6.2.2.cmml" type="integer" xref="Thmtheorem1.p1.6.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.6.6.m6.2c">\epsilon\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.6.6.m6.2d">italic_ϵ ∈ ( 0 , 1 )</annotation></semantics></math>, there exists a set of parameters <math alttext="\bm{\theta}" class="ltx_Math" display="inline" id="Thmtheorem1.p1.7.7.m7.1"><semantics id="Thmtheorem1.p1.7.7.m7.1a"><mi id="Thmtheorem1.p1.7.7.m7.1.1" xref="Thmtheorem1.p1.7.7.m7.1.1.cmml">𝛉</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.7.7.m7.1b"><ci id="Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="Thmtheorem1.p1.7.7.m7.1.1">𝛉</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.7.7.m7.1c">\bm{\theta}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.7.7.m7.1d">bold_italic_θ</annotation></semantics></math> such that, for all sequences <math alttext="(x_{t})_{t\geq 1}" class="ltx_Math" display="inline" id="Thmtheorem1.p1.8.8.m8.1"><semantics id="Thmtheorem1.p1.8.8.m8.1a"><msub id="Thmtheorem1.p1.8.8.m8.1.1" xref="Thmtheorem1.p1.8.8.m8.1.1.cmml"><mrow id="Thmtheorem1.p1.8.8.m8.1.1.1.1" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.cmml"><mo id="Thmtheorem1.p1.8.8.m8.1.1.1.1.2" stretchy="false" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.cmml">(</mo><msub id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.cmml"><mi id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.2" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.2.cmml">x</mi><mi id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.3" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.3.cmml">t</mi></msub><mo id="Thmtheorem1.p1.8.8.m8.1.1.1.1.3" stretchy="false" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.cmml">)</mo></mrow><mrow id="Thmtheorem1.p1.8.8.m8.1.1.3" xref="Thmtheorem1.p1.8.8.m8.1.1.3.cmml"><mi id="Thmtheorem1.p1.8.8.m8.1.1.3.2" xref="Thmtheorem1.p1.8.8.m8.1.1.3.2.cmml">t</mi><mo id="Thmtheorem1.p1.8.8.m8.1.1.3.1" xref="Thmtheorem1.p1.8.8.m8.1.1.3.1.cmml">≥</mo><mn id="Thmtheorem1.p1.8.8.m8.1.1.3.3" xref="Thmtheorem1.p1.8.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.8.8.m8.1b"><apply id="Thmtheorem1.p1.8.8.m8.1.1.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="Thmtheorem1.p1.8.8.m8.1.1.2.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1">subscript</csymbol><apply id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1">subscript</csymbol><ci id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.2.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.2">𝑥</ci><ci id="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.3.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.1.1.1.3">𝑡</ci></apply><apply id="Thmtheorem1.p1.8.8.m8.1.1.3.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.3"><geq id="Thmtheorem1.p1.8.8.m8.1.1.3.1.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.3.1"></geq><ci id="Thmtheorem1.p1.8.8.m8.1.1.3.2.cmml" xref="Thmtheorem1.p1.8.8.m8.1.1.3.2">𝑡</ci><cn id="Thmtheorem1.p1.8.8.m8.1.1.3.3.cmml" type="integer" xref="Thmtheorem1.p1.8.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.8.8.m8.1c">(x_{t})_{t\geq 1}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.8.8.m8.1d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t ≥ 1 end_POSTSUBSCRIPT</annotation></semantics></math> and all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="Thmtheorem1.p1.9.9.m9.1"><semantics id="Thmtheorem1.p1.9.9.m9.1a"><mrow id="Thmtheorem1.p1.9.9.m9.1.1" xref="Thmtheorem1.p1.9.9.m9.1.1.cmml"><mi id="Thmtheorem1.p1.9.9.m9.1.1.2" xref="Thmtheorem1.p1.9.9.m9.1.1.2.cmml">t</mi><mo id="Thmtheorem1.p1.9.9.m9.1.1.1" xref="Thmtheorem1.p1.9.9.m9.1.1.1.cmml">≥</mo><mn id="Thmtheorem1.p1.9.9.m9.1.1.3" xref="Thmtheorem1.p1.9.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.9.9.m9.1b"><apply id="Thmtheorem1.p1.9.9.m9.1.1.cmml" xref="Thmtheorem1.p1.9.9.m9.1.1"><geq id="Thmtheorem1.p1.9.9.m9.1.1.1.cmml" xref="Thmtheorem1.p1.9.9.m9.1.1.1"></geq><ci id="Thmtheorem1.p1.9.9.m9.1.1.2.cmml" xref="Thmtheorem1.p1.9.9.m9.1.1.2">𝑡</ci><cn id="Thmtheorem1.p1.9.9.m9.1.1.3.cmml" type="integer" xref="Thmtheorem1.p1.9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.9.9.m9.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.9.9.m9.1d">italic_t ≥ 1</annotation></semantics></math>,</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx5"> <tbody id="S4.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D_{\rm KL}\left(\mathbb{P}_{\beta}^{(1)}(\cdot\mid x_{1}^{t})\|% \mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)\right)\leq\epsilon." class="ltx_math_unparsed" display="inline" id="S4.Ex14.m1.1"><semantics id="S4.Ex14.m1.1a"><mrow id="S4.Ex14.m1.1b"><msub id="S4.Ex14.m1.1.2"><mi id="S4.Ex14.m1.1.2.2">D</mi><mi id="S4.Ex14.m1.1.2.3">KL</mi></msub><mrow id="S4.Ex14.m1.1.3"><mo id="S4.Ex14.m1.1.3.1">(</mo><msubsup id="S4.Ex14.m1.1.3.2"><mi id="S4.Ex14.m1.1.3.2.2.2">ℙ</mi><mi id="S4.Ex14.m1.1.3.2.2.3">β</mi><mrow id="S4.Ex14.m1.1.1.1.3"><mo id="S4.Ex14.m1.1.1.1.3.1" stretchy="false">(</mo><mn id="S4.Ex14.m1.1.1.1.1">1</mn><mo id="S4.Ex14.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.Ex14.m1.1.3.3"><mo id="S4.Ex14.m1.1.3.3.1" stretchy="false">(</mo><mo id="S4.Ex14.m1.1.3.3.2" lspace="0em" rspace="0em">⋅</mo><mo id="S4.Ex14.m1.1.3.3.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S4.Ex14.m1.1.3.3.4"><mi id="S4.Ex14.m1.1.3.3.4.2.2">x</mi><mn id="S4.Ex14.m1.1.3.3.4.2.3">1</mn><mi id="S4.Ex14.m1.1.3.3.4.3">t</mi></msubsup><mo id="S4.Ex14.m1.1.3.3.5" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.1.3.4" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.Ex14.m1.1.3.5"><mi id="S4.Ex14.m1.1.3.5.2">ℙ</mi><mi id="S4.Ex14.m1.1.3.5.3">𝜽</mi></msub><mrow id="S4.Ex14.m1.1.3.6"><mo id="S4.Ex14.m1.1.3.6.1">(</mo><mo id="S4.Ex14.m1.1.3.6.2" lspace="0em" rspace="0em">⋅</mo><mo id="S4.Ex14.m1.1.3.6.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S4.Ex14.m1.1.3.6.4"><mi id="S4.Ex14.m1.1.3.6.4.2.2">x</mi><mn id="S4.Ex14.m1.1.3.6.4.2.3">1</mn><mi id="S4.Ex14.m1.1.3.6.4.3">t</mi></msubsup><mo id="S4.Ex14.m1.1.3.6.5">)</mo></mrow><mo id="S4.Ex14.m1.1.3.7">)</mo></mrow><mo id="S4.Ex14.m1.1.4">≤</mo><mi id="S4.Ex14.m1.1.5">ϵ</mi><mo id="S4.Ex14.m1.1.6" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex14.m1.1c">\displaystyle D_{\rm KL}\left(\mathbb{P}_{\beta}^{(1)}(\cdot\mid x_{1}^{t})\|% \mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)\right)\leq\epsilon.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.1d">italic_D start_POSTSUBSCRIPT roman_KL end_POSTSUBSCRIPT ( blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∥ blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) ≤ italic_ϵ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.4"><span class="ltx_text ltx_font_bold" id="S4.SS2.p2.4.1">Remark.</span> The KL divergence above is precisely the penalty paid in the cross-entropy loss at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_t</annotation></semantics></math> (Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.E2" title="Equation 2 ‣ 2.3 Learning task: next-token prediction ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2</span></a>)) when using the predictor <math alttext="\mathbb{P}_{\bm{\theta}}" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><msub id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml">ℙ</mi><mi id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2">ℙ</ci><ci id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">\mathbb{P}_{\bm{\theta}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> instead of the optimal <math alttext="\mathbb{P}_{\beta}^{(1)}" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><msubsup id="S4.SS2.p2.3.m3.1.2" xref="S4.SS2.p2.3.m3.1.2.cmml"><mi id="S4.SS2.p2.3.m3.1.2.2.2" xref="S4.SS2.p2.3.m3.1.2.2.2.cmml">ℙ</mi><mi id="S4.SS2.p2.3.m3.1.2.2.3" xref="S4.SS2.p2.3.m3.1.2.2.3.cmml">β</mi><mrow id="S4.SS2.p2.3.m3.1.1.1.3" xref="S4.SS2.p2.3.m3.1.2.cmml"><mo id="S4.SS2.p2.3.m3.1.1.1.3.1" stretchy="false" xref="S4.SS2.p2.3.m3.1.2.cmml">(</mo><mn id="S4.SS2.p2.3.m3.1.1.1.1" xref="S4.SS2.p2.3.m3.1.1.1.1.cmml">1</mn><mo id="S4.SS2.p2.3.m3.1.1.1.3.2" stretchy="false" xref="S4.SS2.p2.3.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.2.cmml" xref="S4.SS2.p2.3.m3.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.2">superscript</csymbol><apply id="S4.SS2.p2.3.m3.1.2.2.cmml" xref="S4.SS2.p2.3.m3.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.2.2.1.cmml" xref="S4.SS2.p2.3.m3.1.2">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.2.2.2.cmml" xref="S4.SS2.p2.3.m3.1.2.2.2">ℙ</ci><ci id="S4.SS2.p2.3.m3.1.2.2.3.cmml" xref="S4.SS2.p2.3.m3.1.2.2.3">𝛽</ci></apply><cn id="S4.SS2.p2.3.m3.1.1.1.1.cmml" type="integer" xref="S4.SS2.p2.3.m3.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">\mathbb{P}_{\beta}^{(1)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. In other words, the result implies that the loss of <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mi id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><ci id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">sansserif_MambaZero</annotation></semantics></math> can be made arbitrarily close to that of the optimal.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1">We defer the full proof to § <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2" title="Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">B</span></a> and outline the sketch below.</p> </div> <section class="ltx_subsubsection" id="S4.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.2.1 </span>Proof sketch</h4> <div class="ltx_para" id="S4.SS2.SSS1.p1"> <p class="ltx_p" id="S4.SS2.SSS1.p1.13"><span class="ltx_text ltx_font_bold" id="S4.SS2.SSS1.p1.13.1">Main idea.</span> To build our intuition towards how <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.1.m1.1"><semantics id="S4.SS2.SSS1.p1.1.m1.1a"><mi id="S4.SS2.SSS1.p1.1.m1.1.1" xref="S4.SS2.SSS1.p1.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.1.m1.1b"><ci id="S4.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.1.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.1.m1.1d">sansserif_MambaZero</annotation></semantics></math> can realize the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.2.m2.1"><semantics id="S4.SS2.SSS1.p1.2.m2.1a"><mi id="S4.SS2.SSS1.p1.2.m2.1.1" xref="S4.SS2.SSS1.p1.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.2.m2.1b"><ci id="S4.SS2.SSS1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.2.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.2.m2.1d">italic_β</annotation></semantics></math> counting estimator for first-order Markov sequences, let’s focus on the core <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex13" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a> block. The key observation here is the following: if the state <math alttext="H_{t-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.3.m3.1"><semantics id="S4.SS2.SSS1.p1.3.m3.1a"><msub id="S4.SS2.SSS1.p1.3.m3.1.1" xref="S4.SS2.SSS1.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p1.3.m3.1.1.2" xref="S4.SS2.SSS1.p1.3.m3.1.1.2.cmml">H</mi><mrow id="S4.SS2.SSS1.p1.3.m3.1.1.3" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.cmml"><mi id="S4.SS2.SSS1.p1.3.m3.1.1.3.2" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.2.cmml">t</mi><mo id="S4.SS2.SSS1.p1.3.m3.1.1.3.1" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p1.3.m3.1.1.3.3" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.3.m3.1b"><apply id="S4.SS2.SSS1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.2">𝐻</ci><apply id="S4.SS2.SSS1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.3"><minus id="S4.SS2.SSS1.p1.3.m3.1.1.3.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p1.3.m3.1.1.3.2.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.2">𝑡</ci><cn id="S4.SS2.SSS1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.3.m3.1c">H_{t-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.3.m3.1d">italic_H start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT</annotation></semantics></math> can capture all the transition counts <math alttext="i\to j" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.4.m4.1"><semantics id="S4.SS2.SSS1.p1.4.m4.1a"><mrow id="S4.SS2.SSS1.p1.4.m4.1.1" xref="S4.SS2.SSS1.p1.4.m4.1.1.cmml"><mi id="S4.SS2.SSS1.p1.4.m4.1.1.2" xref="S4.SS2.SSS1.p1.4.m4.1.1.2.cmml">i</mi><mo id="S4.SS2.SSS1.p1.4.m4.1.1.1" stretchy="false" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.p1.4.m4.1.1.3" xref="S4.SS2.SSS1.p1.4.m4.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.4.m4.1b"><apply id="S4.SS2.SSS1.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1"><ci id="S4.SS2.SSS1.p1.4.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.1">→</ci><ci id="S4.SS2.SSS1.p1.4.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.2">𝑖</ci><ci id="S4.SS2.SSS1.p1.4.m4.1.1.3.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.4.m4.1c">i\to j</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.4.m4.1d">italic_i → italic_j</annotation></semantics></math> till <math alttext="x_{1}^{t-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.5.m5.1"><semantics id="S4.SS2.SSS1.p1.5.m5.1a"><msubsup id="S4.SS2.SSS1.p1.5.m5.1.1" xref="S4.SS2.SSS1.p1.5.m5.1.1.cmml"><mi id="S4.SS2.SSS1.p1.5.m5.1.1.2.2" xref="S4.SS2.SSS1.p1.5.m5.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS1.p1.5.m5.1.1.2.3" xref="S4.SS2.SSS1.p1.5.m5.1.1.2.3.cmml">1</mn><mrow id="S4.SS2.SSS1.p1.5.m5.1.1.3" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.cmml"><mi id="S4.SS2.SSS1.p1.5.m5.1.1.3.2" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.2.cmml">t</mi><mo id="S4.SS2.SSS1.p1.5.m5.1.1.3.1" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p1.5.m5.1.1.3.3" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.3.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.5.m5.1b"><apply id="S4.SS2.SSS1.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.5.m5.1.1.1.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.p1.5.m5.1.1.2.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.5.m5.1.1.2.1.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.5.m5.1.1.2.2.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS1.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.5.m5.1.1.2.3">1</cn></apply><apply id="S4.SS2.SSS1.p1.5.m5.1.1.3.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1.3"><minus id="S4.SS2.SSS1.p1.5.m5.1.1.3.1.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p1.5.m5.1.1.3.2.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.2">𝑡</ci><cn id="S4.SS2.SSS1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.5.m5.1c">x_{1}^{t-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.5.m5.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, the new state <math alttext="H_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.6.m6.1"><semantics id="S4.SS2.SSS1.p1.6.m6.1a"><msub id="S4.SS2.SSS1.p1.6.m6.1.1" xref="S4.SS2.SSS1.p1.6.m6.1.1.cmml"><mi id="S4.SS2.SSS1.p1.6.m6.1.1.2" xref="S4.SS2.SSS1.p1.6.m6.1.1.2.cmml">H</mi><mi id="S4.SS2.SSS1.p1.6.m6.1.1.3" xref="S4.SS2.SSS1.p1.6.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.6.m6.1b"><apply id="S4.SS2.SSS1.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.6.m6.1.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.6.m6.1.1.2.cmml" xref="S4.SS2.SSS1.p1.6.m6.1.1.2">𝐻</ci><ci id="S4.SS2.SSS1.p1.6.m6.1.1.3.cmml" xref="S4.SS2.SSS1.p1.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.6.m6.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.6.m6.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> can be updated to account for the current transition <math alttext="x_{t-1}\to x_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.7.m7.1"><semantics id="S4.SS2.SSS1.p1.7.m7.1a"><mrow id="S4.SS2.SSS1.p1.7.m7.1.1" xref="S4.SS2.SSS1.p1.7.m7.1.1.cmml"><msub id="S4.SS2.SSS1.p1.7.m7.1.1.2" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.cmml"><mi id="S4.SS2.SSS1.p1.7.m7.1.1.2.2" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.2.cmml">x</mi><mrow id="S4.SS2.SSS1.p1.7.m7.1.1.2.3" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.cmml"><mi id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.2" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.2.cmml">t</mi><mo id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.1" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.3" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p1.7.m7.1.1.1" stretchy="false" xref="S4.SS2.SSS1.p1.7.m7.1.1.1.cmml">→</mo><msub id="S4.SS2.SSS1.p1.7.m7.1.1.3" xref="S4.SS2.SSS1.p1.7.m7.1.1.3.cmml"><mi id="S4.SS2.SSS1.p1.7.m7.1.1.3.2" xref="S4.SS2.SSS1.p1.7.m7.1.1.3.2.cmml">x</mi><mi id="S4.SS2.SSS1.p1.7.m7.1.1.3.3" xref="S4.SS2.SSS1.p1.7.m7.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.7.m7.1b"><apply id="S4.SS2.SSS1.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1"><ci id="S4.SS2.SSS1.p1.7.m7.1.1.1.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.1">→</ci><apply id="S4.SS2.SSS1.p1.7.m7.1.1.2.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.7.m7.1.1.2.1.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p1.7.m7.1.1.2.2.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.2">𝑥</ci><apply id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3"><minus id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.1.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.1"></minus><ci id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.2.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.2">𝑡</ci><cn id="S4.SS2.SSS1.p1.7.m7.1.1.2.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.7.m7.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.p1.7.m7.1.1.3.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.7.m7.1.1.3.1.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.p1.7.m7.1.1.3.2.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.3.2">𝑥</ci><ci id="S4.SS2.SSS1.p1.7.m7.1.1.3.3.cmml" xref="S4.SS2.SSS1.p1.7.m7.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.7.m7.1c">x_{t-1}\to x_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.7.m7.1d">italic_x start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT → italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> on top of the existing counts, by a suitable choice of <math alttext="a_{t},\widetilde{\bm{x}}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.8.m8.2"><semantics id="S4.SS2.SSS1.p1.8.m8.2a"><mrow id="S4.SS2.SSS1.p1.8.m8.2.2.2" xref="S4.SS2.SSS1.p1.8.m8.2.2.3.cmml"><msub id="S4.SS2.SSS1.p1.8.m8.1.1.1.1" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p1.8.m8.1.1.1.1.2" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1.2.cmml">a</mi><mi id="S4.SS2.SSS1.p1.8.m8.1.1.1.1.3" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p1.8.m8.2.2.2.3" xref="S4.SS2.SSS1.p1.8.m8.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.p1.8.m8.2.2.2.2" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.2" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.cmml">𝒙</mi><mo id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.1" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.3" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.8.m8.2b"><list id="S4.SS2.SSS1.p1.8.m8.2.2.3.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2"><apply id="S4.SS2.SSS1.p1.8.m8.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.8.m8.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.8.m8.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1.2">𝑎</ci><ci id="S4.SS2.SSS1.p1.8.m8.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p1.8.m8.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2"><ci id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.1">~</ci><ci id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.2.2">𝒙</ci></apply><ci id="S4.SS2.SSS1.p1.8.m8.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p1.8.m8.2.2.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.8.m8.2c">a_{t},\widetilde{\bm{x}}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.8.m8.2d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\bm{b}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.9.m9.1"><semantics id="S4.SS2.SSS1.p1.9.m9.1a"><msub id="S4.SS2.SSS1.p1.9.m9.1.1" xref="S4.SS2.SSS1.p1.9.m9.1.1.cmml"><mi id="S4.SS2.SSS1.p1.9.m9.1.1.2" xref="S4.SS2.SSS1.p1.9.m9.1.1.2.cmml">𝒃</mi><mi id="S4.SS2.SSS1.p1.9.m9.1.1.3" xref="S4.SS2.SSS1.p1.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.9.m9.1b"><apply id="S4.SS2.SSS1.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.9.m9.1.1.1.cmml" xref="S4.SS2.SSS1.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.9.m9.1.1.2.cmml" xref="S4.SS2.SSS1.p1.9.m9.1.1.2">𝒃</ci><ci id="S4.SS2.SSS1.p1.9.m9.1.1.3.cmml" xref="S4.SS2.SSS1.p1.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.9.m9.1c">\bm{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.9.m9.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. Then the relevant count information corresponding to the current prefix <math alttext="x_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.10.m10.1"><semantics id="S4.SS2.SSS1.p1.10.m10.1a"><msub id="S4.SS2.SSS1.p1.10.m10.1.1" xref="S4.SS2.SSS1.p1.10.m10.1.1.cmml"><mi id="S4.SS2.SSS1.p1.10.m10.1.1.2" xref="S4.SS2.SSS1.p1.10.m10.1.1.2.cmml">x</mi><mi id="S4.SS2.SSS1.p1.10.m10.1.1.3" xref="S4.SS2.SSS1.p1.10.m10.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.10.m10.1b"><apply id="S4.SS2.SSS1.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.10.m10.1.1.1.cmml" xref="S4.SS2.SSS1.p1.10.m10.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.10.m10.1.1.2.cmml" xref="S4.SS2.SSS1.p1.10.m10.1.1.2">𝑥</ci><ci id="S4.SS2.SSS1.p1.10.m10.1.1.3.cmml" xref="S4.SS2.SSS1.p1.10.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.10.m10.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.10.m10.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> could be read off from the state projection <math alttext="\bm{y}_{t}=H_{t}\bm{c}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.11.m11.1"><semantics id="S4.SS2.SSS1.p1.11.m11.1a"><mrow id="S4.SS2.SSS1.p1.11.m11.1.1" xref="S4.SS2.SSS1.p1.11.m11.1.1.cmml"><msub id="S4.SS2.SSS1.p1.11.m11.1.1.2" xref="S4.SS2.SSS1.p1.11.m11.1.1.2.cmml"><mi id="S4.SS2.SSS1.p1.11.m11.1.1.2.2" xref="S4.SS2.SSS1.p1.11.m11.1.1.2.2.cmml">𝒚</mi><mi id="S4.SS2.SSS1.p1.11.m11.1.1.2.3" xref="S4.SS2.SSS1.p1.11.m11.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p1.11.m11.1.1.1" xref="S4.SS2.SSS1.p1.11.m11.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS1.p1.11.m11.1.1.3" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.cmml"><msub id="S4.SS2.SSS1.p1.11.m11.1.1.3.2" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p1.11.m11.1.1.3.2.2" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2.2.cmml">H</mi><mi id="S4.SS2.SSS1.p1.11.m11.1.1.3.2.3" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p1.11.m11.1.1.3.1" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.SSS1.p1.11.m11.1.1.3.3" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3.cmml"><mi id="S4.SS2.SSS1.p1.11.m11.1.1.3.3.2" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3.2.cmml">𝒄</mi><mi id="S4.SS2.SSS1.p1.11.m11.1.1.3.3.3" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3.3.cmml">t</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.11.m11.1b"><apply id="S4.SS2.SSS1.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1"><eq id="S4.SS2.SSS1.p1.11.m11.1.1.1.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.1"></eq><apply id="S4.SS2.SSS1.p1.11.m11.1.1.2.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.11.m11.1.1.2.1.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p1.11.m11.1.1.2.2.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.2.2">𝒚</ci><ci id="S4.SS2.SSS1.p1.11.m11.1.1.2.3.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.2.3">𝑡</ci></apply><apply id="S4.SS2.SSS1.p1.11.m11.1.1.3.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3"><times id="S4.SS2.SSS1.p1.11.m11.1.1.3.1.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.1"></times><apply id="S4.SS2.SSS1.p1.11.m11.1.1.3.2.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.11.m11.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p1.11.m11.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2.2">𝐻</ci><ci id="S4.SS2.SSS1.p1.11.m11.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.2.3">𝑡</ci></apply><apply id="S4.SS2.SSS1.p1.11.m11.1.1.3.3.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.11.m11.1.1.3.3.1.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3">subscript</csymbol><ci id="S4.SS2.SSS1.p1.11.m11.1.1.3.3.2.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3.2">𝒄</ci><ci id="S4.SS2.SSS1.p1.11.m11.1.1.3.3.3.cmml" xref="S4.SS2.SSS1.p1.11.m11.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.11.m11.1c">\bm{y}_{t}=H_{t}\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.11.m11.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, and be modified to account for <math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.12.m12.1"><semantics id="S4.SS2.SSS1.p1.12.m12.1a"><mi id="S4.SS2.SSS1.p1.12.m12.1.1" xref="S4.SS2.SSS1.p1.12.m12.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.12.m12.1b"><ci id="S4.SS2.SSS1.p1.12.m12.1.1.cmml" xref="S4.SS2.SSS1.p1.12.m12.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.12.m12.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.12.m12.1d">italic_β</annotation></semantics></math>-smoothing via the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex11" title="Equation Linear ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Linear</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex12" title="Equation Prediction ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Prediction</span></a> layers. Buttressing this idea are two key empirical facts, which in fact hold for any <math alttext="k\geq 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.13.m13.1"><semantics id="S4.SS2.SSS1.p1.13.m13.1a"><mrow id="S4.SS2.SSS1.p1.13.m13.1.1" xref="S4.SS2.SSS1.p1.13.m13.1.1.cmml"><mi id="S4.SS2.SSS1.p1.13.m13.1.1.2" xref="S4.SS2.SSS1.p1.13.m13.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS1.p1.13.m13.1.1.1" xref="S4.SS2.SSS1.p1.13.m13.1.1.1.cmml">≥</mo><mn id="S4.SS2.SSS1.p1.13.m13.1.1.3" xref="S4.SS2.SSS1.p1.13.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.13.m13.1b"><apply id="S4.SS2.SSS1.p1.13.m13.1.1.cmml" xref="S4.SS2.SSS1.p1.13.m13.1.1"><geq id="S4.SS2.SSS1.p1.13.m13.1.1.1.cmml" xref="S4.SS2.SSS1.p1.13.m13.1.1.1"></geq><ci id="S4.SS2.SSS1.p1.13.m13.1.1.2.cmml" xref="S4.SS2.SSS1.p1.13.m13.1.1.2">𝑘</ci><cn id="S4.SS2.SSS1.p1.13.m13.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.13.m13.1c">k\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.13.m13.1d">italic_k ≥ 1</annotation></semantics></math>, underpinning our construction:</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p2"> <p class="ltx_p" id="S4.SS2.SSS1.p2.10"><span class="ltx_text ltx_font_bold" id="S4.SS2.SSS1.p2.2.2">(i) State-to-state transition factor <math alttext="a_{t}\approx 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.1.1.m1.1"><semantics id="S4.SS2.SSS1.p2.1.1.m1.1a"><mrow id="S4.SS2.SSS1.p2.1.1.m1.1.1" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.cmml"><msub id="S4.SS2.SSS1.p2.1.1.m1.1.1.2" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p2.1.1.m1.1.1.2.2" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2.2.cmml">a</mi><mi id="S4.SS2.SSS1.p2.1.1.m1.1.1.2.3" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p2.1.1.m1.1.1.1" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.1.cmml">≈</mo><mn id="S4.SS2.SSS1.p2.1.1.m1.1.1.3" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.1.1.m1.1b"><apply id="S4.SS2.SSS1.p2.1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p2.1.1.m1.1.1"><approx id="S4.SS2.SSS1.p2.1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.1"></approx><apply id="S4.SS2.SSS1.p2.1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.1.1.m1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p2.1.1.m1.1.1.2.2.cmml" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2.2">𝑎</ci><ci id="S4.SS2.SSS1.p2.1.1.m1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p2.1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.1.1.m1.1c">a_{t}\approx 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≈ 1</annotation></semantics></math> for all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.2.2.m2.1"><semantics id="S4.SS2.SSS1.p2.2.2.m2.1a"><mrow id="S4.SS2.SSS1.p2.2.2.m2.1.1" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p2.2.2.m2.1.1.2" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.2.cmml">t</mi><mo id="S4.SS2.SSS1.p2.2.2.m2.1.1.1" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.1.cmml">≥</mo><mn id="S4.SS2.SSS1.p2.2.2.m2.1.1.3" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.2.2.m2.1b"><apply id="S4.SS2.SSS1.p2.2.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p2.2.2.m2.1.1"><geq id="S4.SS2.SSS1.p2.2.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.1"></geq><ci id="S4.SS2.SSS1.p2.2.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.2">𝑡</ci><cn id="S4.SS2.SSS1.p2.2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.2.2.m2.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.2.2.m2.1d">italic_t ≥ 1</annotation></semantics></math>.</span> We empirically observe that when <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.3.m1.1"><semantics id="S4.SS2.SSS1.p2.3.m1.1a"><mi id="S4.SS2.SSS1.p2.3.m1.1.1" xref="S4.SS2.SSS1.p2.3.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.3.m1.1b"><ci id="S4.SS2.SSS1.p2.3.m1.1.1.cmml" xref="S4.SS2.SSS1.p2.3.m1.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.3.m1.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.3.m1.1d">sansserif_MambaZero</annotation></semantics></math> model is trained on random first-order Markov data, at convergence we have <math alttext="a_{t}\approx 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.4.m2.1"><semantics id="S4.SS2.SSS1.p2.4.m2.1a"><mrow id="S4.SS2.SSS1.p2.4.m2.1.1" xref="S4.SS2.SSS1.p2.4.m2.1.1.cmml"><msub id="S4.SS2.SSS1.p2.4.m2.1.1.2" xref="S4.SS2.SSS1.p2.4.m2.1.1.2.cmml"><mi id="S4.SS2.SSS1.p2.4.m2.1.1.2.2" xref="S4.SS2.SSS1.p2.4.m2.1.1.2.2.cmml">a</mi><mi id="S4.SS2.SSS1.p2.4.m2.1.1.2.3" xref="S4.SS2.SSS1.p2.4.m2.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p2.4.m2.1.1.1" xref="S4.SS2.SSS1.p2.4.m2.1.1.1.cmml">≈</mo><mn id="S4.SS2.SSS1.p2.4.m2.1.1.3" xref="S4.SS2.SSS1.p2.4.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.4.m2.1b"><apply id="S4.SS2.SSS1.p2.4.m2.1.1.cmml" xref="S4.SS2.SSS1.p2.4.m2.1.1"><approx id="S4.SS2.SSS1.p2.4.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p2.4.m2.1.1.1"></approx><apply id="S4.SS2.SSS1.p2.4.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p2.4.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.4.m2.1.1.2.1.cmml" xref="S4.SS2.SSS1.p2.4.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p2.4.m2.1.1.2.2.cmml" xref="S4.SS2.SSS1.p2.4.m2.1.1.2.2">𝑎</ci><ci id="S4.SS2.SSS1.p2.4.m2.1.1.2.3.cmml" xref="S4.SS2.SSS1.p2.4.m2.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p2.4.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.4.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.4.m2.1c">a_{t}\approx 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.4.m2.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≈ 1</annotation></semantics></math> for all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.5.m3.1"><semantics id="S4.SS2.SSS1.p2.5.m3.1a"><mrow id="S4.SS2.SSS1.p2.5.m3.1.1" xref="S4.SS2.SSS1.p2.5.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p2.5.m3.1.1.2" xref="S4.SS2.SSS1.p2.5.m3.1.1.2.cmml">t</mi><mo id="S4.SS2.SSS1.p2.5.m3.1.1.1" xref="S4.SS2.SSS1.p2.5.m3.1.1.1.cmml">≥</mo><mn id="S4.SS2.SSS1.p2.5.m3.1.1.3" xref="S4.SS2.SSS1.p2.5.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.5.m3.1b"><apply id="S4.SS2.SSS1.p2.5.m3.1.1.cmml" xref="S4.SS2.SSS1.p2.5.m3.1.1"><geq id="S4.SS2.SSS1.p2.5.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p2.5.m3.1.1.1"></geq><ci id="S4.SS2.SSS1.p2.5.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p2.5.m3.1.1.2">𝑡</ci><cn id="S4.SS2.SSS1.p2.5.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.5.m3.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.5.m3.1d">italic_t ≥ 1</annotation></semantics></math> (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.F5" title="Figure 5 ‣ B.1 Empirical insights ‣ Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">5</span></a>). Since <math alttext="a_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.6.m4.1"><semantics id="S4.SS2.SSS1.p2.6.m4.1a"><msub id="S4.SS2.SSS1.p2.6.m4.1.1" xref="S4.SS2.SSS1.p2.6.m4.1.1.cmml"><mi id="S4.SS2.SSS1.p2.6.m4.1.1.2" xref="S4.SS2.SSS1.p2.6.m4.1.1.2.cmml">a</mi><mi id="S4.SS2.SSS1.p2.6.m4.1.1.3" xref="S4.SS2.SSS1.p2.6.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.6.m4.1b"><apply id="S4.SS2.SSS1.p2.6.m4.1.1.cmml" xref="S4.SS2.SSS1.p2.6.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.6.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p2.6.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.6.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p2.6.m4.1.1.2">𝑎</ci><ci id="S4.SS2.SSS1.p2.6.m4.1.1.3.cmml" xref="S4.SS2.SSS1.p2.6.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.6.m4.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.6.m4.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> modulates how much past information flows into the present, <math alttext="a_{t}=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.7.m5.1"><semantics id="S4.SS2.SSS1.p2.7.m5.1a"><mrow id="S4.SS2.SSS1.p2.7.m5.1.1" xref="S4.SS2.SSS1.p2.7.m5.1.1.cmml"><msub id="S4.SS2.SSS1.p2.7.m5.1.1.2" xref="S4.SS2.SSS1.p2.7.m5.1.1.2.cmml"><mi id="S4.SS2.SSS1.p2.7.m5.1.1.2.2" xref="S4.SS2.SSS1.p2.7.m5.1.1.2.2.cmml">a</mi><mi id="S4.SS2.SSS1.p2.7.m5.1.1.2.3" xref="S4.SS2.SSS1.p2.7.m5.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p2.7.m5.1.1.1" xref="S4.SS2.SSS1.p2.7.m5.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p2.7.m5.1.1.3" xref="S4.SS2.SSS1.p2.7.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.7.m5.1b"><apply id="S4.SS2.SSS1.p2.7.m5.1.1.cmml" xref="S4.SS2.SSS1.p2.7.m5.1.1"><eq id="S4.SS2.SSS1.p2.7.m5.1.1.1.cmml" xref="S4.SS2.SSS1.p2.7.m5.1.1.1"></eq><apply id="S4.SS2.SSS1.p2.7.m5.1.1.2.cmml" xref="S4.SS2.SSS1.p2.7.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.7.m5.1.1.2.1.cmml" xref="S4.SS2.SSS1.p2.7.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p2.7.m5.1.1.2.2.cmml" xref="S4.SS2.SSS1.p2.7.m5.1.1.2.2">𝑎</ci><ci id="S4.SS2.SSS1.p2.7.m5.1.1.2.3.cmml" xref="S4.SS2.SSS1.p2.7.m5.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p2.7.m5.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.7.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.7.m5.1c">a_{t}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.7.m5.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math> serves as a natural choice when the states <math alttext="H_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.8.m6.1"><semantics id="S4.SS2.SSS1.p2.8.m6.1a"><msub id="S4.SS2.SSS1.p2.8.m6.1.1" xref="S4.SS2.SSS1.p2.8.m6.1.1.cmml"><mi id="S4.SS2.SSS1.p2.8.m6.1.1.2" xref="S4.SS2.SSS1.p2.8.m6.1.1.2.cmml">H</mi><mi id="S4.SS2.SSS1.p2.8.m6.1.1.3" xref="S4.SS2.SSS1.p2.8.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.8.m6.1b"><apply id="S4.SS2.SSS1.p2.8.m6.1.1.cmml" xref="S4.SS2.SSS1.p2.8.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.8.m6.1.1.1.cmml" xref="S4.SS2.SSS1.p2.8.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.8.m6.1.1.2.cmml" xref="S4.SS2.SSS1.p2.8.m6.1.1.2">𝐻</ci><ci id="S4.SS2.SSS1.p2.8.m6.1.1.3.cmml" xref="S4.SS2.SSS1.p2.8.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.8.m6.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.8.m6.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> carry count information till current tokens, which we would like to update without any diminishing factor. Note that this can be easily achieved by setting either <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.9.m7.1"><semantics id="S4.SS2.SSS1.p2.9.m7.1a"><mi id="S4.SS2.SSS1.p2.9.m7.1.1" xref="S4.SS2.SSS1.p2.9.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.9.m7.1b"><ci id="S4.SS2.SSS1.p2.9.m7.1.1.cmml" xref="S4.SS2.SSS1.p2.9.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.9.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.9.m7.1d">italic_a</annotation></semantics></math> or <math alttext="\Delta_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.10.m8.1"><semantics id="S4.SS2.SSS1.p2.10.m8.1a"><msub id="S4.SS2.SSS1.p2.10.m8.1.1" xref="S4.SS2.SSS1.p2.10.m8.1.1.cmml"><mi id="S4.SS2.SSS1.p2.10.m8.1.1.2" mathvariant="normal" xref="S4.SS2.SSS1.p2.10.m8.1.1.2.cmml">Δ</mi><mi id="S4.SS2.SSS1.p2.10.m8.1.1.3" xref="S4.SS2.SSS1.p2.10.m8.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.10.m8.1b"><apply id="S4.SS2.SSS1.p2.10.m8.1.1.cmml" xref="S4.SS2.SSS1.p2.10.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.10.m8.1.1.1.cmml" xref="S4.SS2.SSS1.p2.10.m8.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.10.m8.1.1.2.cmml" xref="S4.SS2.SSS1.p2.10.m8.1.1.2">Δ</ci><ci id="S4.SS2.SSS1.p2.10.m8.1.1.3.cmml" xref="S4.SS2.SSS1.p2.10.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.10.m8.1c">\Delta_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.10.m8.1d">roman_Δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> to be zero in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex2" title="Equation Input selectivity ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Input selectivity</span></a>, which we empirically observe as well.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p3"> <p class="ltx_p" id="S4.SS2.SSS1.p3.18"><span class="ltx_text ltx_font_bold" id="S4.SS2.SSS1.p3.1.1">(ii) Convolution window <math alttext="w\geq k+1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.1.1.m1.1"><semantics id="S4.SS2.SSS1.p3.1.1.m1.1a"><mrow id="S4.SS2.SSS1.p3.1.1.m1.1.1" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.1.1.m1.1.1.2" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.SSS1.p3.1.1.m1.1.1.1" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.1.cmml">≥</mo><mrow id="S4.SS2.SSS1.p3.1.1.m1.1.1.3" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.2" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.1" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.1.cmml">+</mo><mn id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.3" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.1.1.m1.1b"><apply id="S4.SS2.SSS1.p3.1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p3.1.1.m1.1.1"><geq id="S4.SS2.SSS1.p3.1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.1"></geq><ci id="S4.SS2.SSS1.p3.1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3"><plus id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.1"></plus><ci id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p3.1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p3.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.1.1.m1.1c">w\geq k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.1.1.m1.1d">italic_w ≥ italic_k + 1</annotation></semantics></math>.</span> Recalling that <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.2.m1.1"><semantics id="S4.SS2.SSS1.p3.2.m1.1a"><mi id="S4.SS2.SSS1.p3.2.m1.1.1" xref="S4.SS2.SSS1.p3.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.2.m1.1b"><ci id="S4.SS2.SSS1.p3.2.m1.1.1.cmml" xref="S4.SS2.SSS1.p3.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.2.m1.1d">italic_k</annotation></semantics></math> is the Markov order, we empirically observe that the window that <math alttext="w=k+1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.3.m2.1"><semantics id="S4.SS2.SSS1.p3.3.m2.1a"><mrow id="S4.SS2.SSS1.p3.3.m2.1.1" xref="S4.SS2.SSS1.p3.3.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p3.3.m2.1.1.2" xref="S4.SS2.SSS1.p3.3.m2.1.1.2.cmml">w</mi><mo id="S4.SS2.SSS1.p3.3.m2.1.1.1" xref="S4.SS2.SSS1.p3.3.m2.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS1.p3.3.m2.1.1.3" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.cmml"><mi id="S4.SS2.SSS1.p3.3.m2.1.1.3.2" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p3.3.m2.1.1.3.1" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.1.cmml">+</mo><mn id="S4.SS2.SSS1.p3.3.m2.1.1.3.3" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.3.m2.1b"><apply id="S4.SS2.SSS1.p3.3.m2.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m2.1.1"><eq id="S4.SS2.SSS1.p3.3.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m2.1.1.1"></eq><ci id="S4.SS2.SSS1.p3.3.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p3.3.m2.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p3.3.m2.1.1.3.cmml" xref="S4.SS2.SSS1.p3.3.m2.1.1.3"><plus id="S4.SS2.SSS1.p3.3.m2.1.1.3.1.cmml" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.1"></plus><ci id="S4.SS2.SSS1.p3.3.m2.1.1.3.2.cmml" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p3.3.m2.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p3.3.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.3.m2.1c">w=k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.3.m2.1d">italic_w = italic_k + 1</annotation></semantics></math> is sufficient for the full Mamba to learn the Laplacian smoothing on <math alttext="k^{\text{th}}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.4.m3.1"><semantics id="S4.SS2.SSS1.p3.4.m3.1a"><msup id="S4.SS2.SSS1.p3.4.m3.1.1" xref="S4.SS2.SSS1.p3.4.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p3.4.m3.1.1.2" xref="S4.SS2.SSS1.p3.4.m3.1.1.2.cmml">k</mi><mtext id="S4.SS2.SSS1.p3.4.m3.1.1.3" xref="S4.SS2.SSS1.p3.4.m3.1.1.3a.cmml">th</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.4.m3.1b"><apply id="S4.SS2.SSS1.p3.4.m3.1.1.cmml" xref="S4.SS2.SSS1.p3.4.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.4.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p3.4.m3.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.p3.4.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p3.4.m3.1.1.2">𝑘</ci><ci id="S4.SS2.SSS1.p3.4.m3.1.1.3a.cmml" xref="S4.SS2.SSS1.p3.4.m3.1.1.3"><mtext id="S4.SS2.SSS1.p3.4.m3.1.1.3.cmml" mathsize="70%" xref="S4.SS2.SSS1.p3.4.m3.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.4.m3.1c">k^{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.4.m3.1d">italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT</annotation></semantics></math>-order Markov chains. To understand why, note that in the <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.5.m4.1"><semantics id="S4.SS2.SSS1.p3.5.m4.1a"><mi id="S4.SS2.SSS1.p3.5.m4.1.1" xref="S4.SS2.SSS1.p3.5.m4.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.5.m4.1b"><ci id="S4.SS2.SSS1.p3.5.m4.1.1.cmml" xref="S4.SS2.SSS1.p3.5.m4.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.5.m4.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.5.m4.1d">sansserif_MambaZero</annotation></semantics></math> architecture above, apart from the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex13" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a> block, all remaining equations operate on the current token at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.6.m5.1"><semantics id="S4.SS2.SSS1.p3.6.m5.1a"><mi id="S4.SS2.SSS1.p3.6.m5.1.1" xref="S4.SS2.SSS1.p3.6.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.6.m5.1b"><ci id="S4.SS2.SSS1.p3.6.m5.1.1.cmml" xref="S4.SS2.SSS1.p3.6.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.6.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.6.m5.1d">italic_t</annotation></semantics></math>. In the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex13" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a> block, same as the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex1" title="Equation Mamba ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Mamba</span></a> block except ReLU, the dependency of the output <math alttext="\bm{y}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.7.m6.1"><semantics id="S4.SS2.SSS1.p3.7.m6.1a"><msub id="S4.SS2.SSS1.p3.7.m6.1.1" xref="S4.SS2.SSS1.p3.7.m6.1.1.cmml"><mi id="S4.SS2.SSS1.p3.7.m6.1.1.2" xref="S4.SS2.SSS1.p3.7.m6.1.1.2.cmml">𝒚</mi><mi id="S4.SS2.SSS1.p3.7.m6.1.1.3" xref="S4.SS2.SSS1.p3.7.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.7.m6.1b"><apply id="S4.SS2.SSS1.p3.7.m6.1.1.cmml" xref="S4.SS2.SSS1.p3.7.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.7.m6.1.1.1.cmml" xref="S4.SS2.SSS1.p3.7.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.7.m6.1.1.2.cmml" xref="S4.SS2.SSS1.p3.7.m6.1.1.2">𝒚</ci><ci id="S4.SS2.SSS1.p3.7.m6.1.1.3.cmml" xref="S4.SS2.SSS1.p3.7.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.7.m6.1c">\bm{y}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.7.m6.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> on the previous tokens is due to that of the state <math alttext="H_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.8.m7.1"><semantics id="S4.SS2.SSS1.p3.8.m7.1a"><msub id="S4.SS2.SSS1.p3.8.m7.1.1" xref="S4.SS2.SSS1.p3.8.m7.1.1.cmml"><mi id="S4.SS2.SSS1.p3.8.m7.1.1.2" xref="S4.SS2.SSS1.p3.8.m7.1.1.2.cmml">H</mi><mi id="S4.SS2.SSS1.p3.8.m7.1.1.3" xref="S4.SS2.SSS1.p3.8.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.8.m7.1b"><apply id="S4.SS2.SSS1.p3.8.m7.1.1.cmml" xref="S4.SS2.SSS1.p3.8.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.8.m7.1.1.1.cmml" xref="S4.SS2.SSS1.p3.8.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.8.m7.1.1.2.cmml" xref="S4.SS2.SSS1.p3.8.m7.1.1.2">𝐻</ci><ci id="S4.SS2.SSS1.p3.8.m7.1.1.3.cmml" xref="S4.SS2.SSS1.p3.8.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.8.m7.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.8.m7.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="(\widetilde{\bm{x}}_{t},\bm{b}_{t})" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.9.m8.2"><semantics id="S4.SS2.SSS1.p3.9.m8.2a"><mrow 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id="S4.SS2.SSS1.p3.9.m8.2.2.2.5" stretchy="false" xref="S4.SS2.SSS1.p3.9.m8.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.9.m8.2b"><interval closure="open" id="S4.SS2.SSS1.p3.9.m8.2.2.3.cmml" xref="S4.SS2.SSS1.p3.9.m8.2.2.2"><apply id="S4.SS2.SSS1.p3.9.m8.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.9.m8.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.9.m8.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.9.m8.1.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.p3.9.m8.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.9.m8.1.1.1.1.2"><ci id="S4.SS2.SSS1.p3.9.m8.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p3.9.m8.1.1.1.1.2.1">~</ci><ci id="S4.SS2.SSS1.p3.9.m8.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.p3.9.m8.1.1.1.1.2.2">𝒙</ci></apply><ci id="S4.SS2.SSS1.p3.9.m8.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.9.m8.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS2.SSS1.p3.9.m8.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.9.m8.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.9.m8.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p3.9.m8.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.9.m8.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.9.m8.2.2.2.2.2">𝒃</ci><ci id="S4.SS2.SSS1.p3.9.m8.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p3.9.m8.2.2.2.2.3">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.9.m8.2c">(\widetilde{\bm{x}}_{t},\bm{b}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.9.m8.2d">( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> in the update equation, and of <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.10.m9.1"><semantics id="S4.SS2.SSS1.p3.10.m9.1a"><msub id="S4.SS2.SSS1.p3.10.m9.1.1" xref="S4.SS2.SSS1.p3.10.m9.1.1.cmml"><mi id="S4.SS2.SSS1.p3.10.m9.1.1.2" xref="S4.SS2.SSS1.p3.10.m9.1.1.2.cmml">𝒄</mi><mi id="S4.SS2.SSS1.p3.10.m9.1.1.3" xref="S4.SS2.SSS1.p3.10.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.10.m9.1b"><apply id="S4.SS2.SSS1.p3.10.m9.1.1.cmml" xref="S4.SS2.SSS1.p3.10.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.10.m9.1.1.1.cmml" xref="S4.SS2.SSS1.p3.10.m9.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.10.m9.1.1.2.cmml" xref="S4.SS2.SSS1.p3.10.m9.1.1.2">𝒄</ci><ci id="S4.SS2.SSS1.p3.10.m9.1.1.3.cmml" xref="S4.SS2.SSS1.p3.10.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.10.m9.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.10.m9.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in the state projection. Since <math alttext="(\widetilde{\bm{x}}_{t},\bm{b}_{t},\bm{c}_{t})" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.11.m10.3"><semantics id="S4.SS2.SSS1.p3.11.m10.3a"><mrow id="S4.SS2.SSS1.p3.11.m10.3.3.3" xref="S4.SS2.SSS1.p3.11.m10.3.3.4.cmml"><mo id="S4.SS2.SSS1.p3.11.m10.3.3.3.4" stretchy="false" xref="S4.SS2.SSS1.p3.11.m10.3.3.4.cmml">(</mo><msub id="S4.SS2.SSS1.p3.11.m10.1.1.1.1" xref="S4.SS2.SSS1.p3.11.m10.1.1.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.p3.11.m10.1.1.1.1.2" xref="S4.SS2.SSS1.p3.11.m10.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p3.11.m10.1.1.1.1.2.2" xref="S4.SS2.SSS1.p3.11.m10.1.1.1.1.2.2.cmml">𝒙</mi><mo id="S4.SS2.SSS1.p3.11.m10.1.1.1.1.2.1" xref="S4.SS2.SSS1.p3.11.m10.1.1.1.1.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.p3.11.m10.1.1.1.1.3" xref="S4.SS2.SSS1.p3.11.m10.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p3.11.m10.3.3.3.5" xref="S4.SS2.SSS1.p3.11.m10.3.3.4.cmml">,</mo><msub id="S4.SS2.SSS1.p3.11.m10.2.2.2.2" 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xref="S4.SS2.SSS1.p3.11.m10.3.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS1.p3.11.m10.3.3.3.3.2.cmml" xref="S4.SS2.SSS1.p3.11.m10.3.3.3.3.2">𝒄</ci><ci id="S4.SS2.SSS1.p3.11.m10.3.3.3.3.3.cmml" xref="S4.SS2.SSS1.p3.11.m10.3.3.3.3.3">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.11.m10.3c">(\widetilde{\bm{x}}_{t},\bm{b}_{t},\bm{c}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.11.m10.3d">( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> depend on the past through the convolutions, a window of size <math alttext="k+1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.12.m11.1"><semantics id="S4.SS2.SSS1.p3.12.m11.1a"><mrow id="S4.SS2.SSS1.p3.12.m11.1.1" xref="S4.SS2.SSS1.p3.12.m11.1.1.cmml"><mi id="S4.SS2.SSS1.p3.12.m11.1.1.2" xref="S4.SS2.SSS1.p3.12.m11.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS1.p3.12.m11.1.1.1" xref="S4.SS2.SSS1.p3.12.m11.1.1.1.cmml">+</mo><mn id="S4.SS2.SSS1.p3.12.m11.1.1.3" xref="S4.SS2.SSS1.p3.12.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.12.m11.1b"><apply id="S4.SS2.SSS1.p3.12.m11.1.1.cmml" xref="S4.SS2.SSS1.p3.12.m11.1.1"><plus id="S4.SS2.SSS1.p3.12.m11.1.1.1.cmml" xref="S4.SS2.SSS1.p3.12.m11.1.1.1"></plus><ci id="S4.SS2.SSS1.p3.12.m11.1.1.2.cmml" xref="S4.SS2.SSS1.p3.12.m11.1.1.2">𝑘</ci><cn id="S4.SS2.SSS1.p3.12.m11.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p3.12.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.12.m11.1c">k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.12.m11.1d">italic_k + 1</annotation></semantics></math> enables them to keep track of the current token as well as its length-<math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.13.m12.1"><semantics id="S4.SS2.SSS1.p3.13.m12.1a"><mi id="S4.SS2.SSS1.p3.13.m12.1.1" xref="S4.SS2.SSS1.p3.13.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.13.m12.1b"><ci id="S4.SS2.SSS1.p3.13.m12.1.1.cmml" xref="S4.SS2.SSS1.p3.13.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.13.m12.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.13.m12.1d">italic_k</annotation></semantics></math> prefix, which is necessary to compute the counts needed in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex8" title="Equation Laplacian smoothing ‣ 2.4 Optimal estimator: Laplacian smoothing ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Laplacian smoothing</span></a>. On the other hand, if <math alttext="w_{X},w_{B}\leq k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.14.m13.2"><semantics id="S4.SS2.SSS1.p3.14.m13.2a"><mrow id="S4.SS2.SSS1.p3.14.m13.2.2" xref="S4.SS2.SSS1.p3.14.m13.2.2.cmml"><mrow id="S4.SS2.SSS1.p3.14.m13.2.2.2.2" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.3.cmml"><msub id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.2" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.2.cmml">w</mi><mi id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.3" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.3.cmml">X</mi></msub><mo id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.3" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.2" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.3" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.3.cmml">B</mi></msub></mrow><mo id="S4.SS2.SSS1.p3.14.m13.2.2.3" xref="S4.SS2.SSS1.p3.14.m13.2.2.3.cmml">≤</mo><mi id="S4.SS2.SSS1.p3.14.m13.2.2.4" xref="S4.SS2.SSS1.p3.14.m13.2.2.4.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.14.m13.2b"><apply id="S4.SS2.SSS1.p3.14.m13.2.2.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2"><leq id="S4.SS2.SSS1.p3.14.m13.2.2.3.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.3"></leq><list id="S4.SS2.SSS1.p3.14.m13.2.2.2.3.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2"><apply id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.2">𝑤</ci><ci id="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.14.m13.1.1.1.1.1.3">𝑋</ci></apply><apply id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.2.2.2.3">𝐵</ci></apply></list><ci id="S4.SS2.SSS1.p3.14.m13.2.2.4.cmml" xref="S4.SS2.SSS1.p3.14.m13.2.2.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.14.m13.2c">w_{X},w_{B}\leq k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.14.m13.2d">italic_w start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ≤ italic_k</annotation></semantics></math>, then one can find <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS1.p3.18.2">confusable</em> sequences, i.e. sequences that share the same number of occurrences of all length-<math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.15.m14.1"><semantics id="S4.SS2.SSS1.p3.15.m14.1a"><mi id="S4.SS2.SSS1.p3.15.m14.1.1" xref="S4.SS2.SSS1.p3.15.m14.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.15.m14.1b"><ci id="S4.SS2.SSS1.p3.15.m14.1.1.cmml" xref="S4.SS2.SSS1.p3.15.m14.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.15.m14.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.15.m14.1d">italic_k</annotation></semantics></math> prefixes, but whose counts of the tokens following each prefix is different, resulting in the model’s estimate to deviate from that of the optimal add-<math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.16.m15.1"><semantics id="S4.SS2.SSS1.p3.16.m15.1a"><mi id="S4.SS2.SSS1.p3.16.m15.1.1" xref="S4.SS2.SSS1.p3.16.m15.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.16.m15.1b"><ci id="S4.SS2.SSS1.p3.16.m15.1.1.cmml" xref="S4.SS2.SSS1.p3.16.m15.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.16.m15.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.16.m15.1d">italic_β</annotation></semantics></math>. We refer to § <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS1" title="B.1 Empirical insights ‣ Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">B.1</span></a> for more details. 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id="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3.1.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3.2.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3.2">𝑤</ci><ci id="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3.3.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.3.3.3.3">𝑐</ci></apply></list><apply id="S4.SS2.SSS1.p3.17.m16.3.3.5.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.5"><plus id="S4.SS2.SSS1.p3.17.m16.3.3.5.1.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.5.1"></plus><ci id="S4.SS2.SSS1.p3.17.m16.3.3.5.2.cmml" xref="S4.SS2.SSS1.p3.17.m16.3.3.5.2">𝑘</ci><cn id="S4.SS2.SSS1.p3.17.m16.3.3.5.3.cmml" type="integer" xref="S4.SS2.SSS1.p3.17.m16.3.3.5.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.17.m16.3c">w_{X},w_{B},w_{c}\geq k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.17.m16.3d">italic_w start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ≥ italic_k + 1</annotation></semantics></math> is sufficient, it can be further strengthened to <math alttext="w_{c}=k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.18.m17.1"><semantics id="S4.SS2.SSS1.p3.18.m17.1a"><mrow id="S4.SS2.SSS1.p3.18.m17.1.1" xref="S4.SS2.SSS1.p3.18.m17.1.1.cmml"><msub id="S4.SS2.SSS1.p3.18.m17.1.1.2" xref="S4.SS2.SSS1.p3.18.m17.1.1.2.cmml"><mi id="S4.SS2.SSS1.p3.18.m17.1.1.2.2" xref="S4.SS2.SSS1.p3.18.m17.1.1.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.p3.18.m17.1.1.2.3" xref="S4.SS2.SSS1.p3.18.m17.1.1.2.3.cmml">c</mi></msub><mo id="S4.SS2.SSS1.p3.18.m17.1.1.1" xref="S4.SS2.SSS1.p3.18.m17.1.1.1.cmml">=</mo><mi id="S4.SS2.SSS1.p3.18.m17.1.1.3" xref="S4.SS2.SSS1.p3.18.m17.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.18.m17.1b"><apply id="S4.SS2.SSS1.p3.18.m17.1.1.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1"><eq id="S4.SS2.SSS1.p3.18.m17.1.1.1.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1.1"></eq><apply id="S4.SS2.SSS1.p3.18.m17.1.1.2.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.18.m17.1.1.2.1.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.18.m17.1.1.2.2.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1.2.2">𝑤</ci><ci id="S4.SS2.SSS1.p3.18.m17.1.1.2.3.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.SSS1.p3.18.m17.1.1.3.cmml" xref="S4.SS2.SSS1.p3.18.m17.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.18.m17.1c">w_{c}=k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.18.m17.1d">italic_w start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_k</annotation></semantics></math> (§ <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS1" title="B.1 Empirical insights ‣ Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">B.1</span></a>).</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p4"> <p class="ltx_p" id="S4.SS2.SSS1.p4.1">We now detail our construction for the order-<math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p4.1.m1.1"><semantics id="S4.SS2.SSS1.p4.1.m1.1a"><mn id="S4.SS2.SSS1.p4.1.m1.1.1" xref="S4.SS2.SSS1.p4.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p4.1.m1.1b"><cn id="S4.SS2.SSS1.p4.1.m1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.p4.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p4.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.1.m1.1d">1</annotation></semantics></math> case, capitalizing on these insights.</p> </div> <div class="ltx_para ltx_noindent" id="S4.SS2.SSS1.p5"> <p class="ltx_p" id="S4.SS2.SSS1.p5.19"><span class="ltx_text ltx_font_bold" id="S4.SS2.SSS1.p5.19.1">Construction.</span> Let us fix <math alttext="w=k+1=2" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.1.m1.1"><semantics id="S4.SS2.SSS1.p5.1.m1.1a"><mrow id="S4.SS2.SSS1.p5.1.m1.1.1" xref="S4.SS2.SSS1.p5.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p5.1.m1.1.1.2" xref="S4.SS2.SSS1.p5.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.SSS1.p5.1.m1.1.1.3" xref="S4.SS2.SSS1.p5.1.m1.1.1.3.cmml">=</mo><mrow id="S4.SS2.SSS1.p5.1.m1.1.1.4" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.cmml"><mi id="S4.SS2.SSS1.p5.1.m1.1.1.4.2" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.2.cmml">k</mi><mo id="S4.SS2.SSS1.p5.1.m1.1.1.4.1" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.1.cmml">+</mo><mn id="S4.SS2.SSS1.p5.1.m1.1.1.4.3" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS1.p5.1.m1.1.1.5" xref="S4.SS2.SSS1.p5.1.m1.1.1.5.cmml">=</mo><mn id="S4.SS2.SSS1.p5.1.m1.1.1.6" xref="S4.SS2.SSS1.p5.1.m1.1.1.6.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.1.m1.1b"><apply id="S4.SS2.SSS1.p5.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1"><and id="S4.SS2.SSS1.p5.1.m1.1.1a.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1"></and><apply id="S4.SS2.SSS1.p5.1.m1.1.1b.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1"><eq id="S4.SS2.SSS1.p5.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1.3"></eq><ci id="S4.SS2.SSS1.p5.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p5.1.m1.1.1.4.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1.4"><plus id="S4.SS2.SSS1.p5.1.m1.1.1.4.1.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.1"></plus><ci id="S4.SS2.SSS1.p5.1.m1.1.1.4.2.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.2">𝑘</ci><cn id="S4.SS2.SSS1.p5.1.m1.1.1.4.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.1.m1.1.1.4.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.p5.1.m1.1.1c.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1"><eq id="S4.SS2.SSS1.p5.1.m1.1.1.5.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2502.10178v1#S4.SS2.SSS1.p5.1.m1.1.1.4.cmml" id="S4.SS2.SSS1.p5.1.m1.1.1d.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1"></share><cn id="S4.SS2.SSS1.p5.1.m1.1.1.6.cmml" type="integer" xref="S4.SS2.SSS1.p5.1.m1.1.1.6">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.1.m1.1c">w=k+1=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.1.m1.1d">italic_w = italic_k + 1 = 2</annotation></semantics></math>. Then, <math alttext="\widetilde{\bm{x}}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.2.m2.1"><semantics id="S4.SS2.SSS1.p5.2.m2.1a"><msub id="S4.SS2.SSS1.p5.2.m2.1.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.p5.2.m2.1.1.2" xref="S4.SS2.SSS1.p5.2.m2.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.2.m2.1.1.2.2" xref="S4.SS2.SSS1.p5.2.m2.1.1.2.2.cmml">𝒙</mi><mo id="S4.SS2.SSS1.p5.2.m2.1.1.2.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.p5.2.m2.1.1.3" xref="S4.SS2.SSS1.p5.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.2.m2.1b"><apply id="S4.SS2.SSS1.p5.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.p5.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.2"><ci id="S4.SS2.SSS1.p5.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.2.1">~</ci><ci id="S4.SS2.SSS1.p5.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.2.2">𝒙</ci></apply><ci id="S4.SS2.SSS1.p5.2.m2.1.1.3.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.2.m2.1c">\widetilde{\bm{x}}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.2.m2.1d">over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bm{b}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.3.m3.1"><semantics id="S4.SS2.SSS1.p5.3.m3.1a"><msub id="S4.SS2.SSS1.p5.3.m3.1.1" xref="S4.SS2.SSS1.p5.3.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p5.3.m3.1.1.2" xref="S4.SS2.SSS1.p5.3.m3.1.1.2.cmml">𝒃</mi><mi id="S4.SS2.SSS1.p5.3.m3.1.1.3" xref="S4.SS2.SSS1.p5.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.3.m3.1b"><apply id="S4.SS2.SSS1.p5.3.m3.1.1.cmml" xref="S4.SS2.SSS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.3.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p5.3.m3.1.1.2">𝒃</ci><ci id="S4.SS2.SSS1.p5.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.p5.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.3.m3.1c">\bm{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.3.m3.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> only depend on the current token <math alttext="x_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.4.m4.1"><semantics id="S4.SS2.SSS1.p5.4.m4.1a"><msub id="S4.SS2.SSS1.p5.4.m4.1.1" xref="S4.SS2.SSS1.p5.4.m4.1.1.cmml"><mi id="S4.SS2.SSS1.p5.4.m4.1.1.2" xref="S4.SS2.SSS1.p5.4.m4.1.1.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.4.m4.1.1.3" xref="S4.SS2.SSS1.p5.4.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.4.m4.1b"><apply id="S4.SS2.SSS1.p5.4.m4.1.1.cmml" xref="S4.SS2.SSS1.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.4.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p5.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.4.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p5.4.m4.1.1.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.4.m4.1.1.3.cmml" xref="S4.SS2.SSS1.p5.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.4.m4.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and the previous one <math alttext="x_{t-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.5.m5.1"><semantics id="S4.SS2.SSS1.p5.5.m5.1a"><msub id="S4.SS2.SSS1.p5.5.m5.1.1" xref="S4.SS2.SSS1.p5.5.m5.1.1.cmml"><mi id="S4.SS2.SSS1.p5.5.m5.1.1.2" xref="S4.SS2.SSS1.p5.5.m5.1.1.2.cmml">x</mi><mrow id="S4.SS2.SSS1.p5.5.m5.1.1.3" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.cmml"><mi id="S4.SS2.SSS1.p5.5.m5.1.1.3.2" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.2.cmml">t</mi><mo id="S4.SS2.SSS1.p5.5.m5.1.1.3.1" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p5.5.m5.1.1.3.3" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.5.m5.1b"><apply id="S4.SS2.SSS1.p5.5.m5.1.1.cmml" xref="S4.SS2.SSS1.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.5.m5.1.1.1.cmml" xref="S4.SS2.SSS1.p5.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.5.m5.1.1.2.cmml" xref="S4.SS2.SSS1.p5.5.m5.1.1.2">𝑥</ci><apply id="S4.SS2.SSS1.p5.5.m5.1.1.3.cmml" xref="S4.SS2.SSS1.p5.5.m5.1.1.3"><minus id="S4.SS2.SSS1.p5.5.m5.1.1.3.1.cmml" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p5.5.m5.1.1.3.2.cmml" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.2">𝑡</ci><cn id="S4.SS2.SSS1.p5.5.m5.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.5.m5.1c">x_{t-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.5.m5.1d">italic_x start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT</annotation></semantics></math>, while <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.6.m6.1"><semantics id="S4.SS2.SSS1.p5.6.m6.1a"><msub id="S4.SS2.SSS1.p5.6.m6.1.1" xref="S4.SS2.SSS1.p5.6.m6.1.1.cmml"><mi id="S4.SS2.SSS1.p5.6.m6.1.1.2" xref="S4.SS2.SSS1.p5.6.m6.1.1.2.cmml">𝒄</mi><mi id="S4.SS2.SSS1.p5.6.m6.1.1.3" xref="S4.SS2.SSS1.p5.6.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.6.m6.1b"><apply id="S4.SS2.SSS1.p5.6.m6.1.1.cmml" xref="S4.SS2.SSS1.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.6.m6.1.1.1.cmml" xref="S4.SS2.SSS1.p5.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.6.m6.1.1.2.cmml" xref="S4.SS2.SSS1.p5.6.m6.1.1.2">𝒄</ci><ci id="S4.SS2.SSS1.p5.6.m6.1.1.3.cmml" xref="S4.SS2.SSS1.p5.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.6.m6.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.6.m6.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> only depends on <math alttext="x_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.7.m7.1"><semantics id="S4.SS2.SSS1.p5.7.m7.1a"><msub id="S4.SS2.SSS1.p5.7.m7.1.1" xref="S4.SS2.SSS1.p5.7.m7.1.1.cmml"><mi id="S4.SS2.SSS1.p5.7.m7.1.1.2" xref="S4.SS2.SSS1.p5.7.m7.1.1.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.7.m7.1.1.3" xref="S4.SS2.SSS1.p5.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.7.m7.1b"><apply id="S4.SS2.SSS1.p5.7.m7.1.1.cmml" xref="S4.SS2.SSS1.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.7.m7.1.1.1.cmml" xref="S4.SS2.SSS1.p5.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.7.m7.1.1.2.cmml" xref="S4.SS2.SSS1.p5.7.m7.1.1.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.7.m7.1.1.3.cmml" xref="S4.SS2.SSS1.p5.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.7.m7.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.7.m7.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, <math alttext="\widetilde{\bm{x}}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.8.m8.1"><semantics id="S4.SS2.SSS1.p5.8.m8.1a"><msub id="S4.SS2.SSS1.p5.8.m8.1.1" xref="S4.SS2.SSS1.p5.8.m8.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.p5.8.m8.1.1.2" xref="S4.SS2.SSS1.p5.8.m8.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.8.m8.1.1.2.2" xref="S4.SS2.SSS1.p5.8.m8.1.1.2.2.cmml">𝒙</mi><mo id="S4.SS2.SSS1.p5.8.m8.1.1.2.1" xref="S4.SS2.SSS1.p5.8.m8.1.1.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.p5.8.m8.1.1.3" xref="S4.SS2.SSS1.p5.8.m8.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.8.m8.1b"><apply id="S4.SS2.SSS1.p5.8.m8.1.1.cmml" xref="S4.SS2.SSS1.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.8.m8.1.1.1.cmml" xref="S4.SS2.SSS1.p5.8.m8.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.p5.8.m8.1.1.2.cmml" xref="S4.SS2.SSS1.p5.8.m8.1.1.2"><ci id="S4.SS2.SSS1.p5.8.m8.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.8.m8.1.1.2.1">~</ci><ci id="S4.SS2.SSS1.p5.8.m8.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.8.m8.1.1.2.2">𝒙</ci></apply><ci id="S4.SS2.SSS1.p5.8.m8.1.1.3.cmml" xref="S4.SS2.SSS1.p5.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.8.m8.1c">\widetilde{\bm{x}}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.8.m8.1d">over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bm{b}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.9.m9.1"><semantics id="S4.SS2.SSS1.p5.9.m9.1a"><msub id="S4.SS2.SSS1.p5.9.m9.1.1" xref="S4.SS2.SSS1.p5.9.m9.1.1.cmml"><mi id="S4.SS2.SSS1.p5.9.m9.1.1.2" xref="S4.SS2.SSS1.p5.9.m9.1.1.2.cmml">𝒃</mi><mi id="S4.SS2.SSS1.p5.9.m9.1.1.3" xref="S4.SS2.SSS1.p5.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.9.m9.1b"><apply id="S4.SS2.SSS1.p5.9.m9.1.1.cmml" xref="S4.SS2.SSS1.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.9.m9.1.1.1.cmml" xref="S4.SS2.SSS1.p5.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.9.m9.1.1.2.cmml" xref="S4.SS2.SSS1.p5.9.m9.1.1.2">𝒃</ci><ci id="S4.SS2.SSS1.p5.9.m9.1.1.3.cmml" xref="S4.SS2.SSS1.p5.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.9.m9.1c">\bm{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.9.m9.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> can only take four possible values depending on the last transition in the sequence, whereas <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.10.m10.1"><semantics id="S4.SS2.SSS1.p5.10.m10.1a"><msub id="S4.SS2.SSS1.p5.10.m10.1.1" xref="S4.SS2.SSS1.p5.10.m10.1.1.cmml"><mi id="S4.SS2.SSS1.p5.10.m10.1.1.2" xref="S4.SS2.SSS1.p5.10.m10.1.1.2.cmml">𝒄</mi><mi id="S4.SS2.SSS1.p5.10.m10.1.1.3" xref="S4.SS2.SSS1.p5.10.m10.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.10.m10.1b"><apply id="S4.SS2.SSS1.p5.10.m10.1.1.cmml" xref="S4.SS2.SSS1.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.10.m10.1.1.1.cmml" xref="S4.SS2.SSS1.p5.10.m10.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.10.m10.1.1.2.cmml" xref="S4.SS2.SSS1.p5.10.m10.1.1.2">𝒄</ci><ci id="S4.SS2.SSS1.p5.10.m10.1.1.3.cmml" xref="S4.SS2.SSS1.p5.10.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.10.m10.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.10.m10.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> only two. To ease the notation, we will denote these values by <math alttext="\widetilde{\bm{x}}^{(ij)}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.11.m11.1"><semantics id="S4.SS2.SSS1.p5.11.m11.1a"><msup id="S4.SS2.SSS1.p5.11.m11.1.2" xref="S4.SS2.SSS1.p5.11.m11.1.2.cmml"><mover accent="true" id="S4.SS2.SSS1.p5.11.m11.1.2.2" xref="S4.SS2.SSS1.p5.11.m11.1.2.2.cmml"><mi id="S4.SS2.SSS1.p5.11.m11.1.2.2.2" xref="S4.SS2.SSS1.p5.11.m11.1.2.2.2.cmml">𝒙</mi><mo id="S4.SS2.SSS1.p5.11.m11.1.2.2.1" xref="S4.SS2.SSS1.p5.11.m11.1.2.2.1.cmml">~</mo></mover><mrow id="S4.SS2.SSS1.p5.11.m11.1.1.1.1" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.2" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.2.cmml">i</mi><mo id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.1" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.3" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.3.cmml">j</mi></mrow><mo id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.11.m11.1b"><apply id="S4.SS2.SSS1.p5.11.m11.1.2.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.11.m11.1.2.1.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.2">superscript</csymbol><apply id="S4.SS2.SSS1.p5.11.m11.1.2.2.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.2.2"><ci id="S4.SS2.SSS1.p5.11.m11.1.2.2.1.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.2.2.1">~</ci><ci id="S4.SS2.SSS1.p5.11.m11.1.2.2.2.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.2.2.2">𝒙</ci></apply><apply id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1"><times id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.1"></times><ci id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.2">𝑖</ci><ci id="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p5.11.m11.1.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.11.m11.1c">\widetilde{\bm{x}}^{(ij)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.11.m11.1d">over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_i italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\bm{b}^{(ij)}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.12.m12.1"><semantics id="S4.SS2.SSS1.p5.12.m12.1a"><msup id="S4.SS2.SSS1.p5.12.m12.1.2" xref="S4.SS2.SSS1.p5.12.m12.1.2.cmml"><mi id="S4.SS2.SSS1.p5.12.m12.1.2.2" xref="S4.SS2.SSS1.p5.12.m12.1.2.2.cmml">𝒃</mi><mrow id="S4.SS2.SSS1.p5.12.m12.1.1.1.1" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.2" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.2.cmml">i</mi><mo id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.1" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.3" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.3.cmml">j</mi></mrow><mo id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.12.m12.1b"><apply id="S4.SS2.SSS1.p5.12.m12.1.2.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.12.m12.1.2.1.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.2">superscript</csymbol><ci id="S4.SS2.SSS1.p5.12.m12.1.2.2.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.2.2">𝒃</ci><apply id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1"><times id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.1"></times><ci id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.2">𝑖</ci><ci id="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p5.12.m12.1.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.12.m12.1c">\bm{b}^{(ij)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.12.m12.1d">bold_italic_b start_POSTSUPERSCRIPT ( italic_i italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\bm{C}^{(i)}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.13.m13.1"><semantics id="S4.SS2.SSS1.p5.13.m13.1a"><msup id="S4.SS2.SSS1.p5.13.m13.1.2" xref="S4.SS2.SSS1.p5.13.m13.1.2.cmml"><mi id="S4.SS2.SSS1.p5.13.m13.1.2.2" xref="S4.SS2.SSS1.p5.13.m13.1.2.2.cmml">𝑪</mi><mrow id="S4.SS2.SSS1.p5.13.m13.1.1.1.3" xref="S4.SS2.SSS1.p5.13.m13.1.2.cmml"><mo id="S4.SS2.SSS1.p5.13.m13.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.p5.13.m13.1.2.cmml">(</mo><mi id="S4.SS2.SSS1.p5.13.m13.1.1.1.1" xref="S4.SS2.SSS1.p5.13.m13.1.1.1.1.cmml">i</mi><mo id="S4.SS2.SSS1.p5.13.m13.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.p5.13.m13.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.13.m13.1b"><apply id="S4.SS2.SSS1.p5.13.m13.1.2.cmml" xref="S4.SS2.SSS1.p5.13.m13.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.13.m13.1.2.1.cmml" xref="S4.SS2.SSS1.p5.13.m13.1.2">superscript</csymbol><ci id="S4.SS2.SSS1.p5.13.m13.1.2.2.cmml" xref="S4.SS2.SSS1.p5.13.m13.1.2.2">𝑪</ci><ci id="S4.SS2.SSS1.p5.13.m13.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.13.m13.1.1.1.1">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.13.m13.1c">\bm{C}^{(i)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.13.m13.1d">bold_italic_C start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT</annotation></semantics></math> respectively, for <math alttext="i,j\in\{0,1\}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.14.m14.4"><semantics id="S4.SS2.SSS1.p5.14.m14.4a"><mrow id="S4.SS2.SSS1.p5.14.m14.4.5" xref="S4.SS2.SSS1.p5.14.m14.4.5.cmml"><mrow id="S4.SS2.SSS1.p5.14.m14.4.5.2.2" xref="S4.SS2.SSS1.p5.14.m14.4.5.2.1.cmml"><mi id="S4.SS2.SSS1.p5.14.m14.3.3" xref="S4.SS2.SSS1.p5.14.m14.3.3.cmml">i</mi><mo id="S4.SS2.SSS1.p5.14.m14.4.5.2.2.1" xref="S4.SS2.SSS1.p5.14.m14.4.5.2.1.cmml">,</mo><mi id="S4.SS2.SSS1.p5.14.m14.4.4" xref="S4.SS2.SSS1.p5.14.m14.4.4.cmml">j</mi></mrow><mo id="S4.SS2.SSS1.p5.14.m14.4.5.1" xref="S4.SS2.SSS1.p5.14.m14.4.5.1.cmml">∈</mo><mrow id="S4.SS2.SSS1.p5.14.m14.4.5.3.2" xref="S4.SS2.SSS1.p5.14.m14.4.5.3.1.cmml"><mo id="S4.SS2.SSS1.p5.14.m14.4.5.3.2.1" stretchy="false" xref="S4.SS2.SSS1.p5.14.m14.4.5.3.1.cmml">{</mo><mn id="S4.SS2.SSS1.p5.14.m14.1.1" xref="S4.SS2.SSS1.p5.14.m14.1.1.cmml">0</mn><mo id="S4.SS2.SSS1.p5.14.m14.4.5.3.2.2" xref="S4.SS2.SSS1.p5.14.m14.4.5.3.1.cmml">,</mo><mn id="S4.SS2.SSS1.p5.14.m14.2.2" xref="S4.SS2.SSS1.p5.14.m14.2.2.cmml">1</mn><mo id="S4.SS2.SSS1.p5.14.m14.4.5.3.2.3" stretchy="false" xref="S4.SS2.SSS1.p5.14.m14.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.14.m14.4b"><apply id="S4.SS2.SSS1.p5.14.m14.4.5.cmml" xref="S4.SS2.SSS1.p5.14.m14.4.5"><in id="S4.SS2.SSS1.p5.14.m14.4.5.1.cmml" xref="S4.SS2.SSS1.p5.14.m14.4.5.1"></in><list id="S4.SS2.SSS1.p5.14.m14.4.5.2.1.cmml" xref="S4.SS2.SSS1.p5.14.m14.4.5.2.2"><ci id="S4.SS2.SSS1.p5.14.m14.3.3.cmml" xref="S4.SS2.SSS1.p5.14.m14.3.3">𝑖</ci><ci id="S4.SS2.SSS1.p5.14.m14.4.4.cmml" xref="S4.SS2.SSS1.p5.14.m14.4.4">𝑗</ci></list><set id="S4.SS2.SSS1.p5.14.m14.4.5.3.1.cmml" xref="S4.SS2.SSS1.p5.14.m14.4.5.3.2"><cn id="S4.SS2.SSS1.p5.14.m14.1.1.cmml" type="integer" xref="S4.SS2.SSS1.p5.14.m14.1.1">0</cn><cn id="S4.SS2.SSS1.p5.14.m14.2.2.cmml" type="integer" xref="S4.SS2.SSS1.p5.14.m14.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.14.m14.4c">i,j\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.14.m14.4d">italic_i , italic_j ∈ { 0 , 1 }</annotation></semantics></math>. Additionally, at <math alttext="t=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.15.m15.1"><semantics id="S4.SS2.SSS1.p5.15.m15.1a"><mrow id="S4.SS2.SSS1.p5.15.m15.1.1" xref="S4.SS2.SSS1.p5.15.m15.1.1.cmml"><mi id="S4.SS2.SSS1.p5.15.m15.1.1.2" xref="S4.SS2.SSS1.p5.15.m15.1.1.2.cmml">t</mi><mo id="S4.SS2.SSS1.p5.15.m15.1.1.1" xref="S4.SS2.SSS1.p5.15.m15.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.15.m15.1.1.3" xref="S4.SS2.SSS1.p5.15.m15.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.15.m15.1b"><apply id="S4.SS2.SSS1.p5.15.m15.1.1.cmml" xref="S4.SS2.SSS1.p5.15.m15.1.1"><eq id="S4.SS2.SSS1.p5.15.m15.1.1.1.cmml" xref="S4.SS2.SSS1.p5.15.m15.1.1.1"></eq><ci id="S4.SS2.SSS1.p5.15.m15.1.1.2.cmml" xref="S4.SS2.SSS1.p5.15.m15.1.1.2">𝑡</ci><cn id="S4.SS2.SSS1.p5.15.m15.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.15.m15.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.15.m15.1d">italic_t = 1</annotation></semantics></math>, these terms depend only on the current symbol, taking two additional values each, denoted by <math alttext="\widetilde{\bm{x}}^{(i)},\bm{b}^{(i)}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.16.m16.4"><semantics id="S4.SS2.SSS1.p5.16.m16.4a"><mrow id="S4.SS2.SSS1.p5.16.m16.4.4.2" xref="S4.SS2.SSS1.p5.16.m16.4.4.3.cmml"><msup id="S4.SS2.SSS1.p5.16.m16.3.3.1.1" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.2" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.2.cmml">𝒙</mi><mo id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.1" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.1.cmml">~</mo></mover><mrow id="S4.SS2.SSS1.p5.16.m16.1.1.1.3" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.cmml"><mo id="S4.SS2.SSS1.p5.16.m16.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.cmml">(</mo><mi id="S4.SS2.SSS1.p5.16.m16.1.1.1.1" xref="S4.SS2.SSS1.p5.16.m16.1.1.1.1.cmml">i</mi><mo id="S4.SS2.SSS1.p5.16.m16.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS2.SSS1.p5.16.m16.4.4.2.3" xref="S4.SS2.SSS1.p5.16.m16.4.4.3.cmml">,</mo><msup id="S4.SS2.SSS1.p5.16.m16.4.4.2.2" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2.cmml"><mi id="S4.SS2.SSS1.p5.16.m16.4.4.2.2.2" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2.2.cmml">𝒃</mi><mrow id="S4.SS2.SSS1.p5.16.m16.2.2.1.3" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2.cmml"><mo id="S4.SS2.SSS1.p5.16.m16.2.2.1.3.1" stretchy="false" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2.cmml">(</mo><mi id="S4.SS2.SSS1.p5.16.m16.2.2.1.1" xref="S4.SS2.SSS1.p5.16.m16.2.2.1.1.cmml">i</mi><mo id="S4.SS2.SSS1.p5.16.m16.2.2.1.3.2" stretchy="false" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2.cmml">)</mo></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.16.m16.4b"><list id="S4.SS2.SSS1.p5.16.m16.4.4.3.cmml" xref="S4.SS2.SSS1.p5.16.m16.4.4.2"><apply id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.cmml" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.1.cmml" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.cmml" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2"><ci id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.1">~</ci><ci id="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.16.m16.3.3.1.1.2.2">𝒙</ci></apply><ci id="S4.SS2.SSS1.p5.16.m16.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.16.m16.1.1.1.1">𝑖</ci></apply><apply id="S4.SS2.SSS1.p5.16.m16.4.4.2.2.cmml" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.16.m16.4.4.2.2.1.cmml" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2">superscript</csymbol><ci id="S4.SS2.SSS1.p5.16.m16.4.4.2.2.2.cmml" xref="S4.SS2.SSS1.p5.16.m16.4.4.2.2.2">𝒃</ci><ci id="S4.SS2.SSS1.p5.16.m16.2.2.1.1.cmml" xref="S4.SS2.SSS1.p5.16.m16.2.2.1.1">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.16.m16.4c">\widetilde{\bm{x}}^{(i)},\bm{b}^{(i)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.16.m16.4d">over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT , bold_italic_b start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="n_{ij}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.17.m17.1"><semantics id="S4.SS2.SSS1.p5.17.m17.1a"><msub id="S4.SS2.SSS1.p5.17.m17.1.1" xref="S4.SS2.SSS1.p5.17.m17.1.1.cmml"><mi id="S4.SS2.SSS1.p5.17.m17.1.1.2" xref="S4.SS2.SSS1.p5.17.m17.1.1.2.cmml">n</mi><mrow id="S4.SS2.SSS1.p5.17.m17.1.1.3" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.cmml"><mi id="S4.SS2.SSS1.p5.17.m17.1.1.3.2" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.2.cmml">i</mi><mo id="S4.SS2.SSS1.p5.17.m17.1.1.3.1" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.p5.17.m17.1.1.3.3" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.17.m17.1b"><apply id="S4.SS2.SSS1.p5.17.m17.1.1.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.17.m17.1.1.1.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.17.m17.1.1.2.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1.2">𝑛</ci><apply id="S4.SS2.SSS1.p5.17.m17.1.1.3.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1.3"><times id="S4.SS2.SSS1.p5.17.m17.1.1.3.1.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.1"></times><ci id="S4.SS2.SSS1.p5.17.m17.1.1.3.2.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.2">𝑖</ci><ci id="S4.SS2.SSS1.p5.17.m17.1.1.3.3.cmml" xref="S4.SS2.SSS1.p5.17.m17.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.17.m17.1c">n_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.17.m17.1d">italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> denote the number of transitions <math alttext="i\to j" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.18.m18.1"><semantics id="S4.SS2.SSS1.p5.18.m18.1a"><mrow id="S4.SS2.SSS1.p5.18.m18.1.1" xref="S4.SS2.SSS1.p5.18.m18.1.1.cmml"><mi id="S4.SS2.SSS1.p5.18.m18.1.1.2" xref="S4.SS2.SSS1.p5.18.m18.1.1.2.cmml">i</mi><mo id="S4.SS2.SSS1.p5.18.m18.1.1.1" stretchy="false" xref="S4.SS2.SSS1.p5.18.m18.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.p5.18.m18.1.1.3" xref="S4.SS2.SSS1.p5.18.m18.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.18.m18.1b"><apply id="S4.SS2.SSS1.p5.18.m18.1.1.cmml" xref="S4.SS2.SSS1.p5.18.m18.1.1"><ci id="S4.SS2.SSS1.p5.18.m18.1.1.1.cmml" xref="S4.SS2.SSS1.p5.18.m18.1.1.1">→</ci><ci id="S4.SS2.SSS1.p5.18.m18.1.1.2.cmml" xref="S4.SS2.SSS1.p5.18.m18.1.1.2">𝑖</ci><ci id="S4.SS2.SSS1.p5.18.m18.1.1.3.cmml" xref="S4.SS2.SSS1.p5.18.m18.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.18.m18.1c">i\to j</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.18.m18.1d">italic_i → italic_j</annotation></semantics></math> in the input sequence <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.19.m19.1"><semantics id="S4.SS2.SSS1.p5.19.m19.1a"><msubsup id="S4.SS2.SSS1.p5.19.m19.1.1" xref="S4.SS2.SSS1.p5.19.m19.1.1.cmml"><mi id="S4.SS2.SSS1.p5.19.m19.1.1.2.2" xref="S4.SS2.SSS1.p5.19.m19.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS1.p5.19.m19.1.1.2.3" xref="S4.SS2.SSS1.p5.19.m19.1.1.2.3.cmml">1</mn><mi id="S4.SS2.SSS1.p5.19.m19.1.1.3" xref="S4.SS2.SSS1.p5.19.m19.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.19.m19.1b"><apply id="S4.SS2.SSS1.p5.19.m19.1.1.cmml" xref="S4.SS2.SSS1.p5.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.19.m19.1.1.1.cmml" xref="S4.SS2.SSS1.p5.19.m19.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.p5.19.m19.1.1.2.cmml" xref="S4.SS2.SSS1.p5.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.19.m19.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.19.m19.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.19.m19.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.19.m19.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS1.p5.19.m19.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.19.m19.1.1.2.3">1</cn></apply><ci id="S4.SS2.SSS1.p5.19.m19.1.1.3.cmml" xref="S4.SS2.SSS1.p5.19.m19.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.19.m19.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.19.m19.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. Then, unfolding the state update recursion in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex10" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a>,</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H_{t}=\widetilde{\bm{x}}_{0}\bm{b}_{0}^{\top}+\sum_{ij}n_{ij}\,\widetilde{\bm{% x}}^{(ij)}\bm{b}^{(ij)\top}." class="ltx_Math" display="block" id="S4.Ex15.m1.3"><semantics id="S4.Ex15.m1.3a"><mrow id="S4.Ex15.m1.3.3.1" xref="S4.Ex15.m1.3.3.1.1.cmml"><mrow id="S4.Ex15.m1.3.3.1.1" xref="S4.Ex15.m1.3.3.1.1.cmml"><msub id="S4.Ex15.m1.3.3.1.1.2" xref="S4.Ex15.m1.3.3.1.1.2.cmml"><mi id="S4.Ex15.m1.3.3.1.1.2.2" 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id="S4.Ex15.m1.3c">H_{t}=\widetilde{\bm{x}}_{0}\bm{b}_{0}^{\top}+\sum_{ij}n_{ij}\,\widetilde{\bm{% x}}^{(ij)}\bm{b}^{(ij)\top}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.3d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_i italic_j ) end_POSTSUPERSCRIPT bold_italic_b start_POSTSUPERSCRIPT ( italic_i italic_j ) ⊤ end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p5.52">Thus the output of the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex10" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a> block is</p> <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\bm{o}_{t}=W_{o}\,\widetilde{\bm{x}}_{0}\bm{b}_{0}^{\top}\bm{c}_{t}+\sum_{ij}n% _{ij}\,W_{o}\,\widetilde{\bm{x}}^{(ij)}\bm{b}^{(ij)\top}\bm{c}_{t}." class="ltx_Math" display="block" id="S4.E3.m1.3"><semantics id="S4.E3.m1.3a"><mrow id="S4.E3.m1.3.3.1" xref="S4.E3.m1.3.3.1.1.cmml"><mrow id="S4.E3.m1.3.3.1.1" xref="S4.E3.m1.3.3.1.1.cmml"><msub id="S4.E3.m1.3.3.1.1.2" xref="S4.E3.m1.3.3.1.1.2.cmml"><mi id="S4.E3.m1.3.3.1.1.2.2" xref="S4.E3.m1.3.3.1.1.2.2.cmml">𝒐</mi><mi id="S4.E3.m1.3.3.1.1.2.3" xref="S4.E3.m1.3.3.1.1.2.3.cmml">t</mi></msub><mo 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id="S4.E3.m1.3c">\bm{o}_{t}=W_{o}\,\widetilde{\bm{x}}_{0}\bm{b}_{0}^{\top}\bm{c}_{t}+\sum_{ij}n% _{ij}\,W_{o}\,\widetilde{\bm{x}}^{(ij)}\bm{b}^{(ij)\top}\bm{c}_{t}.</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.3d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_i italic_j ) end_POSTSUPERSCRIPT bold_italic_b start_POSTSUPERSCRIPT ( italic_i italic_j ) ⊤ end_POSTSUPERSCRIPT bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p5.22">While the output in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.E3" title="Equation 3 ‣ 4.2.1 Proof sketch ‣ 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3</span></a>) depends on all the transition counts, in view of <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex8" title="Equation Laplacian smoothing ‣ 2.4 Optimal estimator: Laplacian smoothing ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Laplacian smoothing</span></a>, we ideally want only those counts pertaining to relevant transitions, i.e. if <math alttext="x_{t}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.20.m1.1"><semantics id="S4.SS2.SSS1.p5.20.m1.1a"><mrow id="S4.SS2.SSS1.p5.20.m1.1.1" xref="S4.SS2.SSS1.p5.20.m1.1.1.cmml"><msub id="S4.SS2.SSS1.p5.20.m1.1.1.2" xref="S4.SS2.SSS1.p5.20.m1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.20.m1.1.1.2.2" xref="S4.SS2.SSS1.p5.20.m1.1.1.2.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.20.m1.1.1.2.3" xref="S4.SS2.SSS1.p5.20.m1.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p5.20.m1.1.1.1" xref="S4.SS2.SSS1.p5.20.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.20.m1.1.1.3" xref="S4.SS2.SSS1.p5.20.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.20.m1.1b"><apply id="S4.SS2.SSS1.p5.20.m1.1.1.cmml" xref="S4.SS2.SSS1.p5.20.m1.1.1"><eq id="S4.SS2.SSS1.p5.20.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.20.m1.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.20.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.20.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.20.m1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.20.m1.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.20.m1.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.20.m1.1.1.2.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.20.m1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p5.20.m1.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p5.20.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.20.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.20.m1.1c">x_{t}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.20.m1.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>, the counts <math alttext="n_{00}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.21.m2.1"><semantics id="S4.SS2.SSS1.p5.21.m2.1a"><msub id="S4.SS2.SSS1.p5.21.m2.1.1" xref="S4.SS2.SSS1.p5.21.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p5.21.m2.1.1.2" xref="S4.SS2.SSS1.p5.21.m2.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.21.m2.1.1.3" xref="S4.SS2.SSS1.p5.21.m2.1.1.3.cmml">00</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.21.m2.1b"><apply id="S4.SS2.SSS1.p5.21.m2.1.1.cmml" xref="S4.SS2.SSS1.p5.21.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.21.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p5.21.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.21.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p5.21.m2.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.21.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.21.m2.1.1.3">00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.21.m2.1c">n_{00}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.21.m2.1d">italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="n_{01}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.22.m3.1"><semantics id="S4.SS2.SSS1.p5.22.m3.1a"><msub id="S4.SS2.SSS1.p5.22.m3.1.1" xref="S4.SS2.SSS1.p5.22.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p5.22.m3.1.1.2" xref="S4.SS2.SSS1.p5.22.m3.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.22.m3.1.1.3" xref="S4.SS2.SSS1.p5.22.m3.1.1.3.cmml">01</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.22.m3.1b"><apply id="S4.SS2.SSS1.p5.22.m3.1.1.cmml" xref="S4.SS2.SSS1.p5.22.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.22.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p5.22.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.22.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p5.22.m3.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.22.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.22.m3.1.1.3">01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.22.m3.1c">n_{01}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.22.m3.1d">italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT</annotation></semantics></math>, and vice versa. To this end, we empirically observe that at convergence, the model’s parameters are such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx6"> <tbody id="S4.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bm{b}^{(11)\top}\bm{c}^{(0)}\approx 0\,\,\text{and}\,\,\bm{b}^{(% 00)\top}\bm{c}^{(1)}\approx 0." class="ltx_Math" display="inline" id="S4.Ex16.m1.5"><semantics id="S4.Ex16.m1.5a"><mrow id="S4.Ex16.m1.5.5.1" xref="S4.Ex16.m1.5.5.1.1.cmml"><mrow id="S4.Ex16.m1.5.5.1.1" xref="S4.Ex16.m1.5.5.1.1.cmml"><mrow id="S4.Ex16.m1.5.5.1.1.2" xref="S4.Ex16.m1.5.5.1.1.2.cmml"><msup id="S4.Ex16.m1.5.5.1.1.2.2" xref="S4.Ex16.m1.5.5.1.1.2.2.cmml"><mi id="S4.Ex16.m1.5.5.1.1.2.2.2" xref="S4.Ex16.m1.5.5.1.1.2.2.2.cmml">𝒃</mi><mrow id="S4.Ex16.m1.1.1.1" xref="S4.Ex16.m1.1.1.1.cmml"><mrow id="S4.Ex16.m1.1.1.1.3.2" xref="S4.Ex16.m1.1.1.1.cmml"><mo id="S4.Ex16.m1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex16.m1.1.1.1.cmml">(</mo><mn id="S4.Ex16.m1.1.1.1.1" xref="S4.Ex16.m1.1.1.1.1.cmml">11</mn><mo id="S4.Ex16.m1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex16.m1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex16.m1.1.1.1.4" lspace="0.222em" xref="S4.Ex16.m1.1.1.1.4.cmml">⊤</mo></mrow></msup><mo id="S4.Ex16.m1.5.5.1.1.2.1" xref="S4.Ex16.m1.5.5.1.1.2.1.cmml">⁢</mo><msup id="S4.Ex16.m1.5.5.1.1.2.3" xref="S4.Ex16.m1.5.5.1.1.2.3.cmml"><mi id="S4.Ex16.m1.5.5.1.1.2.3.2" xref="S4.Ex16.m1.5.5.1.1.2.3.2.cmml">𝒄</mi><mrow id="S4.Ex16.m1.2.2.1.3" xref="S4.Ex16.m1.5.5.1.1.2.3.cmml"><mo id="S4.Ex16.m1.2.2.1.3.1" stretchy="false" xref="S4.Ex16.m1.5.5.1.1.2.3.cmml">(</mo><mn id="S4.Ex16.m1.2.2.1.1" xref="S4.Ex16.m1.2.2.1.1.cmml">0</mn><mo id="S4.Ex16.m1.2.2.1.3.2" stretchy="false" xref="S4.Ex16.m1.5.5.1.1.2.3.cmml">)</mo></mrow></msup></mrow><mo id="S4.Ex16.m1.5.5.1.1.3" xref="S4.Ex16.m1.5.5.1.1.3.cmml">≈</mo><mrow 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encoding="application/x-tex" id="S4.Ex16.m1.5c">\displaystyle\bm{b}^{(11)\top}\bm{c}^{(0)}\approx 0\,\,\text{and}\,\,\bm{b}^{(% 00)\top}\bm{c}^{(1)}\approx 0.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.5d">bold_italic_b start_POSTSUPERSCRIPT ( 11 ) ⊤ end_POSTSUPERSCRIPT bold_italic_c start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ≈ 0 and bold_italic_b start_POSTSUPERSCRIPT ( 00 ) ⊤ end_POSTSUPERSCRIPT bold_italic_c start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ≈ 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p5.46">Due to this property, the counts <math alttext="n_{00}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.23.m1.1"><semantics id="S4.SS2.SSS1.p5.23.m1.1a"><msub id="S4.SS2.SSS1.p5.23.m1.1.1" xref="S4.SS2.SSS1.p5.23.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p5.23.m1.1.1.2" xref="S4.SS2.SSS1.p5.23.m1.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.23.m1.1.1.3" xref="S4.SS2.SSS1.p5.23.m1.1.1.3.cmml">00</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.23.m1.1b"><apply id="S4.SS2.SSS1.p5.23.m1.1.1.cmml" xref="S4.SS2.SSS1.p5.23.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.23.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.23.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.23.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.23.m1.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.23.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.23.m1.1.1.3">00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.23.m1.1c">n_{00}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.23.m1.1d">italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="n_{11}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.24.m2.1"><semantics id="S4.SS2.SSS1.p5.24.m2.1a"><msub id="S4.SS2.SSS1.p5.24.m2.1.1" xref="S4.SS2.SSS1.p5.24.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p5.24.m2.1.1.2" xref="S4.SS2.SSS1.p5.24.m2.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.24.m2.1.1.3" xref="S4.SS2.SSS1.p5.24.m2.1.1.3.cmml">11</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.24.m2.1b"><apply id="S4.SS2.SSS1.p5.24.m2.1.1.cmml" xref="S4.SS2.SSS1.p5.24.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.24.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p5.24.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.24.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p5.24.m2.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.24.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.24.m2.1.1.3">11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.24.m2.1c">n_{11}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.24.m2.1d">italic_n start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT</annotation></semantics></math> do not contribute to the output <math alttext="\bm{o}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.25.m3.1"><semantics id="S4.SS2.SSS1.p5.25.m3.1a"><msub id="S4.SS2.SSS1.p5.25.m3.1.1" xref="S4.SS2.SSS1.p5.25.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p5.25.m3.1.1.2" xref="S4.SS2.SSS1.p5.25.m3.1.1.2.cmml">𝒐</mi><mi id="S4.SS2.SSS1.p5.25.m3.1.1.3" xref="S4.SS2.SSS1.p5.25.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.25.m3.1b"><apply id="S4.SS2.SSS1.p5.25.m3.1.1.cmml" xref="S4.SS2.SSS1.p5.25.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.25.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p5.25.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.25.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p5.25.m3.1.1.2">𝒐</ci><ci id="S4.SS2.SSS1.p5.25.m3.1.1.3.cmml" xref="S4.SS2.SSS1.p5.25.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.25.m3.1c">\bm{o}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.25.m3.1d">bold_italic_o start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, respectively when <math alttext="x_{t}=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.26.m4.1"><semantics id="S4.SS2.SSS1.p5.26.m4.1a"><mrow id="S4.SS2.SSS1.p5.26.m4.1.1" xref="S4.SS2.SSS1.p5.26.m4.1.1.cmml"><msub id="S4.SS2.SSS1.p5.26.m4.1.1.2" xref="S4.SS2.SSS1.p5.26.m4.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.26.m4.1.1.2.2" xref="S4.SS2.SSS1.p5.26.m4.1.1.2.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.26.m4.1.1.2.3" xref="S4.SS2.SSS1.p5.26.m4.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p5.26.m4.1.1.1" xref="S4.SS2.SSS1.p5.26.m4.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.26.m4.1.1.3" xref="S4.SS2.SSS1.p5.26.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.26.m4.1b"><apply id="S4.SS2.SSS1.p5.26.m4.1.1.cmml" xref="S4.SS2.SSS1.p5.26.m4.1.1"><eq id="S4.SS2.SSS1.p5.26.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p5.26.m4.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.26.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p5.26.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.26.m4.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.26.m4.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.26.m4.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.26.m4.1.1.2.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.26.m4.1.1.2.3.cmml" xref="S4.SS2.SSS1.p5.26.m4.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p5.26.m4.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.26.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.26.m4.1c">x_{t}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.26.m4.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="x_{t}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.27.m5.1"><semantics id="S4.SS2.SSS1.p5.27.m5.1a"><mrow id="S4.SS2.SSS1.p5.27.m5.1.1" xref="S4.SS2.SSS1.p5.27.m5.1.1.cmml"><msub id="S4.SS2.SSS1.p5.27.m5.1.1.2" xref="S4.SS2.SSS1.p5.27.m5.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.27.m5.1.1.2.2" xref="S4.SS2.SSS1.p5.27.m5.1.1.2.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.27.m5.1.1.2.3" xref="S4.SS2.SSS1.p5.27.m5.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p5.27.m5.1.1.1" xref="S4.SS2.SSS1.p5.27.m5.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.27.m5.1.1.3" xref="S4.SS2.SSS1.p5.27.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.27.m5.1b"><apply id="S4.SS2.SSS1.p5.27.m5.1.1.cmml" xref="S4.SS2.SSS1.p5.27.m5.1.1"><eq id="S4.SS2.SSS1.p5.27.m5.1.1.1.cmml" xref="S4.SS2.SSS1.p5.27.m5.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.27.m5.1.1.2.cmml" xref="S4.SS2.SSS1.p5.27.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.27.m5.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.27.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.27.m5.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.27.m5.1.1.2.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.27.m5.1.1.2.3.cmml" xref="S4.SS2.SSS1.p5.27.m5.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p5.27.m5.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.27.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.27.m5.1c">x_{t}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.27.m5.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>. On the other hand, another key theoretical insight is that the transition counts in binary sequences are strongly correlated. Specifically, <math alttext="n_{01}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.28.m6.1"><semantics id="S4.SS2.SSS1.p5.28.m6.1a"><msub id="S4.SS2.SSS1.p5.28.m6.1.1" xref="S4.SS2.SSS1.p5.28.m6.1.1.cmml"><mi id="S4.SS2.SSS1.p5.28.m6.1.1.2" xref="S4.SS2.SSS1.p5.28.m6.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.28.m6.1.1.3" xref="S4.SS2.SSS1.p5.28.m6.1.1.3.cmml">01</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.28.m6.1b"><apply id="S4.SS2.SSS1.p5.28.m6.1.1.cmml" xref="S4.SS2.SSS1.p5.28.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.28.m6.1.1.1.cmml" xref="S4.SS2.SSS1.p5.28.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.28.m6.1.1.2.cmml" xref="S4.SS2.SSS1.p5.28.m6.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.28.m6.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.28.m6.1.1.3">01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.28.m6.1c">n_{01}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.28.m6.1d">italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="n_{10}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.29.m7.1"><semantics id="S4.SS2.SSS1.p5.29.m7.1a"><msub id="S4.SS2.SSS1.p5.29.m7.1.1" xref="S4.SS2.SSS1.p5.29.m7.1.1.cmml"><mi id="S4.SS2.SSS1.p5.29.m7.1.1.2" xref="S4.SS2.SSS1.p5.29.m7.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.29.m7.1.1.3" xref="S4.SS2.SSS1.p5.29.m7.1.1.3.cmml">10</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.29.m7.1b"><apply id="S4.SS2.SSS1.p5.29.m7.1.1.cmml" xref="S4.SS2.SSS1.p5.29.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.29.m7.1.1.1.cmml" xref="S4.SS2.SSS1.p5.29.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.29.m7.1.1.2.cmml" xref="S4.SS2.SSS1.p5.29.m7.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.29.m7.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.29.m7.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.29.m7.1c">n_{10}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.29.m7.1d">italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT</annotation></semantics></math> are at most one apart: since, every time a transition <math alttext="0\to 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.30.m8.1"><semantics id="S4.SS2.SSS1.p5.30.m8.1a"><mrow id="S4.SS2.SSS1.p5.30.m8.1.1" xref="S4.SS2.SSS1.p5.30.m8.1.1.cmml"><mn id="S4.SS2.SSS1.p5.30.m8.1.1.2" xref="S4.SS2.SSS1.p5.30.m8.1.1.2.cmml">0</mn><mo id="S4.SS2.SSS1.p5.30.m8.1.1.1" stretchy="false" xref="S4.SS2.SSS1.p5.30.m8.1.1.1.cmml">→</mo><mn id="S4.SS2.SSS1.p5.30.m8.1.1.3" xref="S4.SS2.SSS1.p5.30.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.30.m8.1b"><apply id="S4.SS2.SSS1.p5.30.m8.1.1.cmml" xref="S4.SS2.SSS1.p5.30.m8.1.1"><ci id="S4.SS2.SSS1.p5.30.m8.1.1.1.cmml" xref="S4.SS2.SSS1.p5.30.m8.1.1.1">→</ci><cn id="S4.SS2.SSS1.p5.30.m8.1.1.2.cmml" type="integer" xref="S4.SS2.SSS1.p5.30.m8.1.1.2">0</cn><cn id="S4.SS2.SSS1.p5.30.m8.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.30.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.30.m8.1c">0\to 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.30.m8.1d">0 → 1</annotation></semantics></math> occurs, either the sequence is followed by only <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.31.m9.1"><semantics id="S4.SS2.SSS1.p5.31.m9.1a"><mn id="S4.SS2.SSS1.p5.31.m9.1.1" xref="S4.SS2.SSS1.p5.31.m9.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.31.m9.1b"><cn id="S4.SS2.SSS1.p5.31.m9.1.1.cmml" type="integer" xref="S4.SS2.SSS1.p5.31.m9.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.31.m9.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.31.m9.1d">1</annotation></semantics></math>’s until the end, or a subsequent transition <math alttext="1\to 0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.32.m10.1"><semantics id="S4.SS2.SSS1.p5.32.m10.1a"><mrow id="S4.SS2.SSS1.p5.32.m10.1.1" xref="S4.SS2.SSS1.p5.32.m10.1.1.cmml"><mn id="S4.SS2.SSS1.p5.32.m10.1.1.2" xref="S4.SS2.SSS1.p5.32.m10.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS1.p5.32.m10.1.1.1" stretchy="false" xref="S4.SS2.SSS1.p5.32.m10.1.1.1.cmml">→</mo><mn id="S4.SS2.SSS1.p5.32.m10.1.1.3" xref="S4.SS2.SSS1.p5.32.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.32.m10.1b"><apply id="S4.SS2.SSS1.p5.32.m10.1.1.cmml" xref="S4.SS2.SSS1.p5.32.m10.1.1"><ci id="S4.SS2.SSS1.p5.32.m10.1.1.1.cmml" xref="S4.SS2.SSS1.p5.32.m10.1.1.1">→</ci><cn id="S4.SS2.SSS1.p5.32.m10.1.1.2.cmml" type="integer" xref="S4.SS2.SSS1.p5.32.m10.1.1.2">1</cn><cn id="S4.SS2.SSS1.p5.32.m10.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.32.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.32.m10.1c">1\to 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.32.m10.1d">1 → 0</annotation></semantics></math> also occurs, and vice versa. In fact, the precise relationship depends on the initial token <math alttext="x_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.33.m11.1"><semantics id="S4.SS2.SSS1.p5.33.m11.1a"><msub id="S4.SS2.SSS1.p5.33.m11.1.1" xref="S4.SS2.SSS1.p5.33.m11.1.1.cmml"><mi id="S4.SS2.SSS1.p5.33.m11.1.1.2" xref="S4.SS2.SSS1.p5.33.m11.1.1.2.cmml">x</mi><mn id="S4.SS2.SSS1.p5.33.m11.1.1.3" xref="S4.SS2.SSS1.p5.33.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.33.m11.1b"><apply id="S4.SS2.SSS1.p5.33.m11.1.1.cmml" xref="S4.SS2.SSS1.p5.33.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.33.m11.1.1.1.cmml" xref="S4.SS2.SSS1.p5.33.m11.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.33.m11.1.1.2.cmml" xref="S4.SS2.SSS1.p5.33.m11.1.1.2">𝑥</ci><cn id="S4.SS2.SSS1.p5.33.m11.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.33.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.33.m11.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.33.m11.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and on the current token <math alttext="x_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.34.m12.1"><semantics id="S4.SS2.SSS1.p5.34.m12.1a"><msub id="S4.SS2.SSS1.p5.34.m12.1.1" xref="S4.SS2.SSS1.p5.34.m12.1.1.cmml"><mi id="S4.SS2.SSS1.p5.34.m12.1.1.2" xref="S4.SS2.SSS1.p5.34.m12.1.1.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.34.m12.1.1.3" xref="S4.SS2.SSS1.p5.34.m12.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.34.m12.1b"><apply id="S4.SS2.SSS1.p5.34.m12.1.1.cmml" xref="S4.SS2.SSS1.p5.34.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.34.m12.1.1.1.cmml" xref="S4.SS2.SSS1.p5.34.m12.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.34.m12.1.1.2.cmml" xref="S4.SS2.SSS1.p5.34.m12.1.1.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.34.m12.1.1.3.cmml" xref="S4.SS2.SSS1.p5.34.m12.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.34.m12.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.34.m12.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. If <math alttext="x_{1}=x_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.35.m13.1"><semantics id="S4.SS2.SSS1.p5.35.m13.1a"><mrow id="S4.SS2.SSS1.p5.35.m13.1.1" xref="S4.SS2.SSS1.p5.35.m13.1.1.cmml"><msub id="S4.SS2.SSS1.p5.35.m13.1.1.2" xref="S4.SS2.SSS1.p5.35.m13.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.35.m13.1.1.2.2" xref="S4.SS2.SSS1.p5.35.m13.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS1.p5.35.m13.1.1.2.3" xref="S4.SS2.SSS1.p5.35.m13.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS1.p5.35.m13.1.1.1" xref="S4.SS2.SSS1.p5.35.m13.1.1.1.cmml">=</mo><msub id="S4.SS2.SSS1.p5.35.m13.1.1.3" xref="S4.SS2.SSS1.p5.35.m13.1.1.3.cmml"><mi id="S4.SS2.SSS1.p5.35.m13.1.1.3.2" xref="S4.SS2.SSS1.p5.35.m13.1.1.3.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.35.m13.1.1.3.3" xref="S4.SS2.SSS1.p5.35.m13.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.35.m13.1b"><apply id="S4.SS2.SSS1.p5.35.m13.1.1.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1"><eq id="S4.SS2.SSS1.p5.35.m13.1.1.1.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.35.m13.1.1.2.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.35.m13.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.35.m13.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS1.p5.35.m13.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.35.m13.1.1.2.3">1</cn></apply><apply id="S4.SS2.SSS1.p5.35.m13.1.1.3.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.35.m13.1.1.3.1.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.p5.35.m13.1.1.3.2.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.3.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.35.m13.1.1.3.3.cmml" xref="S4.SS2.SSS1.p5.35.m13.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.35.m13.1c">x_{1}=x_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.35.m13.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="n_{01}=n_{10}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.36.m14.1"><semantics id="S4.SS2.SSS1.p5.36.m14.1a"><mrow id="S4.SS2.SSS1.p5.36.m14.1.1" xref="S4.SS2.SSS1.p5.36.m14.1.1.cmml"><msub id="S4.SS2.SSS1.p5.36.m14.1.1.2" xref="S4.SS2.SSS1.p5.36.m14.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.36.m14.1.1.2.2" xref="S4.SS2.SSS1.p5.36.m14.1.1.2.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.36.m14.1.1.2.3" xref="S4.SS2.SSS1.p5.36.m14.1.1.2.3.cmml">01</mn></msub><mo id="S4.SS2.SSS1.p5.36.m14.1.1.1" xref="S4.SS2.SSS1.p5.36.m14.1.1.1.cmml">=</mo><msub id="S4.SS2.SSS1.p5.36.m14.1.1.3" xref="S4.SS2.SSS1.p5.36.m14.1.1.3.cmml"><mi id="S4.SS2.SSS1.p5.36.m14.1.1.3.2" xref="S4.SS2.SSS1.p5.36.m14.1.1.3.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.36.m14.1.1.3.3" xref="S4.SS2.SSS1.p5.36.m14.1.1.3.3.cmml">10</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.36.m14.1b"><apply id="S4.SS2.SSS1.p5.36.m14.1.1.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1"><eq id="S4.SS2.SSS1.p5.36.m14.1.1.1.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.36.m14.1.1.2.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.36.m14.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.36.m14.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.2.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.36.m14.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.36.m14.1.1.2.3">01</cn></apply><apply id="S4.SS2.SSS1.p5.36.m14.1.1.3.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.36.m14.1.1.3.1.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.p5.36.m14.1.1.3.2.cmml" xref="S4.SS2.SSS1.p5.36.m14.1.1.3.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.36.m14.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.36.m14.1.1.3.3">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.36.m14.1c">n_{01}=n_{10}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.36.m14.1d">italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT</annotation></semantics></math>; if <math alttext="x_{1}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.37.m15.1"><semantics id="S4.SS2.SSS1.p5.37.m15.1a"><mrow id="S4.SS2.SSS1.p5.37.m15.1.1" xref="S4.SS2.SSS1.p5.37.m15.1.1.cmml"><msub id="S4.SS2.SSS1.p5.37.m15.1.1.2" xref="S4.SS2.SSS1.p5.37.m15.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.37.m15.1.1.2.2" xref="S4.SS2.SSS1.p5.37.m15.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS1.p5.37.m15.1.1.2.3" xref="S4.SS2.SSS1.p5.37.m15.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS1.p5.37.m15.1.1.1" xref="S4.SS2.SSS1.p5.37.m15.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.37.m15.1.1.3" xref="S4.SS2.SSS1.p5.37.m15.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.37.m15.1b"><apply id="S4.SS2.SSS1.p5.37.m15.1.1.cmml" xref="S4.SS2.SSS1.p5.37.m15.1.1"><eq id="S4.SS2.SSS1.p5.37.m15.1.1.1.cmml" xref="S4.SS2.SSS1.p5.37.m15.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.37.m15.1.1.2.cmml" xref="S4.SS2.SSS1.p5.37.m15.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.37.m15.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.37.m15.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.37.m15.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.37.m15.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS1.p5.37.m15.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.37.m15.1.1.2.3">1</cn></apply><cn id="S4.SS2.SSS1.p5.37.m15.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.37.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.37.m15.1c">x_{1}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.37.m15.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0</annotation></semantics></math> and <math alttext="x_{t}=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.38.m16.1"><semantics id="S4.SS2.SSS1.p5.38.m16.1a"><mrow id="S4.SS2.SSS1.p5.38.m16.1.1" xref="S4.SS2.SSS1.p5.38.m16.1.1.cmml"><msub id="S4.SS2.SSS1.p5.38.m16.1.1.2" xref="S4.SS2.SSS1.p5.38.m16.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.38.m16.1.1.2.2" xref="S4.SS2.SSS1.p5.38.m16.1.1.2.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.38.m16.1.1.2.3" xref="S4.SS2.SSS1.p5.38.m16.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p5.38.m16.1.1.1" xref="S4.SS2.SSS1.p5.38.m16.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.38.m16.1.1.3" xref="S4.SS2.SSS1.p5.38.m16.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.38.m16.1b"><apply id="S4.SS2.SSS1.p5.38.m16.1.1.cmml" xref="S4.SS2.SSS1.p5.38.m16.1.1"><eq id="S4.SS2.SSS1.p5.38.m16.1.1.1.cmml" xref="S4.SS2.SSS1.p5.38.m16.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.38.m16.1.1.2.cmml" xref="S4.SS2.SSS1.p5.38.m16.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.38.m16.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.38.m16.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.38.m16.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.38.m16.1.1.2.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.38.m16.1.1.2.3.cmml" xref="S4.SS2.SSS1.p5.38.m16.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p5.38.m16.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.38.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.38.m16.1c">x_{t}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.38.m16.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math>, then <math alttext="n_{01}=n_{10}+1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.39.m17.1"><semantics id="S4.SS2.SSS1.p5.39.m17.1a"><mrow id="S4.SS2.SSS1.p5.39.m17.1.1" xref="S4.SS2.SSS1.p5.39.m17.1.1.cmml"><msub id="S4.SS2.SSS1.p5.39.m17.1.1.2" xref="S4.SS2.SSS1.p5.39.m17.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.39.m17.1.1.2.2" xref="S4.SS2.SSS1.p5.39.m17.1.1.2.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.39.m17.1.1.2.3" xref="S4.SS2.SSS1.p5.39.m17.1.1.2.3.cmml">01</mn></msub><mo id="S4.SS2.SSS1.p5.39.m17.1.1.1" xref="S4.SS2.SSS1.p5.39.m17.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS1.p5.39.m17.1.1.3" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.cmml"><msub id="S4.SS2.SSS1.p5.39.m17.1.1.3.2" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p5.39.m17.1.1.3.2.2" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.39.m17.1.1.3.2.3" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2.3.cmml">10</mn></msub><mo id="S4.SS2.SSS1.p5.39.m17.1.1.3.1" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.1.cmml">+</mo><mn id="S4.SS2.SSS1.p5.39.m17.1.1.3.3" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.39.m17.1b"><apply id="S4.SS2.SSS1.p5.39.m17.1.1.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1"><eq id="S4.SS2.SSS1.p5.39.m17.1.1.1.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.39.m17.1.1.2.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.39.m17.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.39.m17.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.2.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.39.m17.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.39.m17.1.1.2.3">01</cn></apply><apply id="S4.SS2.SSS1.p5.39.m17.1.1.3.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.3"><plus id="S4.SS2.SSS1.p5.39.m17.1.1.3.1.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.1"></plus><apply id="S4.SS2.SSS1.p5.39.m17.1.1.3.2.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.39.m17.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.39.m17.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.39.m17.1.1.3.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.2.3">10</cn></apply><cn id="S4.SS2.SSS1.p5.39.m17.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.39.m17.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.39.m17.1c">n_{01}=n_{10}+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.39.m17.1d">italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT + 1</annotation></semantics></math>; while if <math alttext="x_{1}=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.40.m18.1"><semantics id="S4.SS2.SSS1.p5.40.m18.1a"><mrow id="S4.SS2.SSS1.p5.40.m18.1.1" xref="S4.SS2.SSS1.p5.40.m18.1.1.cmml"><msub id="S4.SS2.SSS1.p5.40.m18.1.1.2" xref="S4.SS2.SSS1.p5.40.m18.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.40.m18.1.1.2.2" xref="S4.SS2.SSS1.p5.40.m18.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS1.p5.40.m18.1.1.2.3" xref="S4.SS2.SSS1.p5.40.m18.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS1.p5.40.m18.1.1.1" xref="S4.SS2.SSS1.p5.40.m18.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.40.m18.1.1.3" xref="S4.SS2.SSS1.p5.40.m18.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.40.m18.1b"><apply id="S4.SS2.SSS1.p5.40.m18.1.1.cmml" xref="S4.SS2.SSS1.p5.40.m18.1.1"><eq id="S4.SS2.SSS1.p5.40.m18.1.1.1.cmml" xref="S4.SS2.SSS1.p5.40.m18.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.40.m18.1.1.2.cmml" xref="S4.SS2.SSS1.p5.40.m18.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.40.m18.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.40.m18.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.40.m18.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.40.m18.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS1.p5.40.m18.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.40.m18.1.1.2.3">1</cn></apply><cn id="S4.SS2.SSS1.p5.40.m18.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.40.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.40.m18.1c">x_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.40.m18.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="x_{t}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.41.m19.1"><semantics id="S4.SS2.SSS1.p5.41.m19.1a"><mrow id="S4.SS2.SSS1.p5.41.m19.1.1" xref="S4.SS2.SSS1.p5.41.m19.1.1.cmml"><msub id="S4.SS2.SSS1.p5.41.m19.1.1.2" xref="S4.SS2.SSS1.p5.41.m19.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.41.m19.1.1.2.2" xref="S4.SS2.SSS1.p5.41.m19.1.1.2.2.cmml">x</mi><mi id="S4.SS2.SSS1.p5.41.m19.1.1.2.3" xref="S4.SS2.SSS1.p5.41.m19.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p5.41.m19.1.1.1" xref="S4.SS2.SSS1.p5.41.m19.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.41.m19.1.1.3" xref="S4.SS2.SSS1.p5.41.m19.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.41.m19.1b"><apply id="S4.SS2.SSS1.p5.41.m19.1.1.cmml" xref="S4.SS2.SSS1.p5.41.m19.1.1"><eq id="S4.SS2.SSS1.p5.41.m19.1.1.1.cmml" xref="S4.SS2.SSS1.p5.41.m19.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.41.m19.1.1.2.cmml" xref="S4.SS2.SSS1.p5.41.m19.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.41.m19.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.41.m19.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.41.m19.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.41.m19.1.1.2.2">𝑥</ci><ci id="S4.SS2.SSS1.p5.41.m19.1.1.2.3.cmml" xref="S4.SS2.SSS1.p5.41.m19.1.1.2.3">𝑡</ci></apply><cn id="S4.SS2.SSS1.p5.41.m19.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.41.m19.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.41.m19.1c">x_{t}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.41.m19.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>, then <math alttext="n_{10}=n_{01}+1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.42.m20.1"><semantics id="S4.SS2.SSS1.p5.42.m20.1a"><mrow id="S4.SS2.SSS1.p5.42.m20.1.1" xref="S4.SS2.SSS1.p5.42.m20.1.1.cmml"><msub id="S4.SS2.SSS1.p5.42.m20.1.1.2" xref="S4.SS2.SSS1.p5.42.m20.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.42.m20.1.1.2.2" xref="S4.SS2.SSS1.p5.42.m20.1.1.2.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.42.m20.1.1.2.3" xref="S4.SS2.SSS1.p5.42.m20.1.1.2.3.cmml">10</mn></msub><mo id="S4.SS2.SSS1.p5.42.m20.1.1.1" xref="S4.SS2.SSS1.p5.42.m20.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS1.p5.42.m20.1.1.3" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.cmml"><msub id="S4.SS2.SSS1.p5.42.m20.1.1.3.2" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p5.42.m20.1.1.3.2.2" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.42.m20.1.1.3.2.3" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2.3.cmml">01</mn></msub><mo id="S4.SS2.SSS1.p5.42.m20.1.1.3.1" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.1.cmml">+</mo><mn id="S4.SS2.SSS1.p5.42.m20.1.1.3.3" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.42.m20.1b"><apply id="S4.SS2.SSS1.p5.42.m20.1.1.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1"><eq id="S4.SS2.SSS1.p5.42.m20.1.1.1.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.1"></eq><apply id="S4.SS2.SSS1.p5.42.m20.1.1.2.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.42.m20.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.42.m20.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.2.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.42.m20.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.42.m20.1.1.2.3">10</cn></apply><apply id="S4.SS2.SSS1.p5.42.m20.1.1.3.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.3"><plus id="S4.SS2.SSS1.p5.42.m20.1.1.3.1.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.1"></plus><apply id="S4.SS2.SSS1.p5.42.m20.1.1.3.2.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.42.m20.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p5.42.m20.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.42.m20.1.1.3.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.2.3">01</cn></apply><cn id="S4.SS2.SSS1.p5.42.m20.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.42.m20.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.42.m20.1c">n_{10}=n_{01}+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.42.m20.1d">italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 1</annotation></semantics></math>. Therefore, the dependency of the output on <math alttext="n_{01}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.43.m21.1"><semantics id="S4.SS2.SSS1.p5.43.m21.1a"><msub id="S4.SS2.SSS1.p5.43.m21.1.1" xref="S4.SS2.SSS1.p5.43.m21.1.1.cmml"><mi id="S4.SS2.SSS1.p5.43.m21.1.1.2" xref="S4.SS2.SSS1.p5.43.m21.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.43.m21.1.1.3" xref="S4.SS2.SSS1.p5.43.m21.1.1.3.cmml">01</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.43.m21.1b"><apply id="S4.SS2.SSS1.p5.43.m21.1.1.cmml" xref="S4.SS2.SSS1.p5.43.m21.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.43.m21.1.1.1.cmml" xref="S4.SS2.SSS1.p5.43.m21.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.43.m21.1.1.2.cmml" xref="S4.SS2.SSS1.p5.43.m21.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.43.m21.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.43.m21.1.1.3">01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.43.m21.1c">n_{01}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.43.m21.1d">italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT</annotation></semantics></math> is in fact a dependency on <math alttext="n_{10}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.44.m22.1"><semantics id="S4.SS2.SSS1.p5.44.m22.1a"><msub id="S4.SS2.SSS1.p5.44.m22.1.1" xref="S4.SS2.SSS1.p5.44.m22.1.1.cmml"><mi id="S4.SS2.SSS1.p5.44.m22.1.1.2" xref="S4.SS2.SSS1.p5.44.m22.1.1.2.cmml">n</mi><mn id="S4.SS2.SSS1.p5.44.m22.1.1.3" xref="S4.SS2.SSS1.p5.44.m22.1.1.3.cmml">10</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.44.m22.1b"><apply id="S4.SS2.SSS1.p5.44.m22.1.1.cmml" xref="S4.SS2.SSS1.p5.44.m22.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.44.m22.1.1.1.cmml" xref="S4.SS2.SSS1.p5.44.m22.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.44.m22.1.1.2.cmml" xref="S4.SS2.SSS1.p5.44.m22.1.1.2">𝑛</ci><cn id="S4.SS2.SSS1.p5.44.m22.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.44.m22.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.44.m22.1c">n_{10}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.44.m22.1d">italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT</annotation></semantics></math>, and vice versa. As we will see, we can leverage this property to help <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.45.m23.1"><semantics id="S4.SS2.SSS1.p5.45.m23.1a"><mi id="S4.SS2.SSS1.p5.45.m23.1.1" xref="S4.SS2.SSS1.p5.45.m23.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.45.m23.1b"><ci id="S4.SS2.SSS1.p5.45.m23.1.1.cmml" xref="S4.SS2.SSS1.p5.45.m23.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.45.m23.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.45.m23.1d">sansserif_MambaZero</annotation></semantics></math> realize Laplacian smoothing with just two-dimensional embeddings, i.e. <math alttext="d=2" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.46.m24.1"><semantics id="S4.SS2.SSS1.p5.46.m24.1a"><mrow id="S4.SS2.SSS1.p5.46.m24.1.1" xref="S4.SS2.SSS1.p5.46.m24.1.1.cmml"><mi id="S4.SS2.SSS1.p5.46.m24.1.1.2" xref="S4.SS2.SSS1.p5.46.m24.1.1.2.cmml">d</mi><mo id="S4.SS2.SSS1.p5.46.m24.1.1.1" xref="S4.SS2.SSS1.p5.46.m24.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p5.46.m24.1.1.3" xref="S4.SS2.SSS1.p5.46.m24.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.46.m24.1b"><apply id="S4.SS2.SSS1.p5.46.m24.1.1.cmml" xref="S4.SS2.SSS1.p5.46.m24.1.1"><eq id="S4.SS2.SSS1.p5.46.m24.1.1.1.cmml" xref="S4.SS2.SSS1.p5.46.m24.1.1.1"></eq><ci id="S4.SS2.SSS1.p5.46.m24.1.1.2.cmml" xref="S4.SS2.SSS1.p5.46.m24.1.1.2">𝑑</ci><cn id="S4.SS2.SSS1.p5.46.m24.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.46.m24.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.46.m24.1c">d=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.46.m24.1d">italic_d = 2</annotation></semantics></math>. Stitching these facts, the final logits in the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex11" title="Equation Linear ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Linear</span></a> layer depend on the first and current token via</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\mathrm{logit}_{t}=W_{\ell}\,\bm{x}_{t}+W_{\ell}W_{o}\,\widetilde% {\bm{x}}^{(x_{1})}\bm{b}^{(x_{1})^{\top}}\bm{c}^{(x_{t})}+\mathbbm{1}_{\{x_{1}% \neq x_{t}\}}W_{\ell}W_{o}\,\widetilde{\bm{x}}^{(x_{1}x_{t})}\bm{b}^{(x_{1}x_{% t})\top}\bm{c}^{(x_{t})}\\ +\sum_{j}n_{x_{t}j}W_{\ell}W_{o}\,\widetilde{\bm{x}}^{(x_{t}j)}\bm{b}^{(x_{t}j% )\top}\bm{c}_{t}.\end{split}" class="ltx_Math" display="block" id="S4.Ex17.m1.49"><semantics id="S4.Ex17.m1.49a"><mtable displaystyle="true" id="S4.Ex17.m1.49.49.2" rowspacing="0pt"><mtr id="S4.Ex17.m1.49.49.2a"><mtd class="ltx_align_right" columnalign="right" id="S4.Ex17.m1.49.49.2b"><mrow id="S4.Ex17.m1.31.31.31.31.31"><msub id="S4.Ex17.m1.31.31.31.31.31.32"><mi id="S4.Ex17.m1.1.1.1.1.1.1" xref="S4.Ex17.m1.1.1.1.1.1.1.cmml">logit</mi><mi id="S4.Ex17.m1.2.2.2.2.2.2.1" xref="S4.Ex17.m1.2.2.2.2.2.2.1.cmml">t</mi></msub><mo id="S4.Ex17.m1.3.3.3.3.3.3" xref="S4.Ex17.m1.3.3.3.3.3.3.cmml">=</mo><mrow id="S4.Ex17.m1.31.31.31.31.31.33"><mrow id="S4.Ex17.m1.31.31.31.31.31.33.1"><msub id="S4.Ex17.m1.31.31.31.31.31.33.1.2"><mi id="S4.Ex17.m1.4.4.4.4.4.4" xref="S4.Ex17.m1.4.4.4.4.4.4.cmml">W</mi><mi id="S4.Ex17.m1.5.5.5.5.5.5.1" mathvariant="normal" xref="S4.Ex17.m1.5.5.5.5.5.5.1.cmml">ℓ</mi></msub><mo id="S4.Ex17.m1.31.31.31.31.31.33.1.1" xref="S4.Ex17.m1.48.48.1.1.1.cmml">⁢</mo><msub id="S4.Ex17.m1.31.31.31.31.31.33.1.3"><mi id="S4.Ex17.m1.6.6.6.6.6.6" xref="S4.Ex17.m1.6.6.6.6.6.6.cmml">𝒙</mi><mi id="S4.Ex17.m1.7.7.7.7.7.7.1" xref="S4.Ex17.m1.7.7.7.7.7.7.1.cmml">t</mi></msub></mrow><mo id="S4.Ex17.m1.8.8.8.8.8.8" xref="S4.Ex17.m1.8.8.8.8.8.8.cmml">+</mo><mrow id="S4.Ex17.m1.31.31.31.31.31.33.2"><msub id="S4.Ex17.m1.31.31.31.31.31.33.2.2"><mi id="S4.Ex17.m1.9.9.9.9.9.9" xref="S4.Ex17.m1.9.9.9.9.9.9.cmml">W</mi><mi id="S4.Ex17.m1.10.10.10.10.10.10.1" mathvariant="normal" xref="S4.Ex17.m1.10.10.10.10.10.10.1.cmml">ℓ</mi></msub><mo id="S4.Ex17.m1.31.31.31.31.31.33.2.1" xref="S4.Ex17.m1.48.48.1.1.1.cmml">⁢</mo><msub id="S4.Ex17.m1.31.31.31.31.31.33.2.3"><mi id="S4.Ex17.m1.11.11.11.11.11.11" xref="S4.Ex17.m1.11.11.11.11.11.11.cmml">W</mi><mi id="S4.Ex17.m1.12.12.12.12.12.12.1" xref="S4.Ex17.m1.12.12.12.12.12.12.1.cmml">o</mi></msub><mo id="S4.Ex17.m1.31.31.31.31.31.33.2.1a" xref="S4.Ex17.m1.48.48.1.1.1.cmml">⁢</mo><msup id="S4.Ex17.m1.31.31.31.31.31.33.2.4"><mover accent="true" id="S4.Ex17.m1.13.13.13.13.13.13" 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id="S4.Ex17.m1.15.15.15.15.15.15" xref="S4.Ex17.m1.15.15.15.15.15.15.cmml">𝒃</mi><msup id="S4.Ex17.m1.16.16.16.16.16.16.1" xref="S4.Ex17.m1.16.16.16.16.16.16.1.cmml"><mrow id="S4.Ex17.m1.16.16.16.16.16.16.1.1.1" xref="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.cmml"><mo id="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.2" stretchy="false" xref="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.cmml">(</mo><msub id="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1" xref="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.cmml"><mi id="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.2" xref="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.2.cmml">x</mi><mn id="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.3" xref="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.3" stretchy="false" xref="S4.Ex17.m1.16.16.16.16.16.16.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex17.m1.16.16.16.16.16.16.1.3" xref="S4.Ex17.m1.16.16.16.16.16.16.1.3.cmml">⊤</mo></msup></msup><mo id="S4.Ex17.m1.31.31.31.31.31.33.2.1c" 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id="S4.Ex17.m1.20.20.20.20.20.20" xref="S4.Ex17.m1.20.20.20.20.20.20.cmml">𝟙</mn><mrow id="S4.Ex17.m1.21.21.21.21.21.21.1.1" xref="S4.Ex17.m1.21.21.21.21.21.21.1.2.cmml"><mo id="S4.Ex17.m1.21.21.21.21.21.21.1.1.2" stretchy="false" xref="S4.Ex17.m1.21.21.21.21.21.21.1.2.cmml">{</mo><mrow id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.cmml"><msub id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.2" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.2.cmml"><mi id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.2.2" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.2.2.cmml">x</mi><mn id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.2.3" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.1" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.1.cmml">≠</mo><msub id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.3" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.3.cmml"><mi id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.3.2" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.3.2.cmml">x</mi><mi id="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.3.3" xref="S4.Ex17.m1.21.21.21.21.21.21.1.1.1.3.3.cmml">t</mi></msub></mrow><mo id="S4.Ex17.m1.21.21.21.21.21.21.1.1.3" stretchy="false" xref="S4.Ex17.m1.21.21.21.21.21.21.1.2.cmml">}</mo></mrow></msub><mo id="S4.Ex17.m1.31.31.31.31.31.33.3.1" xref="S4.Ex17.m1.48.48.1.1.1.cmml">⁢</mo><msub id="S4.Ex17.m1.31.31.31.31.31.33.3.3"><mi id="S4.Ex17.m1.22.22.22.22.22.22" xref="S4.Ex17.m1.22.22.22.22.22.22.cmml">W</mi><mi id="S4.Ex17.m1.23.23.23.23.23.23.1" mathvariant="normal" xref="S4.Ex17.m1.23.23.23.23.23.23.1.cmml">ℓ</mi></msub><mo id="S4.Ex17.m1.31.31.31.31.31.33.3.1a" xref="S4.Ex17.m1.48.48.1.1.1.cmml">⁢</mo><msub id="S4.Ex17.m1.31.31.31.31.31.33.3.4"><mi id="S4.Ex17.m1.24.24.24.24.24.24" xref="S4.Ex17.m1.24.24.24.24.24.24.cmml">W</mi><mi id="S4.Ex17.m1.25.25.25.25.25.25.1" xref="S4.Ex17.m1.25.25.25.25.25.25.1.cmml">o</mi></msub><mo id="S4.Ex17.m1.31.31.31.31.31.33.3.1b" xref="S4.Ex17.m1.48.48.1.1.1.cmml">⁢</mo><msup id="S4.Ex17.m1.31.31.31.31.31.33.3.5"><mover accent="true" id="S4.Ex17.m1.26.26.26.26.26.26" xref="S4.Ex17.m1.26.26.26.26.26.26.cmml"><mi id="S4.Ex17.m1.26.26.26.26.26.26.2" xref="S4.Ex17.m1.26.26.26.26.26.26.2.cmml">𝒙</mi><mo id="S4.Ex17.m1.26.26.26.26.26.26.1" xref="S4.Ex17.m1.26.26.26.26.26.26.1.cmml">~</mo></mover><mrow id="S4.Ex17.m1.27.27.27.27.27.27.1.1" xref="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.cmml"><mo id="S4.Ex17.m1.27.27.27.27.27.27.1.1.2" stretchy="false" xref="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.cmml">(</mo><mrow id="S4.Ex17.m1.27.27.27.27.27.27.1.1.1" xref="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.cmml"><msub id="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.2" xref="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.2.cmml"><mi id="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.2.2" xref="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.2.2.cmml">x</mi><mn id="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.2.3" xref="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.Ex17.m1.27.27.27.27.27.27.1.1.1.1" 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start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT bold_italic_b start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT bold_italic_c start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT bold_italic_b start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ⊤ end_POSTSUPERSCRIPT bold_italic_c start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_j ) end_POSTSUPERSCRIPT bold_italic_b start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_j ) ⊤ end_POSTSUPERSCRIPT bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT . end_CELL 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xref="S4.SS2.SSS1.p5.48.m2.2.3.1"></in><ci id="S4.SS2.SSS1.p5.48.m2.2.3.2.cmml" xref="S4.SS2.SSS1.p5.48.m2.2.3.2">𝑗</ci><set id="S4.SS2.SSS1.p5.48.m2.2.3.3.1.cmml" xref="S4.SS2.SSS1.p5.48.m2.2.3.3.2"><cn id="S4.SS2.SSS1.p5.48.m2.1.1.cmml" type="integer" xref="S4.SS2.SSS1.p5.48.m2.1.1">0</cn><cn id="S4.SS2.SSS1.p5.48.m2.2.2.cmml" type="integer" xref="S4.SS2.SSS1.p5.48.m2.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.48.m2.2c">j\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.48.m2.2d">italic_j ∈ { 0 , 1 }</annotation></semantics></math>, to be orthogonal, and the other vectors, independent of the counts, to sum up to a vector close to <math alttext="(\beta,\beta)^{\top}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.49.m3.2"><semantics id="S4.SS2.SSS1.p5.49.m3.2a"><msup id="S4.SS2.SSS1.p5.49.m3.2.3" xref="S4.SS2.SSS1.p5.49.m3.2.3.cmml"><mrow id="S4.SS2.SSS1.p5.49.m3.2.3.2.2" 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xref="S4.SS2.SSS1.p5.49.m3.1.1">𝛽</ci><ci id="S4.SS2.SSS1.p5.49.m3.2.2.cmml" xref="S4.SS2.SSS1.p5.49.m3.2.2">𝛽</ci></interval><csymbol cd="latexml" id="S4.SS2.SSS1.p5.49.m3.2.3.3.cmml" xref="S4.SS2.SSS1.p5.49.m3.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.49.m3.2c">(\beta,\beta)^{\top}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.49.m3.2d">( italic_β , italic_β ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. Subsequently, the <math alttext="L_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.50.m4.1"><semantics id="S4.SS2.SSS1.p5.50.m4.1a"><msub id="S4.SS2.SSS1.p5.50.m4.1.1" xref="S4.SS2.SSS1.p5.50.m4.1.1.cmml"><mi id="S4.SS2.SSS1.p5.50.m4.1.1.2" xref="S4.SS2.SSS1.p5.50.m4.1.1.2.cmml">L</mi><mn id="S4.SS2.SSS1.p5.50.m4.1.1.3" xref="S4.SS2.SSS1.p5.50.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.50.m4.1b"><apply id="S4.SS2.SSS1.p5.50.m4.1.1.cmml" xref="S4.SS2.SSS1.p5.50.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.50.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p5.50.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p5.50.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p5.50.m4.1.1.2">𝐿</ci><cn id="S4.SS2.SSS1.p5.50.m4.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.50.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.50.m4.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.50.m4.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> normalization in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex12" title="Equation Prediction ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Prediction</span></a> layer will give a next-token probability estimate, matching that of the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.51.m5.1"><semantics id="S4.SS2.SSS1.p5.51.m5.1a"><mi id="S4.SS2.SSS1.p5.51.m5.1.1" xref="S4.SS2.SSS1.p5.51.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.51.m5.1b"><ci id="S4.SS2.SSS1.p5.51.m5.1.1.cmml" xref="S4.SS2.SSS1.p5.51.m5.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.51.m5.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.51.m5.1d">italic_β</annotation></semantics></math> estimator. We refer to § <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.SS2" title="B.2 Proof of Thm. 1 ‣ Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">B.2</span></a> for full details.</p> </div> </section> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Higher order processes (<math alttext="k>1" class="ltx_Math" display="inline" id="S4.SS3.1.m1.1"><semantics id="S4.SS3.1.m1.1b"><mrow id="S4.SS3.1.m1.1.1" xref="S4.SS3.1.m1.1.1.cmml"><mi id="S4.SS3.1.m1.1.1.2" xref="S4.SS3.1.m1.1.1.2.cmml">k</mi><mo id="S4.SS3.1.m1.1.1.1" xref="S4.SS3.1.m1.1.1.1.cmml">></mo><mn id="S4.SS3.1.m1.1.1.3" xref="S4.SS3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.m1.1c"><apply id="S4.SS3.1.m1.1.1.cmml" xref="S4.SS3.1.m1.1.1"><gt id="S4.SS3.1.m1.1.1.1.cmml" xref="S4.SS3.1.m1.1.1.1"></gt><ci id="S4.SS3.1.m1.1.1.2.cmml" xref="S4.SS3.1.m1.1.1.2">𝑘</ci><cn id="S4.SS3.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.m1.1d">k>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.m1.1e">italic_k > 1</annotation></semantics></math>)</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.2">While empirical evidence suggests that Thm. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#Thmtheorem1" title="Theorem 1 (𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈 represents order-1 Laplacian smoothing). ‣ 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1</span></a> would also hold for <math alttext="k>1" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mrow id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml">k</mi><mo id="S4.SS3.p1.1.m1.1.1.1" xref="S4.SS3.p1.1.m1.1.1.1.cmml">></mo><mn id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><gt id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1.1"></gt><ci id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2">𝑘</ci><cn id="S4.SS3.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">k>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_k > 1</annotation></semantics></math> (Figs. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S1.F1.sf2" title="Figure 1(b) ‣ Figure 1 ‣ 1 Introduction ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1(b)</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3.F3.sf2" title="Figure 3(b) ‣ Figure 3 ‣ 3 Does Mamba learn in-context estimators? ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3(b)</span></a>), theoretically the proof becomes intractable due to the difficulty in tracking the correlations between the transition counts as <math alttext="k" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mi id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><ci id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">italic_k</annotation></semantics></math> increases. Nonetheless, we believe that it is possible to extend the main proof ideas from the first-order case to here, though with a more involved construction, leading to the conjecture:</p> </div> <div class="ltx_theorem ltx_theorem_conjecture" id="Thmconjecture1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmconjecture1.1.1.1">Conjecture 1</span></span><span class="ltx_text ltx_font_bold" id="Thmconjecture1.2.2">.</span> </h6> <div class="ltx_para" id="Thmconjecture1.p1"> <p class="ltx_p" id="Thmconjecture1.p1.6"><span class="ltx_text ltx_font_italic" id="Thmconjecture1.p1.6.6">The canonical <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="Thmconjecture1.p1.1.1.m1.1"><semantics id="Thmconjecture1.p1.1.1.m1.1a"><mi id="Thmconjecture1.p1.1.1.m1.1.1" xref="Thmconjecture1.p1.1.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="Thmconjecture1.p1.1.1.m1.1b"><ci 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id="Thmconjecture1.p1.2.2.m2.1d">italic_β</annotation></semantics></math> estimator, for the binary order-<math alttext="k" class="ltx_Math" display="inline" id="Thmconjecture1.p1.3.3.m3.1"><semantics id="Thmconjecture1.p1.3.3.m3.1a"><mi id="Thmconjecture1.p1.3.3.m3.1.1" xref="Thmconjecture1.p1.3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmconjecture1.p1.3.3.m3.1b"><ci id="Thmconjecture1.p1.3.3.m3.1.1.cmml" xref="Thmconjecture1.p1.3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmconjecture1.p1.3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmconjecture1.p1.3.3.m3.1d">italic_k</annotation></semantics></math> Markov setting, with dimensions <math alttext="d=N=2^{k+1}" class="ltx_Math" display="inline" id="Thmconjecture1.p1.4.4.m4.1"><semantics id="Thmconjecture1.p1.4.4.m4.1a"><mrow id="Thmconjecture1.p1.4.4.m4.1.1" xref="Thmconjecture1.p1.4.4.m4.1.1.cmml"><mi id="Thmconjecture1.p1.4.4.m4.1.1.2" xref="Thmconjecture1.p1.4.4.m4.1.1.2.cmml">d</mi><mo id="Thmconjecture1.p1.4.4.m4.1.1.3" xref="Thmconjecture1.p1.4.4.m4.1.1.3.cmml">=</mo><mi id="Thmconjecture1.p1.4.4.m4.1.1.4" xref="Thmconjecture1.p1.4.4.m4.1.1.4.cmml">N</mi><mo id="Thmconjecture1.p1.4.4.m4.1.1.5" xref="Thmconjecture1.p1.4.4.m4.1.1.5.cmml">=</mo><msup id="Thmconjecture1.p1.4.4.m4.1.1.6" xref="Thmconjecture1.p1.4.4.m4.1.1.6.cmml"><mn id="Thmconjecture1.p1.4.4.m4.1.1.6.2" xref="Thmconjecture1.p1.4.4.m4.1.1.6.2.cmml">2</mn><mrow id="Thmconjecture1.p1.4.4.m4.1.1.6.3" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.cmml"><mi id="Thmconjecture1.p1.4.4.m4.1.1.6.3.2" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.2.cmml">k</mi><mo id="Thmconjecture1.p1.4.4.m4.1.1.6.3.1" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.1.cmml">+</mo><mn id="Thmconjecture1.p1.4.4.m4.1.1.6.3.3" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmconjecture1.p1.4.4.m4.1b"><apply 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id="Thmconjecture1.p1.4.4.m4.1.1.6.1.cmml" xref="Thmconjecture1.p1.4.4.m4.1.1.6">superscript</csymbol><cn id="Thmconjecture1.p1.4.4.m4.1.1.6.2.cmml" type="integer" xref="Thmconjecture1.p1.4.4.m4.1.1.6.2">2</cn><apply id="Thmconjecture1.p1.4.4.m4.1.1.6.3.cmml" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3"><plus id="Thmconjecture1.p1.4.4.m4.1.1.6.3.1.cmml" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.1"></plus><ci id="Thmconjecture1.p1.4.4.m4.1.1.6.3.2.cmml" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.2">𝑘</ci><cn id="Thmconjecture1.p1.4.4.m4.1.1.6.3.3.cmml" type="integer" xref="Thmconjecture1.p1.4.4.m4.1.1.6.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmconjecture1.p1.4.4.m4.1c">d=N=2^{k+1}</annotation><annotation encoding="application/x-llamapun" id="Thmconjecture1.p1.4.4.m4.1d">italic_d = italic_N = 2 start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="e=1" class="ltx_Math" display="inline" id="Thmconjecture1.p1.5.5.m5.1"><semantics id="Thmconjecture1.p1.5.5.m5.1a"><mrow id="Thmconjecture1.p1.5.5.m5.1.1" xref="Thmconjecture1.p1.5.5.m5.1.1.cmml"><mi id="Thmconjecture1.p1.5.5.m5.1.1.2" xref="Thmconjecture1.p1.5.5.m5.1.1.2.cmml">e</mi><mo id="Thmconjecture1.p1.5.5.m5.1.1.1" xref="Thmconjecture1.p1.5.5.m5.1.1.1.cmml">=</mo><mn id="Thmconjecture1.p1.5.5.m5.1.1.3" xref="Thmconjecture1.p1.5.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmconjecture1.p1.5.5.m5.1b"><apply id="Thmconjecture1.p1.5.5.m5.1.1.cmml" xref="Thmconjecture1.p1.5.5.m5.1.1"><eq id="Thmconjecture1.p1.5.5.m5.1.1.1.cmml" xref="Thmconjecture1.p1.5.5.m5.1.1.1"></eq><ci id="Thmconjecture1.p1.5.5.m5.1.1.2.cmml" xref="Thmconjecture1.p1.5.5.m5.1.1.2">𝑒</ci><cn id="Thmconjecture1.p1.5.5.m5.1.1.3.cmml" type="integer" xref="Thmconjecture1.p1.5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmconjecture1.p1.5.5.m5.1c">e=1</annotation><annotation encoding="application/x-llamapun" id="Thmconjecture1.p1.5.5.m5.1d">italic_e = 1</annotation></semantics></math>, and window <math alttext="w=k+1" class="ltx_Math" display="inline" id="Thmconjecture1.p1.6.6.m6.1"><semantics id="Thmconjecture1.p1.6.6.m6.1a"><mrow id="Thmconjecture1.p1.6.6.m6.1.1" xref="Thmconjecture1.p1.6.6.m6.1.1.cmml"><mi id="Thmconjecture1.p1.6.6.m6.1.1.2" xref="Thmconjecture1.p1.6.6.m6.1.1.2.cmml">w</mi><mo id="Thmconjecture1.p1.6.6.m6.1.1.1" xref="Thmconjecture1.p1.6.6.m6.1.1.1.cmml">=</mo><mrow id="Thmconjecture1.p1.6.6.m6.1.1.3" xref="Thmconjecture1.p1.6.6.m6.1.1.3.cmml"><mi id="Thmconjecture1.p1.6.6.m6.1.1.3.2" xref="Thmconjecture1.p1.6.6.m6.1.1.3.2.cmml">k</mi><mo id="Thmconjecture1.p1.6.6.m6.1.1.3.1" xref="Thmconjecture1.p1.6.6.m6.1.1.3.1.cmml">+</mo><mn id="Thmconjecture1.p1.6.6.m6.1.1.3.3" xref="Thmconjecture1.p1.6.6.m6.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmconjecture1.p1.6.6.m6.1b"><apply id="Thmconjecture1.p1.6.6.m6.1.1.cmml" xref="Thmconjecture1.p1.6.6.m6.1.1"><eq id="Thmconjecture1.p1.6.6.m6.1.1.1.cmml" xref="Thmconjecture1.p1.6.6.m6.1.1.1"></eq><ci id="Thmconjecture1.p1.6.6.m6.1.1.2.cmml" xref="Thmconjecture1.p1.6.6.m6.1.1.2">𝑤</ci><apply id="Thmconjecture1.p1.6.6.m6.1.1.3.cmml" xref="Thmconjecture1.p1.6.6.m6.1.1.3"><plus id="Thmconjecture1.p1.6.6.m6.1.1.3.1.cmml" xref="Thmconjecture1.p1.6.6.m6.1.1.3.1"></plus><ci id="Thmconjecture1.p1.6.6.m6.1.1.3.2.cmml" xref="Thmconjecture1.p1.6.6.m6.1.1.3.2">𝑘</ci><cn id="Thmconjecture1.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="Thmconjecture1.p1.6.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmconjecture1.p1.6.6.m6.1c">w=k+1</annotation><annotation encoding="application/x-llamapun" id="Thmconjecture1.p1.6.6.m6.1d">italic_w = italic_k + 1</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.2">We further strengthen this claim by the following matching lower bound, which shows that, with finite bit precision, any recurrent architecture such as <math alttext="\mathsf{Mamba}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">𝖬𝖺𝗆𝖻𝖺</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝖬𝖺𝗆𝖻𝖺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">\mathsf{Mamba}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">sansserif_Mamba</annotation></semantics></math> cannot implement the Laplacian estimator with arbitrarily small error if the hidden dimension does not scale as <math alttext="\Omega(2^{k})" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mrow id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml"><mi id="S4.SS3.p2.2.m2.1.1.3" mathvariant="normal" xref="S4.SS3.p2.2.m2.1.1.3.cmml">Ω</mi><mo id="S4.SS3.p2.2.m2.1.1.2" xref="S4.SS3.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.p2.2.m2.1.1.1.1" xref="S4.SS3.p2.2.m2.1.1.1.1.1.cmml"><mo id="S4.SS3.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S4.SS3.p2.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S4.SS3.p2.2.m2.1.1.1.1.1" xref="S4.SS3.p2.2.m2.1.1.1.1.1.cmml"><mn id="S4.SS3.p2.2.m2.1.1.1.1.1.2" xref="S4.SS3.p2.2.m2.1.1.1.1.1.2.cmml">2</mn><mi id="S4.SS3.p2.2.m2.1.1.1.1.1.3" xref="S4.SS3.p2.2.m2.1.1.1.1.1.3.cmml">k</mi></msup><mo id="S4.SS3.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S4.SS3.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1"><times id="S4.SS3.p2.2.m2.1.1.2.cmml" xref="S4.SS3.p2.2.m2.1.1.2"></times><ci id="S4.SS3.p2.2.m2.1.1.3.cmml" xref="S4.SS3.p2.2.m2.1.1.3">Ω</ci><apply id="S4.SS3.p2.2.m2.1.1.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1">superscript</csymbol><cn id="S4.SS3.p2.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS3.p2.2.m2.1.1.1.1.1.2">2</cn><ci id="S4.SS3.p2.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">\Omega(2^{k})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">roman_Ω ( 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem2.1.1.1">Theorem 2</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem2.p1"> <p class="ltx_p" id="Thmtheorem2.p1.10"><span class="ltx_text ltx_font_italic" id="Thmtheorem2.p1.10.1">Consider a recurrent model of the form</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx7"> <tbody id="S4.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{t}" class="ltx_Math" display="inline" id="S4.Ex18.m1.1"><semantics id="S4.Ex18.m1.1a"><msub id="S4.Ex18.m1.1.1" xref="S4.Ex18.m1.1.1.cmml"><mi id="S4.Ex18.m1.1.1.2" xref="S4.Ex18.m1.1.1.2.cmml">H</mi><mi id="S4.Ex18.m1.1.1.3" xref="S4.Ex18.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Ex18.m1.1b"><apply id="S4.Ex18.m1.1.1.cmml" xref="S4.Ex18.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex18.m1.1.1.1.cmml" xref="S4.Ex18.m1.1.1">subscript</csymbol><ci id="S4.Ex18.m1.1.1.2.cmml" xref="S4.Ex18.m1.1.1.2">𝐻</ci><ci id="S4.Ex18.m1.1.1.3.cmml" xref="S4.Ex18.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex18.m1.1c">\displaystyle H_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex18.m1.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=h(H_{t-1},x_{t})," class="ltx_Math" display="inline" id="S4.Ex18.m2.1"><semantics id="S4.Ex18.m2.1a"><mrow id="S4.Ex18.m2.1.1.1" xref="S4.Ex18.m2.1.1.1.1.cmml"><mrow id="S4.Ex18.m2.1.1.1.1" xref="S4.Ex18.m2.1.1.1.1.cmml"><mi id="S4.Ex18.m2.1.1.1.1.4" xref="S4.Ex18.m2.1.1.1.1.4.cmml"></mi><mo id="S4.Ex18.m2.1.1.1.1.3" xref="S4.Ex18.m2.1.1.1.1.3.cmml">=</mo><mrow id="S4.Ex18.m2.1.1.1.1.2" xref="S4.Ex18.m2.1.1.1.1.2.cmml"><mi id="S4.Ex18.m2.1.1.1.1.2.4" xref="S4.Ex18.m2.1.1.1.1.2.4.cmml">h</mi><mo id="S4.Ex18.m2.1.1.1.1.2.3" xref="S4.Ex18.m2.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S4.Ex18.m2.1.1.1.1.2.2.2" xref="S4.Ex18.m2.1.1.1.1.2.2.3.cmml"><mo id="S4.Ex18.m2.1.1.1.1.2.2.2.3" stretchy="false" xref="S4.Ex18.m2.1.1.1.1.2.2.3.cmml">(</mo><msub id="S4.Ex18.m2.1.1.1.1.1.1.1.1" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex18.m2.1.1.1.1.1.1.1.1.2" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.2.cmml">H</mi><mrow id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.1" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.3" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.Ex18.m2.1.1.1.1.2.2.2.4" xref="S4.Ex18.m2.1.1.1.1.2.2.3.cmml">,</mo><msub id="S4.Ex18.m2.1.1.1.1.2.2.2.2" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2.cmml"><mi id="S4.Ex18.m2.1.1.1.1.2.2.2.2.2" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2.2.cmml">x</mi><mi id="S4.Ex18.m2.1.1.1.1.2.2.2.2.3" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2.3.cmml">t</mi></msub><mo id="S4.Ex18.m2.1.1.1.1.2.2.2.5" stretchy="false" xref="S4.Ex18.m2.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex18.m2.1.1.1.2" xref="S4.Ex18.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex18.m2.1b"><apply id="S4.Ex18.m2.1.1.1.1.cmml" xref="S4.Ex18.m2.1.1.1"><eq id="S4.Ex18.m2.1.1.1.1.3.cmml" xref="S4.Ex18.m2.1.1.1.1.3"></eq><csymbol cd="latexml" id="S4.Ex18.m2.1.1.1.1.4.cmml" xref="S4.Ex18.m2.1.1.1.1.4">absent</csymbol><apply id="S4.Ex18.m2.1.1.1.1.2.cmml" xref="S4.Ex18.m2.1.1.1.1.2"><times id="S4.Ex18.m2.1.1.1.1.2.3.cmml" xref="S4.Ex18.m2.1.1.1.1.2.3"></times><ci id="S4.Ex18.m2.1.1.1.1.2.4.cmml" xref="S4.Ex18.m2.1.1.1.1.2.4">ℎ</ci><interval closure="open" id="S4.Ex18.m2.1.1.1.1.2.2.3.cmml" xref="S4.Ex18.m2.1.1.1.1.2.2.2"><apply id="S4.Ex18.m2.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex18.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex18.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.2">𝐻</ci><apply id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3"><minus id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.1"></minus><ci id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Ex18.m2.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.Ex18.m2.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex18.m2.1.1.1.1.2.2.2.2.1.cmml" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S4.Ex18.m2.1.1.1.1.2.2.2.2.2.cmml" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2.2">𝑥</ci><ci id="S4.Ex18.m2.1.1.1.1.2.2.2.2.3.cmml" xref="S4.Ex18.m2.1.1.1.1.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex18.m2.1c">\displaystyle=h(H_{t-1},x_{t}),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex18.m2.1d">= italic_h ( italic_H start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle y_{t}" class="ltx_Math" display="inline" id="S4.Ex19.m1.1"><semantics id="S4.Ex19.m1.1a"><msub id="S4.Ex19.m1.1.1" xref="S4.Ex19.m1.1.1.cmml"><mi id="S4.Ex19.m1.1.1.2" xref="S4.Ex19.m1.1.1.2.cmml">y</mi><mi id="S4.Ex19.m1.1.1.3" xref="S4.Ex19.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Ex19.m1.1b"><apply id="S4.Ex19.m1.1.1.cmml" xref="S4.Ex19.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex19.m1.1.1.1.cmml" xref="S4.Ex19.m1.1.1">subscript</csymbol><ci id="S4.Ex19.m1.1.1.2.cmml" xref="S4.Ex19.m1.1.1.2">𝑦</ci><ci id="S4.Ex19.m1.1.1.3.cmml" xref="S4.Ex19.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex19.m1.1c">\displaystyle y_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m1.1d">italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)=g(H_{t})," class="ltx_math_unparsed" display="inline" id="S4.Ex19.m2.1"><semantics id="S4.Ex19.m2.1a"><mrow id="S4.Ex19.m2.1b"><mo id="S4.Ex19.m2.1.1">=</mo><msub id="S4.Ex19.m2.1.2"><mi id="S4.Ex19.m2.1.2.2">ℙ</mi><mi id="S4.Ex19.m2.1.2.3">𝜽</mi></msub><mrow id="S4.Ex19.m2.1.3"><mo id="S4.Ex19.m2.1.3.1">(</mo><mo id="S4.Ex19.m2.1.3.2" lspace="0em" rspace="0em">⋅</mo><mo id="S4.Ex19.m2.1.3.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S4.Ex19.m2.1.3.4"><mi id="S4.Ex19.m2.1.3.4.2.2">x</mi><mn id="S4.Ex19.m2.1.3.4.2.3">1</mn><mi id="S4.Ex19.m2.1.3.4.3">t</mi></msubsup><mo id="S4.Ex19.m2.1.3.5">)</mo></mrow><mo id="S4.Ex19.m2.1.4">=</mo><mi id="S4.Ex19.m2.1.5">g</mi><mrow id="S4.Ex19.m2.1.6"><mo id="S4.Ex19.m2.1.6.1" stretchy="false">(</mo><msub id="S4.Ex19.m2.1.6.2"><mi id="S4.Ex19.m2.1.6.2.2">H</mi><mi id="S4.Ex19.m2.1.6.2.3">t</mi></msub><mo id="S4.Ex19.m2.1.6.3" stretchy="false">)</mo></mrow><mo id="S4.Ex19.m2.1.7">,</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex19.m2.1c">\displaystyle=\mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)=g(H_{t}),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m2.1d">= blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = italic_g ( italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem2.p1.9"><span class="ltx_text ltx_font_italic" id="Thmtheorem2.p1.9.9">with transformations <math alttext="(h,g)" class="ltx_Math" display="inline" id="Thmtheorem2.p1.1.1.m1.2"><semantics id="Thmtheorem2.p1.1.1.m1.2a"><mrow id="Thmtheorem2.p1.1.1.m1.2.3.2" xref="Thmtheorem2.p1.1.1.m1.2.3.1.cmml"><mo id="Thmtheorem2.p1.1.1.m1.2.3.2.1" stretchy="false" xref="Thmtheorem2.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="Thmtheorem2.p1.1.1.m1.1.1" xref="Thmtheorem2.p1.1.1.m1.1.1.cmml">h</mi><mo id="Thmtheorem2.p1.1.1.m1.2.3.2.2" xref="Thmtheorem2.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="Thmtheorem2.p1.1.1.m1.2.2" xref="Thmtheorem2.p1.1.1.m1.2.2.cmml">g</mi><mo id="Thmtheorem2.p1.1.1.m1.2.3.2.3" stretchy="false" xref="Thmtheorem2.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.1.1.m1.2b"><interval closure="open" id="Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem2.p1.1.1.m1.2.3.2"><ci id="Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1">ℎ</ci><ci id="Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="Thmtheorem2.p1.1.1.m1.2.2">𝑔</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.1.1.m1.2c">(h,g)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.1.1.m1.2d">( italic_h , italic_g )</annotation></semantics></math>, where <math alttext="H_{t}\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.2.2.m2.1"><semantics id="Thmtheorem2.p1.2.2.m2.1a"><mrow id="Thmtheorem2.p1.2.2.m2.1.1" xref="Thmtheorem2.p1.2.2.m2.1.1.cmml"><msub id="Thmtheorem2.p1.2.2.m2.1.1.2" xref="Thmtheorem2.p1.2.2.m2.1.1.2.cmml"><mi id="Thmtheorem2.p1.2.2.m2.1.1.2.2" xref="Thmtheorem2.p1.2.2.m2.1.1.2.2.cmml">H</mi><mi id="Thmtheorem2.p1.2.2.m2.1.1.2.3" xref="Thmtheorem2.p1.2.2.m2.1.1.2.3.cmml">t</mi></msub><mo id="Thmtheorem2.p1.2.2.m2.1.1.1" xref="Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="Thmtheorem2.p1.2.2.m2.1.1.3" xref="Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mi id="Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.2.2.m2.1b"><apply id="Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1"><in id="Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.1"></in><apply id="Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem2.p1.2.2.m2.1.1.2.1.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="Thmtheorem2.p1.2.2.m2.1.1.2.2.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.2.2">𝐻</ci><ci id="Thmtheorem2.p1.2.2.m2.1.1.2.3.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.2.3">𝑡</ci></apply><apply id="Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.2.2.m2.1c">H_{t}\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.2.2.m2.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and the model has a bit precision of <math alttext="\mathtt{p}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.3.3.m3.1"><semantics id="Thmtheorem2.p1.3.3.m3.1a"><mi id="Thmtheorem2.p1.3.3.m3.1.1" xref="Thmtheorem2.p1.3.3.m3.1.1.cmml">𝚙</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.3.3.m3.1b"><ci id="Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="Thmtheorem2.p1.3.3.m3.1.1">𝚙</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.3.3.m3.1c">\mathtt{p}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.3.3.m3.1d">typewriter_p</annotation></semantics></math>. Suppose that the Markov kernel <math alttext="P" class="ltx_Math" display="inline" id="Thmtheorem2.p1.4.4.m4.1"><semantics id="Thmtheorem2.p1.4.4.m4.1a"><mi id="Thmtheorem2.p1.4.4.m4.1.1" xref="Thmtheorem2.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.4.4.m4.1b"><ci id="Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="Thmtheorem2.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.4.4.m4.1d">italic_P</annotation></semantics></math> is sampled from the Dirichlet prior with <math alttext="\beta=1" class="ltx_Math" display="inline" id="Thmtheorem2.p1.5.5.m5.1"><semantics id="Thmtheorem2.p1.5.5.m5.1a"><mrow id="Thmtheorem2.p1.5.5.m5.1.1" xref="Thmtheorem2.p1.5.5.m5.1.1.cmml"><mi id="Thmtheorem2.p1.5.5.m5.1.1.2" xref="Thmtheorem2.p1.5.5.m5.1.1.2.cmml">β</mi><mo id="Thmtheorem2.p1.5.5.m5.1.1.1" xref="Thmtheorem2.p1.5.5.m5.1.1.1.cmml">=</mo><mn id="Thmtheorem2.p1.5.5.m5.1.1.3" xref="Thmtheorem2.p1.5.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.5.5.m5.1b"><apply id="Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="Thmtheorem2.p1.5.5.m5.1.1"><eq id="Thmtheorem2.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem2.p1.5.5.m5.1.1.1"></eq><ci id="Thmtheorem2.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem2.p1.5.5.m5.1.1.2">𝛽</ci><cn id="Thmtheorem2.p1.5.5.m5.1.1.3.cmml" type="integer" xref="Thmtheorem2.p1.5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.5.5.m5.1c">\beta=1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.5.5.m5.1d">italic_β = 1</annotation></semantics></math>, <math alttext="P\sim\mathrm{Dir}(1\cdot\bm{1})" class="ltx_Math" display="inline" id="Thmtheorem2.p1.6.6.m6.1"><semantics id="Thmtheorem2.p1.6.6.m6.1a"><mrow id="Thmtheorem2.p1.6.6.m6.1.1" xref="Thmtheorem2.p1.6.6.m6.1.1.cmml"><mi id="Thmtheorem2.p1.6.6.m6.1.1.3" xref="Thmtheorem2.p1.6.6.m6.1.1.3.cmml">P</mi><mo id="Thmtheorem2.p1.6.6.m6.1.1.2" xref="Thmtheorem2.p1.6.6.m6.1.1.2.cmml">∼</mo><mrow id="Thmtheorem2.p1.6.6.m6.1.1.1" xref="Thmtheorem2.p1.6.6.m6.1.1.1.cmml"><mi id="Thmtheorem2.p1.6.6.m6.1.1.1.3" xref="Thmtheorem2.p1.6.6.m6.1.1.1.3.cmml">Dir</mi><mo id="Thmtheorem2.p1.6.6.m6.1.1.1.2" xref="Thmtheorem2.p1.6.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.cmml"><mn id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.2" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.2.cmml">1</mn><mo id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.cmml">⋅</mo><mn id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.3" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.6.6.m6.1b"><apply id="Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1"><csymbol cd="latexml" id="Thmtheorem2.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.2">similar-to</csymbol><ci id="Thmtheorem2.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.3">𝑃</ci><apply id="Thmtheorem2.p1.6.6.m6.1.1.1.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.1"><times id="Thmtheorem2.p1.6.6.m6.1.1.1.2.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.1.2"></times><ci id="Thmtheorem2.p1.6.6.m6.1.1.1.3.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.1.3">Dir</ci><apply id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1"><ci id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1">⋅</ci><cn id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.2.cmml" type="integer" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.2">1</cn><cn id="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.6.6.m6.1c">P\sim\mathrm{Dir}(1\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.6.6.m6.1d">italic_P ∼ roman_Dir ( 1 ⋅ bold_1 )</annotation></semantics></math>. Suppose also that the recurrent architecture satisfies the following point-wise guarantee: for any sufficiently large <math alttext="t" class="ltx_Math" display="inline" id="Thmtheorem2.p1.7.7.m7.1"><semantics id="Thmtheorem2.p1.7.7.m7.1a"><mi id="Thmtheorem2.p1.7.7.m7.1.1" xref="Thmtheorem2.p1.7.7.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.7.7.m7.1b"><ci id="Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="Thmtheorem2.p1.7.7.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.7.7.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.7.7.m7.1d">italic_t</annotation></semantics></math>, almost surely over <math alttext="P" class="ltx_Math" display="inline" id="Thmtheorem2.p1.8.8.m8.1"><semantics id="Thmtheorem2.p1.8.8.m8.1a"><mi id="Thmtheorem2.p1.8.8.m8.1.1" xref="Thmtheorem2.p1.8.8.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.8.8.m8.1b"><ci id="Thmtheorem2.p1.8.8.m8.1.1.cmml" xref="Thmtheorem2.p1.8.8.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.8.8.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.8.8.m8.1d">italic_P</annotation></semantics></math> and <math alttext="x_{1}^{t}\sim P" class="ltx_Math" display="inline" id="Thmtheorem2.p1.9.9.m9.1"><semantics id="Thmtheorem2.p1.9.9.m9.1a"><mrow id="Thmtheorem2.p1.9.9.m9.1.1" xref="Thmtheorem2.p1.9.9.m9.1.1.cmml"><msubsup id="Thmtheorem2.p1.9.9.m9.1.1.2" xref="Thmtheorem2.p1.9.9.m9.1.1.2.cmml"><mi id="Thmtheorem2.p1.9.9.m9.1.1.2.2.2" xref="Thmtheorem2.p1.9.9.m9.1.1.2.2.2.cmml">x</mi><mn id="Thmtheorem2.p1.9.9.m9.1.1.2.2.3" xref="Thmtheorem2.p1.9.9.m9.1.1.2.2.3.cmml">1</mn><mi id="Thmtheorem2.p1.9.9.m9.1.1.2.3" xref="Thmtheorem2.p1.9.9.m9.1.1.2.3.cmml">t</mi></msubsup><mo id="Thmtheorem2.p1.9.9.m9.1.1.1" xref="Thmtheorem2.p1.9.9.m9.1.1.1.cmml">∼</mo><mi id="Thmtheorem2.p1.9.9.m9.1.1.3" xref="Thmtheorem2.p1.9.9.m9.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.9.9.m9.1b"><apply id="Thmtheorem2.p1.9.9.m9.1.1.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1"><csymbol cd="latexml" id="Thmtheorem2.p1.9.9.m9.1.1.1.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.1">similar-to</csymbol><apply id="Thmtheorem2.p1.9.9.m9.1.1.2.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem2.p1.9.9.m9.1.1.2.1.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.2">superscript</csymbol><apply id="Thmtheorem2.p1.9.9.m9.1.1.2.2.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem2.p1.9.9.m9.1.1.2.2.1.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.2">subscript</csymbol><ci id="Thmtheorem2.p1.9.9.m9.1.1.2.2.2.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.2.2.2">𝑥</ci><cn id="Thmtheorem2.p1.9.9.m9.1.1.2.2.3.cmml" type="integer" xref="Thmtheorem2.p1.9.9.m9.1.1.2.2.3">1</cn></apply><ci id="Thmtheorem2.p1.9.9.m9.1.1.2.3.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.2.3">𝑡</ci></apply><ci id="Thmtheorem2.p1.9.9.m9.1.1.3.cmml" xref="Thmtheorem2.p1.9.9.m9.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.9.9.m9.1c">x_{1}^{t}\sim P</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.9.9.m9.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ italic_P</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\left\|\mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)-\mathbb{P}_{1}% ^{(k)}(\cdot\mid x_{1}^{t})\right\|_{\infty}\leq\varepsilon." class="ltx_math_unparsed" display="block" id="S4.E4.m1.1"><semantics id="S4.E4.m1.1a"><mrow id="S4.E4.m1.1b"><mo id="S4.E4.m1.1.2" rspace="0.167em" stretchy="true">∥</mo><msub id="S4.E4.m1.1.3"><mi id="S4.E4.m1.1.3.2">ℙ</mi><mi id="S4.E4.m1.1.3.3">𝜽</mi></msub><mrow id="S4.E4.m1.1.4"><mo id="S4.E4.m1.1.4.1">(</mo><mo id="S4.E4.m1.1.4.2" lspace="0em" rspace="0em">⋅</mo><mo id="S4.E4.m1.1.4.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S4.E4.m1.1.4.4"><mi id="S4.E4.m1.1.4.4.2.2">x</mi><mn id="S4.E4.m1.1.4.4.2.3">1</mn><mi id="S4.E4.m1.1.4.4.3">t</mi></msubsup><mo id="S4.E4.m1.1.4.5">)</mo></mrow><mo id="S4.E4.m1.1.5">−</mo><msubsup id="S4.E4.m1.1.6"><mi id="S4.E4.m1.1.6.2.2">ℙ</mi><mn id="S4.E4.m1.1.6.2.3">1</mn><mrow id="S4.E4.m1.1.1.1.3"><mo id="S4.E4.m1.1.1.1.3.1" stretchy="false">(</mo><mi id="S4.E4.m1.1.1.1.1">k</mi><mo id="S4.E4.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.E4.m1.1.7"><mo id="S4.E4.m1.1.7.1" stretchy="false">(</mo><mo id="S4.E4.m1.1.7.2" lspace="0em" rspace="0em">⋅</mo><mo id="S4.E4.m1.1.7.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="S4.E4.m1.1.7.4"><mi id="S4.E4.m1.1.7.4.2.2">x</mi><mn id="S4.E4.m1.1.7.4.2.3">1</mn><mi id="S4.E4.m1.1.7.4.3">t</mi></msubsup><mo id="S4.E4.m1.1.7.5" stretchy="false">)</mo></mrow><msub id="S4.E4.m1.1.8"><mo id="S4.E4.m1.1.8.2" lspace="0em" rspace="0.0835em" stretchy="true">∥</mo><mi id="S4.E4.m1.1.8.3" mathvariant="normal">∞</mi></msub><mo id="S4.E4.m1.1.9" lspace="0.0835em">≤</mo><mi id="S4.E4.m1.1.10">ε</mi><mo id="S4.E4.m1.1.11" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S4.E4.m1.1c">\left\|\mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)-\mathbb{P}_{1}% ^{(k)}(\cdot\mid x_{1}^{t})\right\|_{\infty}\leq\varepsilon.</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.1d">∥ blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) - blackboard_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≤ italic_ε .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem2.p1.11"><span class="ltx_text ltx_font_italic" id="Thmtheorem2.p1.11.1">Then, the recurrent architecture must satisfy</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="d\cdot\mathtt{p}\geq 2^{k}(1-3\varepsilon)\log(1/\varepsilon)." class="ltx_Math" display="block" id="S4.Ex20.m1.2"><semantics id="S4.Ex20.m1.2a"><mrow id="S4.Ex20.m1.2.2.1" xref="S4.Ex20.m1.2.2.1.1.cmml"><mrow id="S4.Ex20.m1.2.2.1.1" xref="S4.Ex20.m1.2.2.1.1.cmml"><mrow id="S4.Ex20.m1.2.2.1.1.4" xref="S4.Ex20.m1.2.2.1.1.4.cmml"><mi id="S4.Ex20.m1.2.2.1.1.4.2" xref="S4.Ex20.m1.2.2.1.1.4.2.cmml">d</mi><mo id="S4.Ex20.m1.2.2.1.1.4.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex20.m1.2.2.1.1.4.1.cmml">⋅</mo><mi id="S4.Ex20.m1.2.2.1.1.4.3" xref="S4.Ex20.m1.2.2.1.1.4.3.cmml">𝚙</mi></mrow><mo id="S4.Ex20.m1.2.2.1.1.3" xref="S4.Ex20.m1.2.2.1.1.3.cmml">≥</mo><mrow id="S4.Ex20.m1.2.2.1.1.2" xref="S4.Ex20.m1.2.2.1.1.2.cmml"><msup id="S4.Ex20.m1.2.2.1.1.2.4" xref="S4.Ex20.m1.2.2.1.1.2.4.cmml"><mn id="S4.Ex20.m1.2.2.1.1.2.4.2" xref="S4.Ex20.m1.2.2.1.1.2.4.2.cmml">2</mn><mi id="S4.Ex20.m1.2.2.1.1.2.4.3" xref="S4.Ex20.m1.2.2.1.1.2.4.3.cmml">k</mi></msup><mo id="S4.Ex20.m1.2.2.1.1.2.3" xref="S4.Ex20.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S4.Ex20.m1.2.2.1.1.1.1.1" xref="S4.Ex20.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S4.Ex20.m1.2.2.1.1.1.1.1.2" 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xref="S4.Ex20.m1.2.2.1.1.2.2.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex20.m1.2c">d\cdot\mathtt{p}\geq 2^{k}(1-3\varepsilon)\log(1/\varepsilon).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex20.m1.2d">italic_d ⋅ typewriter_p ≥ 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( 1 - 3 italic_ε ) roman_log ( 1 / italic_ε ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.3"><span class="ltx_text ltx_font_bold" id="S4.SS3.p3.3.1">Remark.</span> While we used time-invariant functions <math alttext="(h,g)" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.2"><semantics id="S4.SS3.p3.1.m1.2a"><mrow id="S4.SS3.p3.1.m1.2.3.2" xref="S4.SS3.p3.1.m1.2.3.1.cmml"><mo id="S4.SS3.p3.1.m1.2.3.2.1" stretchy="false" xref="S4.SS3.p3.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml">h</mi><mo id="S4.SS3.p3.1.m1.2.3.2.2" xref="S4.SS3.p3.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS3.p3.1.m1.2.2" xref="S4.SS3.p3.1.m1.2.2.cmml">g</mi><mo id="S4.SS3.p3.1.m1.2.3.2.3" stretchy="false" xref="S4.SS3.p3.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.2b"><interval closure="open" id="S4.SS3.p3.1.m1.2.3.1.cmml" xref="S4.SS3.p3.1.m1.2.3.2"><ci id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1">ℎ</ci><ci id="S4.SS3.p3.1.m1.2.2.cmml" xref="S4.SS3.p3.1.m1.2.2">𝑔</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.2c">(h,g)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.2d">( italic_h , italic_g )</annotation></semantics></math> in Thm. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#Thmtheorem2" title="Theorem 2. ‣ 4.3 Higher order processes (𝑘>1) ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2</span></a> for notational simplicity, the proof does not rely on this fact and holds for any general <math alttext="H_{t}=h_{t}(H_{t-1},x_{t})" class="ltx_Math" display="inline" id="S4.SS3.p3.2.m2.2"><semantics id="S4.SS3.p3.2.m2.2a"><mrow id="S4.SS3.p3.2.m2.2.2" xref="S4.SS3.p3.2.m2.2.2.cmml"><msub id="S4.SS3.p3.2.m2.2.2.4" xref="S4.SS3.p3.2.m2.2.2.4.cmml"><mi id="S4.SS3.p3.2.m2.2.2.4.2" xref="S4.SS3.p3.2.m2.2.2.4.2.cmml">H</mi><mi id="S4.SS3.p3.2.m2.2.2.4.3" xref="S4.SS3.p3.2.m2.2.2.4.3.cmml">t</mi></msub><mo id="S4.SS3.p3.2.m2.2.2.3" xref="S4.SS3.p3.2.m2.2.2.3.cmml">=</mo><mrow id="S4.SS3.p3.2.m2.2.2.2" xref="S4.SS3.p3.2.m2.2.2.2.cmml"><msub id="S4.SS3.p3.2.m2.2.2.2.4" xref="S4.SS3.p3.2.m2.2.2.2.4.cmml"><mi id="S4.SS3.p3.2.m2.2.2.2.4.2" xref="S4.SS3.p3.2.m2.2.2.2.4.2.cmml">h</mi><mi id="S4.SS3.p3.2.m2.2.2.2.4.3" xref="S4.SS3.p3.2.m2.2.2.2.4.3.cmml">t</mi></msub><mo id="S4.SS3.p3.2.m2.2.2.2.3" 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id="S4.SS3.p3.2.m2.2.2.2.2.2.2.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p3.2.m2.2.2.2.2.2.2.1.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS3.p3.2.m2.2.2.2.2.2.2.2.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.2.2.2">𝑥</ci><ci id="S4.SS3.p3.2.m2.2.2.2.2.2.2.3.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.2.m2.2c">H_{t}=h_{t}(H_{t-1},x_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.2.m2.2d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="y_{t}=g_{t}(H_{t})" class="ltx_Math" display="inline" id="S4.SS3.p3.3.m3.1"><semantics id="S4.SS3.p3.3.m3.1a"><mrow 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xref="S4.SS3.p3.3.m3.1.1.1.1.1.1.2.cmml">H</mi><mi id="S4.SS3.p3.3.m3.1.1.1.1.1.1.3" xref="S4.SS3.p3.3.m3.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS3.p3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.p3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.3.m3.1b"><apply id="S4.SS3.p3.3.m3.1.1.cmml" xref="S4.SS3.p3.3.m3.1.1"><eq id="S4.SS3.p3.3.m3.1.1.2.cmml" xref="S4.SS3.p3.3.m3.1.1.2"></eq><apply id="S4.SS3.p3.3.m3.1.1.3.cmml" xref="S4.SS3.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p3.3.m3.1.1.3.1.cmml" xref="S4.SS3.p3.3.m3.1.1.3">subscript</csymbol><ci id="S4.SS3.p3.3.m3.1.1.3.2.cmml" xref="S4.SS3.p3.3.m3.1.1.3.2">𝑦</ci><ci id="S4.SS3.p3.3.m3.1.1.3.3.cmml" xref="S4.SS3.p3.3.m3.1.1.3.3">𝑡</ci></apply><apply id="S4.SS3.p3.3.m3.1.1.1.cmml" xref="S4.SS3.p3.3.m3.1.1.1"><times id="S4.SS3.p3.3.m3.1.1.1.2.cmml" xref="S4.SS3.p3.3.m3.1.1.1.2"></times><apply id="S4.SS3.p3.3.m3.1.1.1.3.cmml" xref="S4.SS3.p3.3.m3.1.1.1.3"><csymbol 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)</annotation></semantics></math>, which subsumes Mamba.</p> </div> <div class="ltx_para" id="S4.SS3.p4"> <p class="ltx_p" id="S4.SS3.p4.1">We provide the intuition behind the proof here. We defer the full proof and additional details to App. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A3" title="Appendix C Proof of Thm. 2 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">C</span></a>.</p> </div> <div class="ltx_para" id="S4.SS3.p5"> <p class="ltx_p" id="S4.SS3.p5.17"><span class="ltx_text ltx_font_bold" id="S4.SS3.p5.17.1">Intuition.</span> The intuition behind this result is in the manner in which the recurrent architecture carries out computation: by proceeding sequentially and compressing the information from the sequence it has seen thus far at some time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS3.p5.1.m1.1"><semantics id="S4.SS3.p5.1.m1.1a"><mi id="S4.SS3.p5.1.m1.1.1" xref="S4.SS3.p5.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.1.m1.1b"><ci id="S4.SS3.p5.1.m1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.1.m1.1d">italic_t</annotation></semantics></math> into a small hidden vector, the model does not know what the next <math alttext="k" class="ltx_Math" display="inline" id="S4.SS3.p5.2.m2.1"><semantics id="S4.SS3.p5.2.m2.1a"><mi id="S4.SS3.p5.2.m2.1.1" xref="S4.SS3.p5.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.2.m2.1b"><ci id="S4.SS3.p5.2.m2.1.1.cmml" xref="S4.SS3.p5.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.2.m2.1d">italic_k</annotation></semantics></math> tokens will be: the knowledge of this is vital to be able to compute the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS3.p5.3.m3.1"><semantics id="S4.SS3.p5.3.m3.1a"><mi id="S4.SS3.p5.3.m3.1.1" xref="S4.SS3.p5.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.3.m3.1b"><ci id="S4.SS3.p5.3.m3.1.1.cmml" xref="S4.SS3.p5.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.3.m3.1d">italic_β</annotation></semantics></math> estimator at time <math alttext="t+k+1" class="ltx_Math" display="inline" id="S4.SS3.p5.4.m4.1"><semantics id="S4.SS3.p5.4.m4.1a"><mrow id="S4.SS3.p5.4.m4.1.1" xref="S4.SS3.p5.4.m4.1.1.cmml"><mi id="S4.SS3.p5.4.m4.1.1.2" xref="S4.SS3.p5.4.m4.1.1.2.cmml">t</mi><mo id="S4.SS3.p5.4.m4.1.1.1" xref="S4.SS3.p5.4.m4.1.1.1.cmml">+</mo><mi id="S4.SS3.p5.4.m4.1.1.3" xref="S4.SS3.p5.4.m4.1.1.3.cmml">k</mi><mo id="S4.SS3.p5.4.m4.1.1.1a" xref="S4.SS3.p5.4.m4.1.1.1.cmml">+</mo><mn id="S4.SS3.p5.4.m4.1.1.4" xref="S4.SS3.p5.4.m4.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.4.m4.1b"><apply id="S4.SS3.p5.4.m4.1.1.cmml" xref="S4.SS3.p5.4.m4.1.1"><plus id="S4.SS3.p5.4.m4.1.1.1.cmml" xref="S4.SS3.p5.4.m4.1.1.1"></plus><ci id="S4.SS3.p5.4.m4.1.1.2.cmml" xref="S4.SS3.p5.4.m4.1.1.2">𝑡</ci><ci id="S4.SS3.p5.4.m4.1.1.3.cmml" xref="S4.SS3.p5.4.m4.1.1.3">𝑘</ci><cn id="S4.SS3.p5.4.m4.1.1.4.cmml" type="integer" xref="S4.SS3.p5.4.m4.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.4.m4.1c">t+k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.4.m4.1d">italic_t + italic_k + 1</annotation></semantics></math> with a small memory footprint. Indeed, when the identity of the next <math alttext="k" class="ltx_Math" display="inline" id="S4.SS3.p5.5.m5.1"><semantics id="S4.SS3.p5.5.m5.1a"><mi id="S4.SS3.p5.5.m5.1.1" xref="S4.SS3.p5.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.5.m5.1b"><ci id="S4.SS3.p5.5.m5.1.1.cmml" xref="S4.SS3.p5.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.5.m5.1d">italic_k</annotation></semantics></math> tokens changes, the output of the model at time <math alttext="t+k+1" class="ltx_Math" display="inline" id="S4.SS3.p5.6.m6.1"><semantics id="S4.SS3.p5.6.m6.1a"><mrow id="S4.SS3.p5.6.m6.1.1" xref="S4.SS3.p5.6.m6.1.1.cmml"><mi id="S4.SS3.p5.6.m6.1.1.2" xref="S4.SS3.p5.6.m6.1.1.2.cmml">t</mi><mo id="S4.SS3.p5.6.m6.1.1.1" xref="S4.SS3.p5.6.m6.1.1.1.cmml">+</mo><mi id="S4.SS3.p5.6.m6.1.1.3" xref="S4.SS3.p5.6.m6.1.1.3.cmml">k</mi><mo id="S4.SS3.p5.6.m6.1.1.1a" xref="S4.SS3.p5.6.m6.1.1.1.cmml">+</mo><mn id="S4.SS3.p5.6.m6.1.1.4" xref="S4.SS3.p5.6.m6.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.6.m6.1b"><apply id="S4.SS3.p5.6.m6.1.1.cmml" xref="S4.SS3.p5.6.m6.1.1"><plus id="S4.SS3.p5.6.m6.1.1.1.cmml" xref="S4.SS3.p5.6.m6.1.1.1"></plus><ci id="S4.SS3.p5.6.m6.1.1.2.cmml" xref="S4.SS3.p5.6.m6.1.1.2">𝑡</ci><ci id="S4.SS3.p5.6.m6.1.1.3.cmml" xref="S4.SS3.p5.6.m6.1.1.3">𝑘</ci><cn id="S4.SS3.p5.6.m6.1.1.4.cmml" type="integer" xref="S4.SS3.p5.6.m6.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.6.m6.1c">t+k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.6.m6.1d">italic_t + italic_k + 1</annotation></semantics></math> must look drastically different (as the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS3.p5.7.m7.1"><semantics id="S4.SS3.p5.7.m7.1a"><mi id="S4.SS3.p5.7.m7.1.1" xref="S4.SS3.p5.7.m7.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.7.m7.1b"><ci id="S4.SS3.p5.7.m7.1.1.cmml" xref="S4.SS3.p5.7.m7.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.7.m7.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.7.m7.1d">italic_β</annotation></semantics></math> estimator corresponds to approximately evaluating <math alttext="\mathbb{P}(\cdot|i_{1}^{k})" class="ltx_math_unparsed" display="inline" id="S4.SS3.p5.8.m8.1"><semantics id="S4.SS3.p5.8.m8.1a"><mrow id="S4.SS3.p5.8.m8.1b"><mi id="S4.SS3.p5.8.m8.1.1">ℙ</mi><mrow id="S4.SS3.p5.8.m8.1.2"><mo id="S4.SS3.p5.8.m8.1.2.1" stretchy="false">(</mo><mo id="S4.SS3.p5.8.m8.1.2.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="S4.SS3.p5.8.m8.1.2.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="S4.SS3.p5.8.m8.1.2.4"><mi id="S4.SS3.p5.8.m8.1.2.4.2.2">i</mi><mn id="S4.SS3.p5.8.m8.1.2.4.2.3">1</mn><mi id="S4.SS3.p5.8.m8.1.2.4.3">k</mi></msubsup><mo id="S4.SS3.p5.8.m8.1.2.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS3.p5.8.m8.1c">\mathbb{P}(\cdot|i_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.8.m8.1d">blackboard_P ( ⋅ | italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>, which are unrelated distributions under different choices of <math alttext="i_{1}^{k}" class="ltx_Math" display="inline" id="S4.SS3.p5.9.m9.1"><semantics id="S4.SS3.p5.9.m9.1a"><msubsup id="S4.SS3.p5.9.m9.1.1" xref="S4.SS3.p5.9.m9.1.1.cmml"><mi id="S4.SS3.p5.9.m9.1.1.2.2" xref="S4.SS3.p5.9.m9.1.1.2.2.cmml">i</mi><mn id="S4.SS3.p5.9.m9.1.1.2.3" xref="S4.SS3.p5.9.m9.1.1.2.3.cmml">1</mn><mi id="S4.SS3.p5.9.m9.1.1.3" xref="S4.SS3.p5.9.m9.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.9.m9.1b"><apply id="S4.SS3.p5.9.m9.1.1.cmml" xref="S4.SS3.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.9.m9.1.1.1.cmml" xref="S4.SS3.p5.9.m9.1.1">superscript</csymbol><apply id="S4.SS3.p5.9.m9.1.1.2.cmml" xref="S4.SS3.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.9.m9.1.1.2.1.cmml" xref="S4.SS3.p5.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p5.9.m9.1.1.2.2.cmml" xref="S4.SS3.p5.9.m9.1.1.2.2">𝑖</ci><cn id="S4.SS3.p5.9.m9.1.1.2.3.cmml" type="integer" xref="S4.SS3.p5.9.m9.1.1.2.3">1</cn></apply><ci id="S4.SS3.p5.9.m9.1.1.3.cmml" xref="S4.SS3.p5.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.9.m9.1c">i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.9.m9.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>). There are <math alttext="\sim 2^{2^{k}}" class="ltx_Math" display="inline" id="S4.SS3.p5.10.m10.1"><semantics id="S4.SS3.p5.10.m10.1a"><mrow id="S4.SS3.p5.10.m10.1.1" xref="S4.SS3.p5.10.m10.1.1.cmml"><mi id="S4.SS3.p5.10.m10.1.1.2" xref="S4.SS3.p5.10.m10.1.1.2.cmml"></mi><mo id="S4.SS3.p5.10.m10.1.1.1" xref="S4.SS3.p5.10.m10.1.1.1.cmml">∼</mo><msup id="S4.SS3.p5.10.m10.1.1.3" xref="S4.SS3.p5.10.m10.1.1.3.cmml"><mn id="S4.SS3.p5.10.m10.1.1.3.2" xref="S4.SS3.p5.10.m10.1.1.3.2.cmml">2</mn><msup id="S4.SS3.p5.10.m10.1.1.3.3" xref="S4.SS3.p5.10.m10.1.1.3.3.cmml"><mn id="S4.SS3.p5.10.m10.1.1.3.3.2" xref="S4.SS3.p5.10.m10.1.1.3.3.2.cmml">2</mn><mi id="S4.SS3.p5.10.m10.1.1.3.3.3" xref="S4.SS3.p5.10.m10.1.1.3.3.3.cmml">k</mi></msup></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.10.m10.1b"><apply id="S4.SS3.p5.10.m10.1.1.cmml" xref="S4.SS3.p5.10.m10.1.1"><csymbol cd="latexml" id="S4.SS3.p5.10.m10.1.1.1.cmml" xref="S4.SS3.p5.10.m10.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.p5.10.m10.1.1.2.cmml" xref="S4.SS3.p5.10.m10.1.1.2">absent</csymbol><apply id="S4.SS3.p5.10.m10.1.1.3.cmml" xref="S4.SS3.p5.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.10.m10.1.1.3.1.cmml" xref="S4.SS3.p5.10.m10.1.1.3">superscript</csymbol><cn id="S4.SS3.p5.10.m10.1.1.3.2.cmml" type="integer" xref="S4.SS3.p5.10.m10.1.1.3.2">2</cn><apply id="S4.SS3.p5.10.m10.1.1.3.3.cmml" xref="S4.SS3.p5.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p5.10.m10.1.1.3.3.1.cmml" xref="S4.SS3.p5.10.m10.1.1.3.3">superscript</csymbol><cn id="S4.SS3.p5.10.m10.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.p5.10.m10.1.1.3.3.2">2</cn><ci id="S4.SS3.p5.10.m10.1.1.3.3.3.cmml" xref="S4.SS3.p5.10.m10.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.10.m10.1c">\sim 2^{2^{k}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.10.m10.1d">∼ 2 start_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> possible values the set <math alttext="P=\{\mathbb{P}(\cdot|i_{1}^{k}):i_{1}^{k}\in\{0,1\}^{k}\}" class="ltx_math_unparsed" display="inline" id="S4.SS3.p5.11.m11.1"><semantics id="S4.SS3.p5.11.m11.1a"><mrow id="S4.SS3.p5.11.m11.1b"><mi id="S4.SS3.p5.11.m11.1.1">P</mi><mo id="S4.SS3.p5.11.m11.1.2">=</mo><mrow id="S4.SS3.p5.11.m11.1.3"><mo id="S4.SS3.p5.11.m11.1.3.1" stretchy="false">{</mo><mi id="S4.SS3.p5.11.m11.1.3.2">ℙ</mi><mrow id="S4.SS3.p5.11.m11.1.3.3"><mo id="S4.SS3.p5.11.m11.1.3.3.1" stretchy="false">(</mo><mo id="S4.SS3.p5.11.m11.1.3.3.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="S4.SS3.p5.11.m11.1.3.3.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="S4.SS3.p5.11.m11.1.3.3.4"><mi id="S4.SS3.p5.11.m11.1.3.3.4.2.2">i</mi><mn id="S4.SS3.p5.11.m11.1.3.3.4.2.3">1</mn><mi id="S4.SS3.p5.11.m11.1.3.3.4.3">k</mi></msubsup><mo id="S4.SS3.p5.11.m11.1.3.3.5" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S4.SS3.p5.11.m11.1.3.4" rspace="0.278em">:</mo><msubsup id="S4.SS3.p5.11.m11.1.3.5"><mi id="S4.SS3.p5.11.m11.1.3.5.2.2">i</mi><mn id="S4.SS3.p5.11.m11.1.3.5.2.3">1</mn><mi id="S4.SS3.p5.11.m11.1.3.5.3">k</mi></msubsup><mo id="S4.SS3.p5.11.m11.1.3.6">∈</mo><msup id="S4.SS3.p5.11.m11.1.3.7"><mrow id="S4.SS3.p5.11.m11.1.3.7.2"><mo id="S4.SS3.p5.11.m11.1.3.7.2.1" stretchy="false">{</mo><mn id="S4.SS3.p5.11.m11.1.3.7.2.2">0</mn><mo id="S4.SS3.p5.11.m11.1.3.7.2.3">,</mo><mn id="S4.SS3.p5.11.m11.1.3.7.2.4">1</mn><mo id="S4.SS3.p5.11.m11.1.3.7.2.5" stretchy="false">}</mo></mrow><mi id="S4.SS3.p5.11.m11.1.3.7.3">k</mi></msup><mo id="S4.SS3.p5.11.m11.1.3.8" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS3.p5.11.m11.1c">P=\{\mathbb{P}(\cdot|i_{1}^{k}):i_{1}^{k}\in\{0,1\}^{k}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.11.m11.1d">italic_P = { blackboard_P ( ⋅ | italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT }</annotation></semantics></math> can take. But when <math alttext="d" class="ltx_Math" display="inline" id="S4.SS3.p5.12.m12.1"><semantics id="S4.SS3.p5.12.m12.1a"><mi id="S4.SS3.p5.12.m12.1.1" xref="S4.SS3.p5.12.m12.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.12.m12.1b"><ci id="S4.SS3.p5.12.m12.1.1.cmml" xref="S4.SS3.p5.12.m12.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.12.m12.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.12.m12.1d">italic_d</annotation></semantics></math> and <math alttext="\mathtt{p}" class="ltx_Math" display="inline" id="S4.SS3.p5.13.m13.1"><semantics id="S4.SS3.p5.13.m13.1a"><mi id="S4.SS3.p5.13.m13.1.1" xref="S4.SS3.p5.13.m13.1.1.cmml">𝚙</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.13.m13.1b"><ci id="S4.SS3.p5.13.m13.1.1.cmml" xref="S4.SS3.p5.13.m13.1.1">𝚙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.13.m13.1c">\mathtt{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.13.m13.1d">typewriter_p</annotation></semantics></math> are small, the output of the model just cannot take so many values: it can realize at most <math alttext="2^{d\mathtt{p}}" class="ltx_Math" display="inline" id="S4.SS3.p5.14.m14.1"><semantics id="S4.SS3.p5.14.m14.1a"><msup id="S4.SS3.p5.14.m14.1.1" xref="S4.SS3.p5.14.m14.1.1.cmml"><mn id="S4.SS3.p5.14.m14.1.1.2" xref="S4.SS3.p5.14.m14.1.1.2.cmml">2</mn><mrow id="S4.SS3.p5.14.m14.1.1.3" xref="S4.SS3.p5.14.m14.1.1.3.cmml"><mi id="S4.SS3.p5.14.m14.1.1.3.2" xref="S4.SS3.p5.14.m14.1.1.3.2.cmml">d</mi><mo id="S4.SS3.p5.14.m14.1.1.3.1" xref="S4.SS3.p5.14.m14.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p5.14.m14.1.1.3.3" xref="S4.SS3.p5.14.m14.1.1.3.3.cmml">𝚙</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.14.m14.1b"><apply id="S4.SS3.p5.14.m14.1.1.cmml" xref="S4.SS3.p5.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.14.m14.1.1.1.cmml" xref="S4.SS3.p5.14.m14.1.1">superscript</csymbol><cn id="S4.SS3.p5.14.m14.1.1.2.cmml" type="integer" xref="S4.SS3.p5.14.m14.1.1.2">2</cn><apply id="S4.SS3.p5.14.m14.1.1.3.cmml" xref="S4.SS3.p5.14.m14.1.1.3"><times id="S4.SS3.p5.14.m14.1.1.3.1.cmml" xref="S4.SS3.p5.14.m14.1.1.3.1"></times><ci id="S4.SS3.p5.14.m14.1.1.3.2.cmml" xref="S4.SS3.p5.14.m14.1.1.3.2">𝑑</ci><ci id="S4.SS3.p5.14.m14.1.1.3.3.cmml" xref="S4.SS3.p5.14.m14.1.1.3.3">𝚙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.14.m14.1c">2^{d\mathtt{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.14.m14.1d">2 start_POSTSUPERSCRIPT italic_d typewriter_p end_POSTSUPERSCRIPT</annotation></semantics></math> possible sets. In other words, in order to succeed, the recurrent architecture is essentially forced to keep track of the number of occurrences of each <math alttext="i_{1}^{k}\in\{0,1\}^{k}" class="ltx_Math" display="inline" id="S4.SS3.p5.15.m15.2"><semantics id="S4.SS3.p5.15.m15.2a"><mrow id="S4.SS3.p5.15.m15.2.3" xref="S4.SS3.p5.15.m15.2.3.cmml"><msubsup id="S4.SS3.p5.15.m15.2.3.2" xref="S4.SS3.p5.15.m15.2.3.2.cmml"><mi id="S4.SS3.p5.15.m15.2.3.2.2.2" xref="S4.SS3.p5.15.m15.2.3.2.2.2.cmml">i</mi><mn id="S4.SS3.p5.15.m15.2.3.2.2.3" xref="S4.SS3.p5.15.m15.2.3.2.2.3.cmml">1</mn><mi id="S4.SS3.p5.15.m15.2.3.2.3" xref="S4.SS3.p5.15.m15.2.3.2.3.cmml">k</mi></msubsup><mo id="S4.SS3.p5.15.m15.2.3.1" xref="S4.SS3.p5.15.m15.2.3.1.cmml">∈</mo><msup id="S4.SS3.p5.15.m15.2.3.3" xref="S4.SS3.p5.15.m15.2.3.3.cmml"><mrow id="S4.SS3.p5.15.m15.2.3.3.2.2" xref="S4.SS3.p5.15.m15.2.3.3.2.1.cmml"><mo id="S4.SS3.p5.15.m15.2.3.3.2.2.1" stretchy="false" xref="S4.SS3.p5.15.m15.2.3.3.2.1.cmml">{</mo><mn id="S4.SS3.p5.15.m15.1.1" xref="S4.SS3.p5.15.m15.1.1.cmml">0</mn><mo id="S4.SS3.p5.15.m15.2.3.3.2.2.2" xref="S4.SS3.p5.15.m15.2.3.3.2.1.cmml">,</mo><mn id="S4.SS3.p5.15.m15.2.2" xref="S4.SS3.p5.15.m15.2.2.cmml">1</mn><mo id="S4.SS3.p5.15.m15.2.3.3.2.2.3" stretchy="false" xref="S4.SS3.p5.15.m15.2.3.3.2.1.cmml">}</mo></mrow><mi id="S4.SS3.p5.15.m15.2.3.3.3" xref="S4.SS3.p5.15.m15.2.3.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.15.m15.2b"><apply id="S4.SS3.p5.15.m15.2.3.cmml" xref="S4.SS3.p5.15.m15.2.3"><in id="S4.SS3.p5.15.m15.2.3.1.cmml" xref="S4.SS3.p5.15.m15.2.3.1"></in><apply id="S4.SS3.p5.15.m15.2.3.2.cmml" xref="S4.SS3.p5.15.m15.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p5.15.m15.2.3.2.1.cmml" xref="S4.SS3.p5.15.m15.2.3.2">superscript</csymbol><apply id="S4.SS3.p5.15.m15.2.3.2.2.cmml" xref="S4.SS3.p5.15.m15.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p5.15.m15.2.3.2.2.1.cmml" xref="S4.SS3.p5.15.m15.2.3.2">subscript</csymbol><ci 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Chains"><span class="ltx_text ltx_ref_tag">Mamba</span></a> enabling selectivity is the state-transition factor <math alttext="a_{t}" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.1"><semantics id="S5.SS1.p1.1.m1.1a"><msub id="S5.SS1.p1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.cmml"><mi id="S5.SS1.p1.1.m1.1.1.2" xref="S5.SS1.p1.1.m1.1.1.2.cmml">a</mi><mi id="S5.SS1.p1.1.m1.1.1.3" xref="S5.SS1.p1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.1b"><apply id="S5.SS1.p1.1.m1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.p1.1.m1.1.1.2.cmml" xref="S5.SS1.p1.1.m1.1.1.2">𝑎</ci><ci id="S5.SS1.p1.1.m1.1.1.3.cmml" xref="S5.SS1.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, that controls the flow of information from the past state <math alttext="H_{t-1}" class="ltx_Math" display="inline" id="S5.SS1.p1.2.m2.1"><semantics id="S5.SS1.p1.2.m2.1a"><msub id="S5.SS1.p1.2.m2.1.1" xref="S5.SS1.p1.2.m2.1.1.cmml"><mi id="S5.SS1.p1.2.m2.1.1.2" xref="S5.SS1.p1.2.m2.1.1.2.cmml">H</mi><mrow id="S5.SS1.p1.2.m2.1.1.3" xref="S5.SS1.p1.2.m2.1.1.3.cmml"><mi id="S5.SS1.p1.2.m2.1.1.3.2" xref="S5.SS1.p1.2.m2.1.1.3.2.cmml">t</mi><mo id="S5.SS1.p1.2.m2.1.1.3.1" xref="S5.SS1.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="S5.SS1.p1.2.m2.1.1.3.3" xref="S5.SS1.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.2.m2.1b"><apply id="S5.SS1.p1.2.m2.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.2.m2.1.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.p1.2.m2.1.1.2.cmml" xref="S5.SS1.p1.2.m2.1.1.2">𝐻</ci><apply id="S5.SS1.p1.2.m2.1.1.3.cmml" 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the current state, and hence the output, whereas <math alttext="a_{t}=0" class="ltx_Math" display="inline" id="S5.SS1.p1.5.m5.1"><semantics id="S5.SS1.p1.5.m5.1a"><mrow id="S5.SS1.p1.5.m5.1.1" xref="S5.SS1.p1.5.m5.1.1.cmml"><msub id="S5.SS1.p1.5.m5.1.1.2" xref="S5.SS1.p1.5.m5.1.1.2.cmml"><mi id="S5.SS1.p1.5.m5.1.1.2.2" xref="S5.SS1.p1.5.m5.1.1.2.2.cmml">a</mi><mi id="S5.SS1.p1.5.m5.1.1.2.3" xref="S5.SS1.p1.5.m5.1.1.2.3.cmml">t</mi></msub><mo id="S5.SS1.p1.5.m5.1.1.1" xref="S5.SS1.p1.5.m5.1.1.1.cmml">=</mo><mn id="S5.SS1.p1.5.m5.1.1.3" xref="S5.SS1.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.5.m5.1b"><apply id="S5.SS1.p1.5.m5.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1"><eq id="S5.SS1.p1.5.m5.1.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1.1"></eq><apply id="S5.SS1.p1.5.m5.1.1.2.cmml" xref="S5.SS1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p1.5.m5.1.1.2.1.cmml" xref="S5.SS1.p1.5.m5.1.1.2">subscript</csymbol><ci id="S5.SS1.p1.5.m5.1.1.2.2.cmml" xref="S5.SS1.p1.5.m5.1.1.2.2">𝑎</ci><ci id="S5.SS1.p1.5.m5.1.1.2.3.cmml" xref="S5.SS1.p1.5.m5.1.1.2.3">𝑡</ci></apply><cn id="S5.SS1.p1.5.m5.1.1.3.cmml" type="integer" xref="S5.SS1.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.5.m5.1c">a_{t}=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math> completely ignores the past. In the Markovian setting considered so far, the role of <math alttext="a_{t}" class="ltx_Math" display="inline" id="S5.SS1.p1.6.m6.1"><semantics id="S5.SS1.p1.6.m6.1a"><msub id="S5.SS1.p1.6.m6.1.1" xref="S5.SS1.p1.6.m6.1.1.cmml"><mi id="S5.SS1.p1.6.m6.1.1.2" xref="S5.SS1.p1.6.m6.1.1.2.cmml">a</mi><mi id="S5.SS1.p1.6.m6.1.1.3" xref="S5.SS1.p1.6.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.6.m6.1b"><apply id="S5.SS1.p1.6.m6.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.1.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="S5.SS1.p1.6.m6.1.1.2.cmml" xref="S5.SS1.p1.6.m6.1.1.2">𝑎</ci><ci id="S5.SS1.p1.6.m6.1.1.3.cmml" xref="S5.SS1.p1.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.6.m6.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.6.m6.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> has largely been dormant: <math alttext="a_{t}\approx 1" class="ltx_Math" display="inline" id="S5.SS1.p1.7.m7.1"><semantics id="S5.SS1.p1.7.m7.1a"><mrow id="S5.SS1.p1.7.m7.1.1" xref="S5.SS1.p1.7.m7.1.1.cmml"><msub id="S5.SS1.p1.7.m7.1.1.2" xref="S5.SS1.p1.7.m7.1.1.2.cmml"><mi id="S5.SS1.p1.7.m7.1.1.2.2" xref="S5.SS1.p1.7.m7.1.1.2.2.cmml">a</mi><mi id="S5.SS1.p1.7.m7.1.1.2.3" xref="S5.SS1.p1.7.m7.1.1.2.3.cmml">t</mi></msub><mo id="S5.SS1.p1.7.m7.1.1.1" xref="S5.SS1.p1.7.m7.1.1.1.cmml">≈</mo><mn id="S5.SS1.p1.7.m7.1.1.3" xref="S5.SS1.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.7.m7.1b"><apply id="S5.SS1.p1.7.m7.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1"><approx id="S5.SS1.p1.7.m7.1.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1.1"></approx><apply id="S5.SS1.p1.7.m7.1.1.2.cmml" xref="S5.SS1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p1.7.m7.1.1.2.1.cmml" xref="S5.SS1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S5.SS1.p1.7.m7.1.1.2.2.cmml" xref="S5.SS1.p1.7.m7.1.1.2.2">𝑎</ci><ci id="S5.SS1.p1.7.m7.1.1.2.3.cmml" xref="S5.SS1.p1.7.m7.1.1.2.3">𝑡</ci></apply><cn id="S5.SS1.p1.7.m7.1.1.3.cmml" type="integer" xref="S5.SS1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.7.m7.1c">a_{t}\approx 1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.7.m7.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≈ 1</annotation></semantics></math> for all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="S5.SS1.p1.8.m8.1"><semantics id="S5.SS1.p1.8.m8.1a"><mrow id="S5.SS1.p1.8.m8.1.1" xref="S5.SS1.p1.8.m8.1.1.cmml"><mi id="S5.SS1.p1.8.m8.1.1.2" xref="S5.SS1.p1.8.m8.1.1.2.cmml">t</mi><mo id="S5.SS1.p1.8.m8.1.1.1" xref="S5.SS1.p1.8.m8.1.1.1.cmml">≥</mo><mn id="S5.SS1.p1.8.m8.1.1.3" xref="S5.SS1.p1.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.8.m8.1b"><apply id="S5.SS1.p1.8.m8.1.1.cmml" xref="S5.SS1.p1.8.m8.1.1"><geq id="S5.SS1.p1.8.m8.1.1.1.cmml" xref="S5.SS1.p1.8.m8.1.1.1"></geq><ci id="S5.SS1.p1.8.m8.1.1.2.cmml" xref="S5.SS1.p1.8.m8.1.1.2">𝑡</ci><cn id="S5.SS1.p1.8.m8.1.1.3.cmml" type="integer" xref="S5.SS1.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.8.m8.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.8.m8.1d">italic_t ≥ 1</annotation></semantics></math>, as the optimal Laplacian predictor requires counts of all transitions, demanding the use of full past (Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS2" title="4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">4.2</span></a>). To better highlight this selectivity mechanism, we consider a non-Markovian process, where the role of <math alttext="a_{t}" class="ltx_Math" display="inline" id="S5.SS1.p1.9.m9.1"><semantics id="S5.SS1.p1.9.m9.1a"><msub id="S5.SS1.p1.9.m9.1.1" xref="S5.SS1.p1.9.m9.1.1.cmml"><mi id="S5.SS1.p1.9.m9.1.1.2" xref="S5.SS1.p1.9.m9.1.1.2.cmml">a</mi><mi id="S5.SS1.p1.9.m9.1.1.3" xref="S5.SS1.p1.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.9.m9.1b"><apply id="S5.SS1.p1.9.m9.1.1.cmml" xref="S5.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.9.m9.1.1.1.cmml" xref="S5.SS1.p1.9.m9.1.1">subscript</csymbol><ci id="S5.SS1.p1.9.m9.1.1.2.cmml" xref="S5.SS1.p1.9.m9.1.1.2">𝑎</ci><ci id="S5.SS1.p1.9.m9.1.1.3.cmml" xref="S5.SS1.p1.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.9.m9.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.9.m9.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> becomes fundamental. Specifically, we focus on the <em class="ltx_emph ltx_font_italic" id="S5.SS1.p1.12.1">switching Markov process</em>, where we add a <em class="ltx_emph ltx_font_italic" id="S5.SS1.p1.12.2">switch</em> token to the binary alphabet, i.e. we consider <math alttext="\mathcal{X}=\{0,1,S\}" class="ltx_Math" display="inline" id="S5.SS1.p1.10.m10.3"><semantics id="S5.SS1.p1.10.m10.3a"><mrow id="S5.SS1.p1.10.m10.3.4" xref="S5.SS1.p1.10.m10.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p1.10.m10.3.4.2" xref="S5.SS1.p1.10.m10.3.4.2.cmml">𝒳</mi><mo id="S5.SS1.p1.10.m10.3.4.1" xref="S5.SS1.p1.10.m10.3.4.1.cmml">=</mo><mrow id="S5.SS1.p1.10.m10.3.4.3.2" xref="S5.SS1.p1.10.m10.3.4.3.1.cmml"><mo id="S5.SS1.p1.10.m10.3.4.3.2.1" stretchy="false" xref="S5.SS1.p1.10.m10.3.4.3.1.cmml">{</mo><mn id="S5.SS1.p1.10.m10.1.1" xref="S5.SS1.p1.10.m10.1.1.cmml">0</mn><mo id="S5.SS1.p1.10.m10.3.4.3.2.2" xref="S5.SS1.p1.10.m10.3.4.3.1.cmml">,</mo><mn id="S5.SS1.p1.10.m10.2.2" xref="S5.SS1.p1.10.m10.2.2.cmml">1</mn><mo id="S5.SS1.p1.10.m10.3.4.3.2.3" xref="S5.SS1.p1.10.m10.3.4.3.1.cmml">,</mo><mi id="S5.SS1.p1.10.m10.3.3" xref="S5.SS1.p1.10.m10.3.3.cmml">S</mi><mo id="S5.SS1.p1.10.m10.3.4.3.2.4" stretchy="false" xref="S5.SS1.p1.10.m10.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.10.m10.3b"><apply id="S5.SS1.p1.10.m10.3.4.cmml" xref="S5.SS1.p1.10.m10.3.4"><eq id="S5.SS1.p1.10.m10.3.4.1.cmml" xref="S5.SS1.p1.10.m10.3.4.1"></eq><ci id="S5.SS1.p1.10.m10.3.4.2.cmml" xref="S5.SS1.p1.10.m10.3.4.2">𝒳</ci><set id="S5.SS1.p1.10.m10.3.4.3.1.cmml" xref="S5.SS1.p1.10.m10.3.4.3.2"><cn id="S5.SS1.p1.10.m10.1.1.cmml" type="integer" xref="S5.SS1.p1.10.m10.1.1">0</cn><cn id="S5.SS1.p1.10.m10.2.2.cmml" type="integer" xref="S5.SS1.p1.10.m10.2.2">1</cn><ci id="S5.SS1.p1.10.m10.3.3.cmml" xref="S5.SS1.p1.10.m10.3.3">𝑆</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.10.m10.3c">\mathcal{X}=\{0,1,S\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.10.m10.3d">caligraphic_X = { 0 , 1 , italic_S }</annotation></semantics></math>. The key difference here compared to the random Markov generation in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.SS1" title="2.1 Input data: Random Markov chains ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2.1</span></a> is that until we hit switch token, the sequence follows the same binary Markov chain, but once the switch state is reached, the next symbols follow a newly sampled Markov chain, independent of the previous one. The switch tokens are sampled according to a parallel i.i.d. Bernoulli process with probability <math alttext="p_{\rm switch}" class="ltx_Math" display="inline" id="S5.SS1.p1.11.m11.1"><semantics id="S5.SS1.p1.11.m11.1a"><msub id="S5.SS1.p1.11.m11.1.1" xref="S5.SS1.p1.11.m11.1.1.cmml"><mi id="S5.SS1.p1.11.m11.1.1.2" xref="S5.SS1.p1.11.m11.1.1.2.cmml">p</mi><mi id="S5.SS1.p1.11.m11.1.1.3" xref="S5.SS1.p1.11.m11.1.1.3.cmml">switch</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.11.m11.1b"><apply id="S5.SS1.p1.11.m11.1.1.cmml" xref="S5.SS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m11.1.1.1.cmml" xref="S5.SS1.p1.11.m11.1.1">subscript</csymbol><ci id="S5.SS1.p1.11.m11.1.1.2.cmml" xref="S5.SS1.p1.11.m11.1.1.2">𝑝</ci><ci id="S5.SS1.p1.11.m11.1.1.3.cmml" xref="S5.SS1.p1.11.m11.1.1.3">switch</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.11.m11.1c">p_{\rm switch}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.11.m11.1d">italic_p start_POSTSUBSCRIPT roman_switch end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="0.01" class="ltx_Math" display="inline" id="S5.SS1.p1.12.m12.1"><semantics id="S5.SS1.p1.12.m12.1a"><mn id="S5.SS1.p1.12.m12.1.1" xref="S5.SS1.p1.12.m12.1.1.cmml">0.01</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.12.m12.1b"><cn id="S5.SS1.p1.12.m12.1.1.cmml" type="float" xref="S5.SS1.p1.12.m12.1.1">0.01</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.12.m12.1c">0.01</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.12.m12.1d">0.01</annotation></semantics></math> in our experiments). More formally, the process consists of the following steps:</p> <ol class="ltx_enumerate" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.1">Initialize <math alttext="t=1" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mrow id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml"><mi id="S5.I1.i1.p1.1.m1.1.1.2" xref="S5.I1.i1.p1.1.m1.1.1.2.cmml">t</mi><mo id="S5.I1.i1.p1.1.m1.1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S5.I1.i1.p1.1.m1.1.1.3" xref="S5.I1.i1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><apply id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1"><eq id="S5.I1.i1.p1.1.m1.1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1"></eq><ci id="S5.I1.i1.p1.1.m1.1.1.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.2">𝑡</ci><cn id="S5.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_t = 1</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.3">Draw a binary Markov kernel <math alttext="P" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><mi id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><ci id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.1d">italic_P</annotation></semantics></math> with each conditional distribution <math alttext="P_{i_{1}^{k}}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><msub id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.2" xref="S5.I1.i2.p1.2.m2.1.1.2.cmml">P</mi><msubsup id="S5.I1.i2.p1.2.m2.1.1.3" xref="S5.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.3.2.2" xref="S5.I1.i2.p1.2.m2.1.1.3.2.2.cmml">i</mi><mn id="S5.I1.i2.p1.2.m2.1.1.3.2.3" xref="S5.I1.i2.p1.2.m2.1.1.3.2.3.cmml">1</mn><mi id="S5.I1.i2.p1.2.m2.1.1.3.3" xref="S5.I1.i2.p1.2.m2.1.1.3.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><apply id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.1.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i2.p1.2.m2.1.1.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2">𝑃</ci><apply id="S5.I1.i2.p1.2.m2.1.1.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3">superscript</csymbol><apply id="S5.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.1.1.3.2.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.I1.i2.p1.2.m2.1.1.3.2.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.2.2">𝑖</ci><cn id="S5.I1.i2.p1.2.m2.1.1.3.2.3.cmml" type="integer" xref="S5.I1.i2.p1.2.m2.1.1.3.2.3">1</cn></apply><ci id="S5.I1.i2.p1.2.m2.1.1.3.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">P_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> sampled i.i.d. from <math alttext="\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.1"><semantics id="S5.I1.i2.p1.3.m3.1a"><mrow id="S5.I1.i2.p1.3.m3.1.1" xref="S5.I1.i2.p1.3.m3.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.1.1.3" xref="S5.I1.i2.p1.3.m3.1.1.3.cmml">Dir</mi><mo id="S5.I1.i2.p1.3.m3.1.1.2" xref="S5.I1.i2.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i2.p1.3.m3.1.1.1.1" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S5.I1.i2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S5.I1.i2.p1.3.m3.1.1.1.1.1" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.1.1.1.1.1.2" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.2.cmml">β</mi><mo id="S5.I1.i2.p1.3.m3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">⋅</mo><mn id="S5.I1.i2.p1.3.m3.1.1.1.1.1.3" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="S5.I1.i2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m3.1b"><apply id="S5.I1.i2.p1.3.m3.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1"><times id="S5.I1.i2.p1.3.m3.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2"></times><ci id="S5.I1.i2.p1.3.m3.1.1.3.cmml" xref="S5.I1.i2.p1.3.m3.1.1.3">Dir</ci><apply id="S5.I1.i2.p1.3.m3.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1.1"><ci id="S5.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.1">⋅</ci><ci id="S5.I1.i2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.2">𝛽</ci><cn id="S5.I1.i2.p1.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S5.I1.i2.p1.3.m3.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.1c">\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.1d">roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.6">Let <math alttext="x_{t}=S" class="ltx_Math" display="inline" id="S5.I1.i3.p1.1.m1.1"><semantics id="S5.I1.i3.p1.1.m1.1a"><mrow id="S5.I1.i3.p1.1.m1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.cmml"><msub id="S5.I1.i3.p1.1.m1.1.1.2" xref="S5.I1.i3.p1.1.m1.1.1.2.cmml"><mi id="S5.I1.i3.p1.1.m1.1.1.2.2" xref="S5.I1.i3.p1.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.I1.i3.p1.1.m1.1.1.2.3" xref="S5.I1.i3.p1.1.m1.1.1.2.3.cmml">t</mi></msub><mo id="S5.I1.i3.p1.1.m1.1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.1.cmml">=</mo><mi id="S5.I1.i3.p1.1.m1.1.1.3" xref="S5.I1.i3.p1.1.m1.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.1.m1.1b"><apply id="S5.I1.i3.p1.1.m1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1"><eq id="S5.I1.i3.p1.1.m1.1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1.1"></eq><apply id="S5.I1.i3.p1.1.m1.1.1.2.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.1.m1.1.1.2.2.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2.2">𝑥</ci><ci id="S5.I1.i3.p1.1.m1.1.1.2.3.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2.3">𝑡</ci></apply><ci id="S5.I1.i3.p1.1.m1.1.1.3.cmml" xref="S5.I1.i3.p1.1.m1.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.1.m1.1c">x_{t}=S</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_S</annotation></semantics></math> with probability <math alttext="p_{\rm switch}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.2.m2.1"><semantics id="S5.I1.i3.p1.2.m2.1a"><msub id="S5.I1.i3.p1.2.m2.1.1" xref="S5.I1.i3.p1.2.m2.1.1.cmml"><mi id="S5.I1.i3.p1.2.m2.1.1.2" xref="S5.I1.i3.p1.2.m2.1.1.2.cmml">p</mi><mi id="S5.I1.i3.p1.2.m2.1.1.3" xref="S5.I1.i3.p1.2.m2.1.1.3.cmml">switch</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.2.m2.1b"><apply id="S5.I1.i3.p1.2.m2.1.1.cmml" xref="S5.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.2.m2.1.1.1.cmml" xref="S5.I1.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.2.m2.1.1.2.cmml" xref="S5.I1.i3.p1.2.m2.1.1.2">𝑝</ci><ci id="S5.I1.i3.p1.2.m2.1.1.3.cmml" xref="S5.I1.i3.p1.2.m2.1.1.3">switch</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.2.m2.1c">p_{\rm switch}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_switch end_POSTSUBSCRIPT</annotation></semantics></math>, or sample <math alttext="x_{t}\sim P_{x_{t-k+1}^{t}}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.3.m3.1"><semantics id="S5.I1.i3.p1.3.m3.1a"><mrow id="S5.I1.i3.p1.3.m3.1.1" xref="S5.I1.i3.p1.3.m3.1.1.cmml"><msub id="S5.I1.i3.p1.3.m3.1.1.2" xref="S5.I1.i3.p1.3.m3.1.1.2.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.2.2" xref="S5.I1.i3.p1.3.m3.1.1.2.2.cmml">x</mi><mi id="S5.I1.i3.p1.3.m3.1.1.2.3" xref="S5.I1.i3.p1.3.m3.1.1.2.3.cmml">t</mi></msub><mo id="S5.I1.i3.p1.3.m3.1.1.1" xref="S5.I1.i3.p1.3.m3.1.1.1.cmml">∼</mo><msub id="S5.I1.i3.p1.3.m3.1.1.3" xref="S5.I1.i3.p1.3.m3.1.1.3.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.3.2" xref="S5.I1.i3.p1.3.m3.1.1.3.2.cmml">P</mi><msubsup id="S5.I1.i3.p1.3.m3.1.1.3.3" xref="S5.I1.i3.p1.3.m3.1.1.3.3.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.3.3.2.2" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.2.cmml">x</mi><mrow id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.cmml"><mrow id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.2" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.2.cmml">t</mi><mo id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.1" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.1.cmml">−</mo><mi id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.3" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.3.cmml">k</mi></mrow><mo id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.1" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.1.cmml">+</mo><mn id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.3" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.3.cmml">1</mn></mrow><mi id="S5.I1.i3.p1.3.m3.1.1.3.3.3" xref="S5.I1.i3.p1.3.m3.1.1.3.3.3.cmml">t</mi></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.3.m3.1b"><apply id="S5.I1.i3.p1.3.m3.1.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1"><csymbol cd="latexml" id="S5.I1.i3.p1.3.m3.1.1.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.1">similar-to</csymbol><apply id="S5.I1.i3.p1.3.m3.1.1.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.3.m3.1.1.2.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2.2">𝑥</ci><ci id="S5.I1.i3.p1.3.m3.1.1.2.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2.3">𝑡</ci></apply><apply id="S5.I1.i3.p1.3.m3.1.1.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.3.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.I1.i3.p1.3.m3.1.1.3.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.2">𝑃</ci><apply id="S5.I1.i3.p1.3.m3.1.1.3.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.3.3.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3">superscript</csymbol><apply id="S5.I1.i3.p1.3.m3.1.1.3.3.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.3.3.2.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S5.I1.i3.p1.3.m3.1.1.3.3.2.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.2">𝑥</ci><apply id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3"><plus id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.1"></plus><apply id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2"><minus id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.1"></minus><ci id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.2">𝑡</ci><ci id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.2.3">𝑘</ci></apply><cn id="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.3.cmml" type="integer" xref="S5.I1.i3.p1.3.m3.1.1.3.3.2.3.3">1</cn></apply></apply><ci id="S5.I1.i3.p1.3.m3.1.1.3.3.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.3.m3.1c">x_{t}\sim P_{x_{t-k+1}^{t}}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∼ italic_P start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with probability <math alttext="1-p_{\rm switch}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.4.m4.1"><semantics id="S5.I1.i3.p1.4.m4.1a"><mrow id="S5.I1.i3.p1.4.m4.1.1" xref="S5.I1.i3.p1.4.m4.1.1.cmml"><mn id="S5.I1.i3.p1.4.m4.1.1.2" xref="S5.I1.i3.p1.4.m4.1.1.2.cmml">1</mn><mo id="S5.I1.i3.p1.4.m4.1.1.1" xref="S5.I1.i3.p1.4.m4.1.1.1.cmml">−</mo><msub id="S5.I1.i3.p1.4.m4.1.1.3" xref="S5.I1.i3.p1.4.m4.1.1.3.cmml"><mi id="S5.I1.i3.p1.4.m4.1.1.3.2" xref="S5.I1.i3.p1.4.m4.1.1.3.2.cmml">p</mi><mi id="S5.I1.i3.p1.4.m4.1.1.3.3" xref="S5.I1.i3.p1.4.m4.1.1.3.3.cmml">switch</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.4.m4.1b"><apply id="S5.I1.i3.p1.4.m4.1.1.cmml" xref="S5.I1.i3.p1.4.m4.1.1"><minus id="S5.I1.i3.p1.4.m4.1.1.1.cmml" xref="S5.I1.i3.p1.4.m4.1.1.1"></minus><cn id="S5.I1.i3.p1.4.m4.1.1.2.cmml" type="integer" xref="S5.I1.i3.p1.4.m4.1.1.2">1</cn><apply id="S5.I1.i3.p1.4.m4.1.1.3.cmml" xref="S5.I1.i3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.4.m4.1.1.3.1.cmml" xref="S5.I1.i3.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.I1.i3.p1.4.m4.1.1.3.2.cmml" xref="S5.I1.i3.p1.4.m4.1.1.3.2">𝑝</ci><ci id="S5.I1.i3.p1.4.m4.1.1.3.3.cmml" xref="S5.I1.i3.p1.4.m4.1.1.3.3">switch</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.4.m4.1c">1-p_{\rm switch}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.4.m4.1d">1 - italic_p start_POSTSUBSCRIPT roman_switch end_POSTSUBSCRIPT</annotation></semantics></math> (the first <math alttext="k" class="ltx_Math" display="inline" id="S5.I1.i3.p1.5.m5.1"><semantics id="S5.I1.i3.p1.5.m5.1a"><mi id="S5.I1.i3.p1.5.m5.1.1" xref="S5.I1.i3.p1.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.5.m5.1b"><ci id="S5.I1.i3.p1.5.m5.1.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.5.m5.1d">italic_k</annotation></semantics></math> samples after each switch token are sampled from <math alttext="\mathrm{Unif}(\{0,1\})" class="ltx_Math" display="inline" id="S5.I1.i3.p1.6.m6.3"><semantics id="S5.I1.i3.p1.6.m6.3a"><mrow id="S5.I1.i3.p1.6.m6.3.3" xref="S5.I1.i3.p1.6.m6.3.3.cmml"><mi id="S5.I1.i3.p1.6.m6.3.3.3" xref="S5.I1.i3.p1.6.m6.3.3.3.cmml">Unif</mi><mo id="S5.I1.i3.p1.6.m6.3.3.2" xref="S5.I1.i3.p1.6.m6.3.3.2.cmml">⁢</mo><mrow id="S5.I1.i3.p1.6.m6.3.3.1.1" xref="S5.I1.i3.p1.6.m6.3.3.cmml"><mo id="S5.I1.i3.p1.6.m6.3.3.1.1.2" stretchy="false" xref="S5.I1.i3.p1.6.m6.3.3.cmml">(</mo><mrow id="S5.I1.i3.p1.6.m6.3.3.1.1.1.2" xref="S5.I1.i3.p1.6.m6.3.3.1.1.1.1.cmml"><mo id="S5.I1.i3.p1.6.m6.3.3.1.1.1.2.1" stretchy="false" xref="S5.I1.i3.p1.6.m6.3.3.1.1.1.1.cmml">{</mo><mn id="S5.I1.i3.p1.6.m6.1.1" xref="S5.I1.i3.p1.6.m6.1.1.cmml">0</mn><mo id="S5.I1.i3.p1.6.m6.3.3.1.1.1.2.2" xref="S5.I1.i3.p1.6.m6.3.3.1.1.1.1.cmml">,</mo><mn id="S5.I1.i3.p1.6.m6.2.2" xref="S5.I1.i3.p1.6.m6.2.2.cmml">1</mn><mo id="S5.I1.i3.p1.6.m6.3.3.1.1.1.2.3" stretchy="false" xref="S5.I1.i3.p1.6.m6.3.3.1.1.1.1.cmml">}</mo></mrow><mo id="S5.I1.i3.p1.6.m6.3.3.1.1.3" stretchy="false" xref="S5.I1.i3.p1.6.m6.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.6.m6.3b"><apply id="S5.I1.i3.p1.6.m6.3.3.cmml" xref="S5.I1.i3.p1.6.m6.3.3"><times id="S5.I1.i3.p1.6.m6.3.3.2.cmml" xref="S5.I1.i3.p1.6.m6.3.3.2"></times><ci id="S5.I1.i3.p1.6.m6.3.3.3.cmml" xref="S5.I1.i3.p1.6.m6.3.3.3">Unif</ci><set id="S5.I1.i3.p1.6.m6.3.3.1.1.1.1.cmml" xref="S5.I1.i3.p1.6.m6.3.3.1.1.1.2"><cn id="S5.I1.i3.p1.6.m6.1.1.cmml" type="integer" xref="S5.I1.i3.p1.6.m6.1.1">0</cn><cn id="S5.I1.i3.p1.6.m6.2.2.cmml" type="integer" xref="S5.I1.i3.p1.6.m6.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.6.m6.3c">\mathrm{Unif}(\{0,1\})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.6.m6.3d">roman_Unif ( { 0 , 1 } )</annotation></semantics></math>).</p> </div> </li> <li class="ltx_item" id="S5.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S5.I1.i4.p1"> <p class="ltx_p" id="S5.I1.i4.p1.4">If <math alttext="x_{t}=S" class="ltx_Math" display="inline" id="S5.I1.i4.p1.1.m1.1"><semantics id="S5.I1.i4.p1.1.m1.1a"><mrow id="S5.I1.i4.p1.1.m1.1.1" xref="S5.I1.i4.p1.1.m1.1.1.cmml"><msub id="S5.I1.i4.p1.1.m1.1.1.2" xref="S5.I1.i4.p1.1.m1.1.1.2.cmml"><mi id="S5.I1.i4.p1.1.m1.1.1.2.2" xref="S5.I1.i4.p1.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.I1.i4.p1.1.m1.1.1.2.3" xref="S5.I1.i4.p1.1.m1.1.1.2.3.cmml">t</mi></msub><mo id="S5.I1.i4.p1.1.m1.1.1.1" xref="S5.I1.i4.p1.1.m1.1.1.1.cmml">=</mo><mi id="S5.I1.i4.p1.1.m1.1.1.3" xref="S5.I1.i4.p1.1.m1.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.1.m1.1b"><apply id="S5.I1.i4.p1.1.m1.1.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1"><eq id="S5.I1.i4.p1.1.m1.1.1.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1.1"></eq><apply id="S5.I1.i4.p1.1.m1.1.1.2.cmml" xref="S5.I1.i4.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i4.p1.1.m1.1.1.2.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1.2">subscript</csymbol><ci id="S5.I1.i4.p1.1.m1.1.1.2.2.cmml" xref="S5.I1.i4.p1.1.m1.1.1.2.2">𝑥</ci><ci id="S5.I1.i4.p1.1.m1.1.1.2.3.cmml" xref="S5.I1.i4.p1.1.m1.1.1.2.3">𝑡</ci></apply><ci id="S5.I1.i4.p1.1.m1.1.1.3.cmml" xref="S5.I1.i4.p1.1.m1.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.1.m1.1c">x_{t}=S</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_S</annotation></semantics></math>, set <math alttext="t=t+1" class="ltx_Math" display="inline" id="S5.I1.i4.p1.2.m2.1"><semantics id="S5.I1.i4.p1.2.m2.1a"><mrow id="S5.I1.i4.p1.2.m2.1.1" xref="S5.I1.i4.p1.2.m2.1.1.cmml"><mi id="S5.I1.i4.p1.2.m2.1.1.2" xref="S5.I1.i4.p1.2.m2.1.1.2.cmml">t</mi><mo id="S5.I1.i4.p1.2.m2.1.1.1" xref="S5.I1.i4.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S5.I1.i4.p1.2.m2.1.1.3" xref="S5.I1.i4.p1.2.m2.1.1.3.cmml"><mi id="S5.I1.i4.p1.2.m2.1.1.3.2" xref="S5.I1.i4.p1.2.m2.1.1.3.2.cmml">t</mi><mo id="S5.I1.i4.p1.2.m2.1.1.3.1" xref="S5.I1.i4.p1.2.m2.1.1.3.1.cmml">+</mo><mn id="S5.I1.i4.p1.2.m2.1.1.3.3" xref="S5.I1.i4.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.2.m2.1b"><apply id="S5.I1.i4.p1.2.m2.1.1.cmml" xref="S5.I1.i4.p1.2.m2.1.1"><eq id="S5.I1.i4.p1.2.m2.1.1.1.cmml" xref="S5.I1.i4.p1.2.m2.1.1.1"></eq><ci id="S5.I1.i4.p1.2.m2.1.1.2.cmml" xref="S5.I1.i4.p1.2.m2.1.1.2">𝑡</ci><apply id="S5.I1.i4.p1.2.m2.1.1.3.cmml" xref="S5.I1.i4.p1.2.m2.1.1.3"><plus id="S5.I1.i4.p1.2.m2.1.1.3.1.cmml" xref="S5.I1.i4.p1.2.m2.1.1.3.1"></plus><ci id="S5.I1.i4.p1.2.m2.1.1.3.2.cmml" xref="S5.I1.i4.p1.2.m2.1.1.3.2">𝑡</ci><cn id="S5.I1.i4.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S5.I1.i4.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.2.m2.1c">t=t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.2.m2.1d">italic_t = italic_t + 1</annotation></semantics></math> and go to step 2; if <math alttext="x_{t}\neq S" class="ltx_Math" display="inline" id="S5.I1.i4.p1.3.m3.1"><semantics id="S5.I1.i4.p1.3.m3.1a"><mrow id="S5.I1.i4.p1.3.m3.1.1" xref="S5.I1.i4.p1.3.m3.1.1.cmml"><msub id="S5.I1.i4.p1.3.m3.1.1.2" xref="S5.I1.i4.p1.3.m3.1.1.2.cmml"><mi id="S5.I1.i4.p1.3.m3.1.1.2.2" xref="S5.I1.i4.p1.3.m3.1.1.2.2.cmml">x</mi><mi id="S5.I1.i4.p1.3.m3.1.1.2.3" xref="S5.I1.i4.p1.3.m3.1.1.2.3.cmml">t</mi></msub><mo id="S5.I1.i4.p1.3.m3.1.1.1" xref="S5.I1.i4.p1.3.m3.1.1.1.cmml">≠</mo><mi id="S5.I1.i4.p1.3.m3.1.1.3" xref="S5.I1.i4.p1.3.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.3.m3.1b"><apply id="S5.I1.i4.p1.3.m3.1.1.cmml" xref="S5.I1.i4.p1.3.m3.1.1"><neq id="S5.I1.i4.p1.3.m3.1.1.1.cmml" xref="S5.I1.i4.p1.3.m3.1.1.1"></neq><apply id="S5.I1.i4.p1.3.m3.1.1.2.cmml" xref="S5.I1.i4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i4.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i4.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.I1.i4.p1.3.m3.1.1.2.2.cmml" xref="S5.I1.i4.p1.3.m3.1.1.2.2">𝑥</ci><ci id="S5.I1.i4.p1.3.m3.1.1.2.3.cmml" xref="S5.I1.i4.p1.3.m3.1.1.2.3">𝑡</ci></apply><ci id="S5.I1.i4.p1.3.m3.1.1.3.cmml" xref="S5.I1.i4.p1.3.m3.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.3.m3.1c">x_{t}\neq S</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≠ italic_S</annotation></semantics></math>, set <math alttext="t=t+1" class="ltx_Math" display="inline" id="S5.I1.i4.p1.4.m4.1"><semantics id="S5.I1.i4.p1.4.m4.1a"><mrow id="S5.I1.i4.p1.4.m4.1.1" xref="S5.I1.i4.p1.4.m4.1.1.cmml"><mi id="S5.I1.i4.p1.4.m4.1.1.2" xref="S5.I1.i4.p1.4.m4.1.1.2.cmml">t</mi><mo id="S5.I1.i4.p1.4.m4.1.1.1" xref="S5.I1.i4.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S5.I1.i4.p1.4.m4.1.1.3" xref="S5.I1.i4.p1.4.m4.1.1.3.cmml"><mi id="S5.I1.i4.p1.4.m4.1.1.3.2" xref="S5.I1.i4.p1.4.m4.1.1.3.2.cmml">t</mi><mo id="S5.I1.i4.p1.4.m4.1.1.3.1" xref="S5.I1.i4.p1.4.m4.1.1.3.1.cmml">+</mo><mn id="S5.I1.i4.p1.4.m4.1.1.3.3" xref="S5.I1.i4.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.4.m4.1b"><apply id="S5.I1.i4.p1.4.m4.1.1.cmml" xref="S5.I1.i4.p1.4.m4.1.1"><eq id="S5.I1.i4.p1.4.m4.1.1.1.cmml" xref="S5.I1.i4.p1.4.m4.1.1.1"></eq><ci id="S5.I1.i4.p1.4.m4.1.1.2.cmml" xref="S5.I1.i4.p1.4.m4.1.1.2">𝑡</ci><apply id="S5.I1.i4.p1.4.m4.1.1.3.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3"><plus id="S5.I1.i4.p1.4.m4.1.1.3.1.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3.1"></plus><ci id="S5.I1.i4.p1.4.m4.1.1.3.2.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3.2">𝑡</ci><cn id="S5.I1.i4.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.I1.i4.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.4.m4.1c">t=t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.4.m4.1d">italic_t = italic_t + 1</annotation></semantics></math> and go to step 3.</p> </div> </li> </ol> </div> <figure class="ltx_figure" id="S5.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S5.F4.sf1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="626" id="S5.F4.sf1.g1" src="x5.png" width="830"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell 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ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S5.F4.sf2.5.2.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S5.F4.sf2.3.1" style="font-size:90%;">Value of <math alttext="a_{t}" class="ltx_Math" display="inline" id="S5.F4.sf2.3.1.m1.1"><semantics id="S5.F4.sf2.3.1.m1.1b"><msub id="S5.F4.sf2.3.1.m1.1.1" xref="S5.F4.sf2.3.1.m1.1.1.cmml"><mi id="S5.F4.sf2.3.1.m1.1.1.2" xref="S5.F4.sf2.3.1.m1.1.1.2.cmml">a</mi><mi id="S5.F4.sf2.3.1.m1.1.1.3" xref="S5.F4.sf2.3.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.F4.sf2.3.1.m1.1c"><apply id="S5.F4.sf2.3.1.m1.1.1.cmml" xref="S5.F4.sf2.3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.F4.sf2.3.1.m1.1.1.1.cmml" xref="S5.F4.sf2.3.1.m1.1.1">subscript</csymbol><ci id="S5.F4.sf2.3.1.m1.1.1.2.cmml" xref="S5.F4.sf2.3.1.m1.1.1.2">𝑎</ci><ci id="S5.F4.sf2.3.1.m1.1.1.3.cmml" xref="S5.F4.sf2.3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F4.sf2.3.1.m1.1d">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.F4.sf2.3.1.m1.1e">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> across positions</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S5.F4.6.3.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S5.F4.4.2" style="font-size:90%;">Single-layer Mamba on data generated from the switching Markov process with <math alttext="p_{\rm switch}=0.01" class="ltx_Math" display="inline" id="S5.F4.3.1.m1.1"><semantics id="S5.F4.3.1.m1.1b"><mrow id="S5.F4.3.1.m1.1.1" xref="S5.F4.3.1.m1.1.1.cmml"><msub id="S5.F4.3.1.m1.1.1.2" xref="S5.F4.3.1.m1.1.1.2.cmml"><mi id="S5.F4.3.1.m1.1.1.2.2" xref="S5.F4.3.1.m1.1.1.2.2.cmml">p</mi><mi id="S5.F4.3.1.m1.1.1.2.3" xref="S5.F4.3.1.m1.1.1.2.3.cmml">switch</mi></msub><mo id="S5.F4.3.1.m1.1.1.1" xref="S5.F4.3.1.m1.1.1.1.cmml">=</mo><mn id="S5.F4.3.1.m1.1.1.3" xref="S5.F4.3.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.F4.3.1.m1.1c"><apply id="S5.F4.3.1.m1.1.1.cmml" xref="S5.F4.3.1.m1.1.1"><eq id="S5.F4.3.1.m1.1.1.1.cmml" xref="S5.F4.3.1.m1.1.1.1"></eq><apply id="S5.F4.3.1.m1.1.1.2.cmml" xref="S5.F4.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.F4.3.1.m1.1.1.2.1.cmml" xref="S5.F4.3.1.m1.1.1.2">subscript</csymbol><ci id="S5.F4.3.1.m1.1.1.2.2.cmml" xref="S5.F4.3.1.m1.1.1.2.2">𝑝</ci><ci id="S5.F4.3.1.m1.1.1.2.3.cmml" xref="S5.F4.3.1.m1.1.1.2.3">switch</ci></apply><cn id="S5.F4.3.1.m1.1.1.3.cmml" type="float" xref="S5.F4.3.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F4.3.1.m1.1d">p_{\rm switch}=0.01</annotation><annotation encoding="application/x-llamapun" id="S5.F4.3.1.m1.1e">italic_p start_POSTSUBSCRIPT roman_switch end_POSTSUBSCRIPT = 0.01</annotation></semantics></math>. The red vertical lines mark the positions of switch tokens. Figure (a) shows that the model’s prediction follows very precisely that of the optimal estimator also in this more complex scenario. Figure (b) highlights the selectivity process of the model: every time a switch token appears, the model erases all information about the past by setting <math alttext="a_{t}=0" class="ltx_Math" display="inline" id="S5.F4.4.2.m2.1"><semantics id="S5.F4.4.2.m2.1b"><mrow id="S5.F4.4.2.m2.1.1" xref="S5.F4.4.2.m2.1.1.cmml"><msub id="S5.F4.4.2.m2.1.1.2" xref="S5.F4.4.2.m2.1.1.2.cmml"><mi id="S5.F4.4.2.m2.1.1.2.2" xref="S5.F4.4.2.m2.1.1.2.2.cmml">a</mi><mi id="S5.F4.4.2.m2.1.1.2.3" xref="S5.F4.4.2.m2.1.1.2.3.cmml">t</mi></msub><mo id="S5.F4.4.2.m2.1.1.1" xref="S5.F4.4.2.m2.1.1.1.cmml">=</mo><mn id="S5.F4.4.2.m2.1.1.3" xref="S5.F4.4.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.F4.4.2.m2.1c"><apply id="S5.F4.4.2.m2.1.1.cmml" xref="S5.F4.4.2.m2.1.1"><eq id="S5.F4.4.2.m2.1.1.1.cmml" xref="S5.F4.4.2.m2.1.1.1"></eq><apply id="S5.F4.4.2.m2.1.1.2.cmml" xref="S5.F4.4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.F4.4.2.m2.1.1.2.1.cmml" xref="S5.F4.4.2.m2.1.1.2">subscript</csymbol><ci id="S5.F4.4.2.m2.1.1.2.2.cmml" xref="S5.F4.4.2.m2.1.1.2.2">𝑎</ci><ci id="S5.F4.4.2.m2.1.1.2.3.cmml" xref="S5.F4.4.2.m2.1.1.2.3">𝑡</ci></apply><cn id="S5.F4.4.2.m2.1.1.3.cmml" type="integer" xref="S5.F4.4.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F4.4.2.m2.1d">a_{t}=0</annotation><annotation encoding="application/x-llamapun" id="S5.F4.4.2.m2.1e">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></figcaption> </figure> <div class="ltx_para ltx_noindent" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.4"><span class="ltx_text ltx_font_bold" id="S5.SS1.p2.4.1">Mamba learns the optimal estimator.</span> With this data model, the optimal prediction strategy is to use the add-<math alttext="\beta" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.1"><semantics id="S5.SS1.p2.1.m1.1a"><mi id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.1b"><ci id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.1d">italic_β</annotation></semantics></math> estimator in between two switch tokens, and reset the transition counts every time a switch occurs. Indeed, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S5.F4" title="Figure 4 ‣ 5.1 Switching Markov model ‣ 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">4</span></a> illustrates that Mamba implements precisely this strategy, closely tracking the switching events via the transition factor <math alttext="a_{t}" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.1"><semantics id="S5.SS1.p2.2.m2.1a"><msub id="S5.SS1.p2.2.m2.1.1" xref="S5.SS1.p2.2.m2.1.1.cmml"><mi id="S5.SS1.p2.2.m2.1.1.2" xref="S5.SS1.p2.2.m2.1.1.2.cmml">a</mi><mi id="S5.SS1.p2.2.m2.1.1.3" xref="S5.SS1.p2.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.1b"><apply id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.p2.2.m2.1.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.p2.2.m2.1.1.2.cmml" xref="S5.SS1.p2.2.m2.1.1.2">𝑎</ci><ci id="S5.SS1.p2.2.m2.1.1.3.cmml" xref="S5.SS1.p2.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>: it sets <math alttext="a_{t}" class="ltx_Math" display="inline" id="S5.SS1.p2.3.m3.1"><semantics id="S5.SS1.p2.3.m3.1a"><msub id="S5.SS1.p2.3.m3.1.1" xref="S5.SS1.p2.3.m3.1.1.cmml"><mi id="S5.SS1.p2.3.m3.1.1.2" xref="S5.SS1.p2.3.m3.1.1.2.cmml">a</mi><mi id="S5.SS1.p2.3.m3.1.1.3" xref="S5.SS1.p2.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.3.m3.1b"><apply id="S5.SS1.p2.3.m3.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS1.p2.3.m3.1.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S5.SS1.p2.3.m3.1.1.2.cmml" xref="S5.SS1.p2.3.m3.1.1.2">𝑎</ci><ci id="S5.SS1.p2.3.m3.1.1.3.cmml" xref="S5.SS1.p2.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.3.m3.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> to be zero whenever <math alttext="x_{t}=S" class="ltx_Math" display="inline" id="S5.SS1.p2.4.m4.1"><semantics id="S5.SS1.p2.4.m4.1a"><mrow id="S5.SS1.p2.4.m4.1.1" xref="S5.SS1.p2.4.m4.1.1.cmml"><msub id="S5.SS1.p2.4.m4.1.1.2" xref="S5.SS1.p2.4.m4.1.1.2.cmml"><mi id="S5.SS1.p2.4.m4.1.1.2.2" xref="S5.SS1.p2.4.m4.1.1.2.2.cmml">x</mi><mi id="S5.SS1.p2.4.m4.1.1.2.3" xref="S5.SS1.p2.4.m4.1.1.2.3.cmml">t</mi></msub><mo id="S5.SS1.p2.4.m4.1.1.1" xref="S5.SS1.p2.4.m4.1.1.1.cmml">=</mo><mi id="S5.SS1.p2.4.m4.1.1.3" xref="S5.SS1.p2.4.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.4.m4.1b"><apply id="S5.SS1.p2.4.m4.1.1.cmml" xref="S5.SS1.p2.4.m4.1.1"><eq id="S5.SS1.p2.4.m4.1.1.1.cmml" xref="S5.SS1.p2.4.m4.1.1.1"></eq><apply id="S5.SS1.p2.4.m4.1.1.2.cmml" xref="S5.SS1.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p2.4.m4.1.1.2.1.cmml" xref="S5.SS1.p2.4.m4.1.1.2">subscript</csymbol><ci id="S5.SS1.p2.4.m4.1.1.2.2.cmml" xref="S5.SS1.p2.4.m4.1.1.2.2">𝑥</ci><ci id="S5.SS1.p2.4.m4.1.1.2.3.cmml" xref="S5.SS1.p2.4.m4.1.1.2.3">𝑡</ci></apply><ci id="S5.SS1.p2.4.m4.1.1.3.cmml" xref="S5.SS1.p2.4.m4.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.4.m4.1c">x_{t}=S</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_S</annotation></semantics></math> and to one otherwise. This enables the model to zero out the transition counts at every switch event, so that it can estimate the statistics of the new chain from scratch.</p> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>Natural language modeling</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.5">To test the generality of our finding that convolution plays key role on Markovian data (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S3.F3" title="Figure 3 ‣ 3 Does Mamba learn in-context estimators? ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">3</span></a>), we conduct experiments on the language modeling task using the WikiText-103 dataset with a sequence length of <math alttext="256" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><mn id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml">256</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><cn id="S5.SS2.p1.1.m1.1.1.cmml" type="integer" xref="S5.SS2.p1.1.m1.1.1">256</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">256</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">256</annotation></semantics></math>. We use a standard 2-layer Transformer consisting of attention and MLP modules with a model dimension of <math alttext="256" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.1"><semantics id="S5.SS2.p1.2.m2.1a"><mn id="S5.SS2.p1.2.m2.1.1" xref="S5.SS2.p1.2.m2.1.1.cmml">256</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.1b"><cn id="S5.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S5.SS2.p1.2.m2.1.1">256</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.1c">256</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.1d">256</annotation></semantics></math>. To ensure a comparable parameter count, we stack the Mamba-<math alttext="2" class="ltx_Math" display="inline" id="S5.SS2.p1.3.m3.1"><semantics id="S5.SS2.p1.3.m3.1a"><mn id="S5.SS2.p1.3.m3.1.1" xref="S5.SS2.p1.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.3.m3.1b"><cn id="S5.SS2.p1.3.m3.1.1.cmml" type="integer" xref="S5.SS2.p1.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.3.m3.1c">2</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.3.m3.1d">2</annotation></semantics></math> cell across four layers, following the design in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">dao2024transformers</span>]</cite>. By adding or removing convolution in both these models, we obtain the results shown in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2502.10178v1#S5.T1" title="Table 1 ‣ 5.2 Natural language modeling ‣ 5 Beyond Markov ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Table 1</span></a>. The results illustrate that while convolution significantly enhances the performance of Mamba-2, reducing perplexity from <math alttext="30.68" class="ltx_Math" display="inline" id="S5.SS2.p1.4.m4.1"><semantics id="S5.SS2.p1.4.m4.1a"><mn id="S5.SS2.p1.4.m4.1.1" xref="S5.SS2.p1.4.m4.1.1.cmml">30.68</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.4.m4.1b"><cn id="S5.SS2.p1.4.m4.1.1.cmml" type="float" xref="S5.SS2.p1.4.m4.1.1">30.68</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.4.m4.1c">30.68</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.4.m4.1d">30.68</annotation></semantics></math> to <math alttext="27.55" class="ltx_Math" display="inline" id="S5.SS2.p1.5.m5.1"><semantics id="S5.SS2.p1.5.m5.1a"><mn id="S5.SS2.p1.5.m5.1.1" xref="S5.SS2.p1.5.m5.1.1.cmml">27.55</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.5.m5.1b"><cn id="S5.SS2.p1.5.m5.1.1.cmml" type="float" xref="S5.SS2.p1.5.m5.1.1">27.55</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.5.m5.1c">27.55</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.5.m5.1d">27.55</annotation></semantics></math>, the improvement for the Transformer is marginal. This highlights the fundamental role of convolution even on the complex language modeling tasks.</p> </div> <figure class="ltx_table" id="S5.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S5.T1.15.2.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S5.T1.2.1" style="font-size:90%;">Perplexity results on the WikiText-<math alttext="103" class="ltx_Math" display="inline" id="S5.T1.2.1.m1.1"><semantics id="S5.T1.2.1.m1.1b"><mn id="S5.T1.2.1.m1.1.1" xref="S5.T1.2.1.m1.1.1.cmml">103</mn><annotation-xml encoding="MathML-Content" id="S5.T1.2.1.m1.1c"><cn id="S5.T1.2.1.m1.1.1.cmml" type="integer" xref="S5.T1.2.1.m1.1.1">103</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.2.1.m1.1d">103</annotation><annotation encoding="application/x-llamapun" id="S5.T1.2.1.m1.1e">103</annotation></semantics></math> dataset for both models. (w/o conv) denotes the absence of convolution, while (w/ conv) indicates its use.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S5.T1.13"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T1.3.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S5.T1.3.1.2"><span class="ltx_text ltx_font_bold" id="S5.T1.3.1.2.1">Model</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="S5.T1.3.1.3"><span class="ltx_text ltx_font_bold" id="S5.T1.3.1.3.1"># Params. (M)</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="S5.T1.3.1.1"><span class="ltx_text ltx_font_bold" id="S5.T1.3.1.1.1">Perplexity (<math alttext="\downarrow" class="ltx_Math" display="inline" id="S5.T1.3.1.1.1.m1.1"><semantics id="S5.T1.3.1.1.1.m1.1a"><mo id="S5.T1.3.1.1.1.m1.1.1" stretchy="false" xref="S5.T1.3.1.1.1.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S5.T1.3.1.1.1.m1.1b"><ci id="S5.T1.3.1.1.1.m1.1.1.cmml" xref="S5.T1.3.1.1.1.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.3.1.1.1.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S5.T1.3.1.1.1.m1.1d">↓</annotation></semantics></math>)</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T1.6.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S5.T1.4.2.1">Mamba-<math alttext="2" class="ltx_Math" display="inline" id="S5.T1.4.2.1.m1.1"><semantics id="S5.T1.4.2.1.m1.1a"><mn id="S5.T1.4.2.1.m1.1.1" xref="S5.T1.4.2.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S5.T1.4.2.1.m1.1b"><cn id="S5.T1.4.2.1.m1.1.1.cmml" type="integer" xref="S5.T1.4.2.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.4.2.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S5.T1.4.2.1.m1.1d">2</annotation></semantics></math> (w/o conv)</th> <td class="ltx_td ltx_align_left ltx_border_t" id="S5.T1.5.3.2"><math alttext="14.53" class="ltx_Math" display="inline" id="S5.T1.5.3.2.m1.1"><semantics id="S5.T1.5.3.2.m1.1a"><mn id="S5.T1.5.3.2.m1.1.1" xref="S5.T1.5.3.2.m1.1.1.cmml">14.53</mn><annotation-xml encoding="MathML-Content" id="S5.T1.5.3.2.m1.1b"><cn id="S5.T1.5.3.2.m1.1.1.cmml" type="float" xref="S5.T1.5.3.2.m1.1.1">14.53</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.5.3.2.m1.1c">14.53</annotation><annotation encoding="application/x-llamapun" id="S5.T1.5.3.2.m1.1d">14.53</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S5.T1.6.4.3"><math alttext="30.68" class="ltx_Math" display="inline" id="S5.T1.6.4.3.m1.1"><semantics id="S5.T1.6.4.3.m1.1a"><mn id="S5.T1.6.4.3.m1.1.1" xref="S5.T1.6.4.3.m1.1.1.cmml">30.68</mn><annotation-xml encoding="MathML-Content" id="S5.T1.6.4.3.m1.1b"><cn id="S5.T1.6.4.3.m1.1.1.cmml" type="float" xref="S5.T1.6.4.3.m1.1.1">30.68</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.6.4.3.m1.1c">30.68</annotation><annotation encoding="application/x-llamapun" id="S5.T1.6.4.3.m1.1d">30.68</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.9.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T1.7.5.1">Mamba-<math alttext="2" class="ltx_Math" display="inline" id="S5.T1.7.5.1.m1.1"><semantics id="S5.T1.7.5.1.m1.1a"><mn id="S5.T1.7.5.1.m1.1.1" xref="S5.T1.7.5.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S5.T1.7.5.1.m1.1b"><cn id="S5.T1.7.5.1.m1.1.1.cmml" type="integer" xref="S5.T1.7.5.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.7.5.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S5.T1.7.5.1.m1.1d">2</annotation></semantics></math> (w/conv)</th> <td class="ltx_td ltx_align_left" id="S5.T1.8.6.2"><math alttext="14.54" class="ltx_Math" display="inline" id="S5.T1.8.6.2.m1.1"><semantics id="S5.T1.8.6.2.m1.1a"><mn id="S5.T1.8.6.2.m1.1.1" xref="S5.T1.8.6.2.m1.1.1.cmml">14.54</mn><annotation-xml encoding="MathML-Content" id="S5.T1.8.6.2.m1.1b"><cn id="S5.T1.8.6.2.m1.1.1.cmml" type="float" xref="S5.T1.8.6.2.m1.1.1">14.54</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.8.6.2.m1.1c">14.54</annotation><annotation encoding="application/x-llamapun" id="S5.T1.8.6.2.m1.1d">14.54</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S5.T1.9.7.3"><math alttext="\bf{27.55}" class="ltx_Math" display="inline" id="S5.T1.9.7.3.m1.1"><semantics id="S5.T1.9.7.3.m1.1a"><mn class="ltx_mathvariant_bold" id="S5.T1.9.7.3.m1.1.1" mathvariant="bold" xref="S5.T1.9.7.3.m1.1.1.cmml">27.55</mn><annotation-xml encoding="MathML-Content" id="S5.T1.9.7.3.m1.1b"><cn id="S5.T1.9.7.3.m1.1.1.cmml" type="float" xref="S5.T1.9.7.3.m1.1.1">27.55</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.9.7.3.m1.1c">\bf{27.55}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.9.7.3.m1.1d">bold_27.55</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.11.9"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S5.T1.11.9.3">Transformer (w/o conv)</th> <td class="ltx_td ltx_align_left ltx_border_t" id="S5.T1.10.8.1"><math alttext="14.46" class="ltx_Math" display="inline" id="S5.T1.10.8.1.m1.1"><semantics id="S5.T1.10.8.1.m1.1a"><mn id="S5.T1.10.8.1.m1.1.1" xref="S5.T1.10.8.1.m1.1.1.cmml">14.46</mn><annotation-xml encoding="MathML-Content" id="S5.T1.10.8.1.m1.1b"><cn id="S5.T1.10.8.1.m1.1.1.cmml" type="float" xref="S5.T1.10.8.1.m1.1.1">14.46</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.10.8.1.m1.1c">14.46</annotation><annotation encoding="application/x-llamapun" id="S5.T1.10.8.1.m1.1d">14.46</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_t" id="S5.T1.11.9.2"><math alttext="29.28" class="ltx_Math" display="inline" id="S5.T1.11.9.2.m1.1"><semantics id="S5.T1.11.9.2.m1.1a"><mn id="S5.T1.11.9.2.m1.1.1" xref="S5.T1.11.9.2.m1.1.1.cmml">29.28</mn><annotation-xml encoding="MathML-Content" id="S5.T1.11.9.2.m1.1b"><cn id="S5.T1.11.9.2.m1.1.1.cmml" type="float" xref="S5.T1.11.9.2.m1.1.1">29.28</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.11.9.2.m1.1c">29.28</annotation><annotation encoding="application/x-llamapun" id="S5.T1.11.9.2.m1.1d">29.28</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.13.11"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb" id="S5.T1.13.11.3">Transformer (w/ conv)</th> <td class="ltx_td ltx_align_left ltx_border_bb" id="S5.T1.12.10.1"><math alttext="14.46" class="ltx_Math" display="inline" id="S5.T1.12.10.1.m1.1"><semantics id="S5.T1.12.10.1.m1.1a"><mn id="S5.T1.12.10.1.m1.1.1" xref="S5.T1.12.10.1.m1.1.1.cmml">14.46</mn><annotation-xml encoding="MathML-Content" id="S5.T1.12.10.1.m1.1b"><cn id="S5.T1.12.10.1.m1.1.1.cmml" type="float" xref="S5.T1.12.10.1.m1.1.1">14.46</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.12.10.1.m1.1c">14.46</annotation><annotation encoding="application/x-llamapun" id="S5.T1.12.10.1.m1.1d">14.46</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_bb" id="S5.T1.13.11.2"><math alttext="\bf{28.67}" class="ltx_Math" display="inline" id="S5.T1.13.11.2.m1.1"><semantics id="S5.T1.13.11.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S5.T1.13.11.2.m1.1.1" mathvariant="bold" xref="S5.T1.13.11.2.m1.1.1.cmml">28.67</mn><annotation-xml encoding="MathML-Content" id="S5.T1.13.11.2.m1.1b"><cn id="S5.T1.13.11.2.m1.1.1.cmml" type="float" xref="S5.T1.13.11.2.m1.1.1">28.67</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.13.11.2.m1.1c">\bf{28.67}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.13.11.2.m1.1d">bold_28.67</annotation></semantics></math></td> </tr> </tbody> </table> </figure> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusion</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">Structured state space sequence models (SSMs) and Selective SSMs such as Mamba have shown remarkable inference speed-ups over transformers while achieving comparable or superior performance on complex language modeling tasks. In this paper, we studied in-context learning (ICL) capabilities of Mamba on Markov chains and show that, unlike transformers, even a single-layer Mamba efficiently learns the in-context Laplacian smoothing estimator. To explain this, we theoretically characterized the representation capacity of Mamba, which revealed the fundamental role of convolution in enabling it. We provided empirical results that align with our theoretical insights. Extending our results to deeper Mamba models and understanding the role of depth are some interesting future directions.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">The work was supported in part by the Swiss National Science Foundation under Grant 200364.</p> </div> <div class="ltx_para" id="Sx1.p2"> <span class="ltx_ERROR undefined" id="Sx1.p2.1">\printbibliography</span> </div> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Preliminaries on Laplacian smoothing</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.6">Laplacian smoothing is a mature and well understood topic. An account can be found, e.g., in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">merhav1998</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">CesaBL:2006</span>]</cite>, with some recent updates in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">BondaschiG:24isit</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">bondaschi1</span>]</cite>. For the sake of completeness, we provide a brief outline of how it applies to our context. For <math alttext="k" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><mi id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><ci id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">italic_k</annotation></semantics></math>-th order Markov data, at every time instant <math alttext="t" class="ltx_Math" display="inline" id="A1.p1.2.m2.1"><semantics id="A1.p1.2.m2.1a"><mi id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><ci id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">italic_t</annotation></semantics></math>, the Laplacian add-<math alttext="\beta" class="ltx_Math" display="inline" id="A1.p1.3.m3.1"><semantics id="A1.p1.3.m3.1a"><mi id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A1.p1.3.m3.1b"><ci id="A1.p1.3.m3.1.1.cmml" xref="A1.p1.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m3.1d">italic_β</annotation></semantics></math> estimator applied to the subsequence of tokens with the same context <math alttext="i_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A1.p1.4.m4.1"><semantics id="A1.p1.4.m4.1a"><mrow id="A1.p1.4.m4.1.1" xref="A1.p1.4.m4.1.1.cmml"><msubsup id="A1.p1.4.m4.1.1.2" xref="A1.p1.4.m4.1.1.2.cmml"><mi id="A1.p1.4.m4.1.1.2.2.2" xref="A1.p1.4.m4.1.1.2.2.2.cmml">i</mi><mn id="A1.p1.4.m4.1.1.2.2.3" xref="A1.p1.4.m4.1.1.2.2.3.cmml">1</mn><mi id="A1.p1.4.m4.1.1.2.3" xref="A1.p1.4.m4.1.1.2.3.cmml">k</mi></msubsup><mo id="A1.p1.4.m4.1.1.1" xref="A1.p1.4.m4.1.1.1.cmml">∈</mo><msup id="A1.p1.4.m4.1.1.3" xref="A1.p1.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p1.4.m4.1.1.3.2" xref="A1.p1.4.m4.1.1.3.2.cmml">𝒳</mi><mi id="A1.p1.4.m4.1.1.3.3" xref="A1.p1.4.m4.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.4.m4.1b"><apply id="A1.p1.4.m4.1.1.cmml" xref="A1.p1.4.m4.1.1"><in id="A1.p1.4.m4.1.1.1.cmml" xref="A1.p1.4.m4.1.1.1"></in><apply id="A1.p1.4.m4.1.1.2.cmml" xref="A1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="A1.p1.4.m4.1.1.2.1.cmml" xref="A1.p1.4.m4.1.1.2">superscript</csymbol><apply id="A1.p1.4.m4.1.1.2.2.cmml" xref="A1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="A1.p1.4.m4.1.1.2.2.1.cmml" xref="A1.p1.4.m4.1.1.2">subscript</csymbol><ci id="A1.p1.4.m4.1.1.2.2.2.cmml" xref="A1.p1.4.m4.1.1.2.2.2">𝑖</ci><cn id="A1.p1.4.m4.1.1.2.2.3.cmml" type="integer" xref="A1.p1.4.m4.1.1.2.2.3">1</cn></apply><ci id="A1.p1.4.m4.1.1.2.3.cmml" xref="A1.p1.4.m4.1.1.2.3">𝑘</ci></apply><apply id="A1.p1.4.m4.1.1.3.cmml" xref="A1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="A1.p1.4.m4.1.1.3.1.cmml" xref="A1.p1.4.m4.1.1.3">superscript</csymbol><ci id="A1.p1.4.m4.1.1.3.2.cmml" xref="A1.p1.4.m4.1.1.3.2">𝒳</ci><ci id="A1.p1.4.m4.1.1.3.3.cmml" xref="A1.p1.4.m4.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.4.m4.1c">i_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.4.m4.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> as the current one is the predictor that minimizes the Bayesian cross-entropy loss in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.E2" title="Equation 2 ‣ 2.3 Learning task: next-token prediction ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2</span></a>), when the Markov kernel is sampled according to the product Dirichlet distribution <math alttext="\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="A1.p1.5.m5.1"><semantics id="A1.p1.5.m5.1a"><mrow id="A1.p1.5.m5.1.1" xref="A1.p1.5.m5.1.1.cmml"><mi id="A1.p1.5.m5.1.1.3" xref="A1.p1.5.m5.1.1.3.cmml">Dir</mi><mo id="A1.p1.5.m5.1.1.2" xref="A1.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="A1.p1.5.m5.1.1.1.1" xref="A1.p1.5.m5.1.1.1.1.1.cmml"><mo id="A1.p1.5.m5.1.1.1.1.2" stretchy="false" xref="A1.p1.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="A1.p1.5.m5.1.1.1.1.1" xref="A1.p1.5.m5.1.1.1.1.1.cmml"><mi id="A1.p1.5.m5.1.1.1.1.1.2" xref="A1.p1.5.m5.1.1.1.1.1.2.cmml">β</mi><mo id="A1.p1.5.m5.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="A1.p1.5.m5.1.1.1.1.1.1.cmml">⋅</mo><mn id="A1.p1.5.m5.1.1.1.1.1.3" xref="A1.p1.5.m5.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="A1.p1.5.m5.1.1.1.1.3" stretchy="false" xref="A1.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.5.m5.1b"><apply id="A1.p1.5.m5.1.1.cmml" xref="A1.p1.5.m5.1.1"><times id="A1.p1.5.m5.1.1.2.cmml" xref="A1.p1.5.m5.1.1.2"></times><ci id="A1.p1.5.m5.1.1.3.cmml" xref="A1.p1.5.m5.1.1.3">Dir</ci><apply id="A1.p1.5.m5.1.1.1.1.1.cmml" xref="A1.p1.5.m5.1.1.1.1"><ci id="A1.p1.5.m5.1.1.1.1.1.1.cmml" xref="A1.p1.5.m5.1.1.1.1.1.1">⋅</ci><ci id="A1.p1.5.m5.1.1.1.1.1.2.cmml" xref="A1.p1.5.m5.1.1.1.1.1.2">𝛽</ci><cn id="A1.p1.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="A1.p1.5.m5.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.5.m5.1c">\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="A1.p1.5.m5.1d">roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math>. We first give an intuition of why this is the case, and we provide a full proof at the end of the section. We consider the binary case <math alttext="\mathcal{X}=\{0,1\}" class="ltx_Math" display="inline" id="A1.p1.6.m6.2"><semantics id="A1.p1.6.m6.2a"><mrow id="A1.p1.6.m6.2.3" xref="A1.p1.6.m6.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p1.6.m6.2.3.2" xref="A1.p1.6.m6.2.3.2.cmml">𝒳</mi><mo id="A1.p1.6.m6.2.3.1" xref="A1.p1.6.m6.2.3.1.cmml">=</mo><mrow id="A1.p1.6.m6.2.3.3.2" xref="A1.p1.6.m6.2.3.3.1.cmml"><mo id="A1.p1.6.m6.2.3.3.2.1" stretchy="false" xref="A1.p1.6.m6.2.3.3.1.cmml">{</mo><mn id="A1.p1.6.m6.1.1" xref="A1.p1.6.m6.1.1.cmml">0</mn><mo id="A1.p1.6.m6.2.3.3.2.2" xref="A1.p1.6.m6.2.3.3.1.cmml">,</mo><mn id="A1.p1.6.m6.2.2" xref="A1.p1.6.m6.2.2.cmml">1</mn><mo id="A1.p1.6.m6.2.3.3.2.3" stretchy="false" xref="A1.p1.6.m6.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.6.m6.2b"><apply id="A1.p1.6.m6.2.3.cmml" xref="A1.p1.6.m6.2.3"><eq id="A1.p1.6.m6.2.3.1.cmml" xref="A1.p1.6.m6.2.3.1"></eq><ci id="A1.p1.6.m6.2.3.2.cmml" xref="A1.p1.6.m6.2.3.2">𝒳</ci><set id="A1.p1.6.m6.2.3.3.1.cmml" xref="A1.p1.6.m6.2.3.3.2"><cn id="A1.p1.6.m6.1.1.cmml" type="integer" xref="A1.p1.6.m6.1.1">0</cn><cn id="A1.p1.6.m6.2.2.cmml" type="integer" xref="A1.p1.6.m6.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.6.m6.2c">\mathcal{X}=\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.6.m6.2d">caligraphic_X = { 0 , 1 }</annotation></semantics></math>, but the results can be extended to arbitrary finite alphabets.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.17">Consider a given sequence <math alttext="(x_{t})_{t=1}^{T}" class="ltx_Math" display="inline" id="A1.p2.1.m1.1"><semantics id="A1.p2.1.m1.1a"><msubsup id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml"><mrow id="A1.p2.1.m1.1.1.1.1.1" xref="A1.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="A1.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="A1.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.1.m1.1.1.1.1.1.1" xref="A1.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="A1.p2.1.m1.1.1.1.1.1.1.2" xref="A1.p2.1.m1.1.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.1.m1.1.1.1.1.1.1.3" xref="A1.p2.1.m1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="A1.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.p2.1.m1.1.1.1.3" xref="A1.p2.1.m1.1.1.1.3.cmml"><mi id="A1.p2.1.m1.1.1.1.3.2" xref="A1.p2.1.m1.1.1.1.3.2.cmml">t</mi><mo id="A1.p2.1.m1.1.1.1.3.1" xref="A1.p2.1.m1.1.1.1.3.1.cmml">=</mo><mn id="A1.p2.1.m1.1.1.1.3.3" xref="A1.p2.1.m1.1.1.1.3.3.cmml">1</mn></mrow><mi id="A1.p2.1.m1.1.1.3" xref="A1.p2.1.m1.1.1.3.cmml">T</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.1b"><apply id="A1.p2.1.m1.1.1.cmml" xref="A1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A1.p2.1.m1.1.1.2.cmml" xref="A1.p2.1.m1.1.1">superscript</csymbol><apply id="A1.p2.1.m1.1.1.1.cmml" xref="A1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A1.p2.1.m1.1.1.1.2.cmml" xref="A1.p2.1.m1.1.1">subscript</csymbol><apply id="A1.p2.1.m1.1.1.1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1.1.1">subscript</csymbol><ci id="A1.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="A1.p2.1.m1.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="A1.p2.1.m1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="A1.p2.1.m1.1.1.1.3.cmml" xref="A1.p2.1.m1.1.1.1.3"><eq id="A1.p2.1.m1.1.1.1.3.1.cmml" xref="A1.p2.1.m1.1.1.1.3.1"></eq><ci id="A1.p2.1.m1.1.1.1.3.2.cmml" xref="A1.p2.1.m1.1.1.1.3.2">𝑡</ci><cn id="A1.p2.1.m1.1.1.1.3.3.cmml" type="integer" xref="A1.p2.1.m1.1.1.1.3.3">1</cn></apply></apply><ci id="A1.p2.1.m1.1.1.3.cmml" xref="A1.p2.1.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.1c">(x_{t})_{t=1}^{T}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.1.m1.1d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. For every length-<math alttext="k" class="ltx_Math" display="inline" id="A1.p2.2.m2.1"><semantics id="A1.p2.2.m2.1a"><mi id="A1.p2.2.m2.1.1" xref="A1.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.p2.2.m2.1b"><ci id="A1.p2.2.m2.1.1.cmml" xref="A1.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.p2.2.m2.1d">italic_k</annotation></semantics></math> context <math alttext="i_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A1.p2.3.m3.1"><semantics id="A1.p2.3.m3.1a"><mrow id="A1.p2.3.m3.1.1" xref="A1.p2.3.m3.1.1.cmml"><msubsup id="A1.p2.3.m3.1.1.2" xref="A1.p2.3.m3.1.1.2.cmml"><mi id="A1.p2.3.m3.1.1.2.2.2" xref="A1.p2.3.m3.1.1.2.2.2.cmml">i</mi><mn id="A1.p2.3.m3.1.1.2.2.3" xref="A1.p2.3.m3.1.1.2.2.3.cmml">1</mn><mi id="A1.p2.3.m3.1.1.2.3" xref="A1.p2.3.m3.1.1.2.3.cmml">k</mi></msubsup><mo id="A1.p2.3.m3.1.1.1" xref="A1.p2.3.m3.1.1.1.cmml">∈</mo><msup id="A1.p2.3.m3.1.1.3" xref="A1.p2.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p2.3.m3.1.1.3.2" xref="A1.p2.3.m3.1.1.3.2.cmml">𝒳</mi><mi id="A1.p2.3.m3.1.1.3.3" xref="A1.p2.3.m3.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.3.m3.1b"><apply id="A1.p2.3.m3.1.1.cmml" xref="A1.p2.3.m3.1.1"><in id="A1.p2.3.m3.1.1.1.cmml" xref="A1.p2.3.m3.1.1.1"></in><apply id="A1.p2.3.m3.1.1.2.cmml" xref="A1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.p2.3.m3.1.1.2.1.cmml" xref="A1.p2.3.m3.1.1.2">superscript</csymbol><apply id="A1.p2.3.m3.1.1.2.2.cmml" xref="A1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.p2.3.m3.1.1.2.2.1.cmml" xref="A1.p2.3.m3.1.1.2">subscript</csymbol><ci id="A1.p2.3.m3.1.1.2.2.2.cmml" xref="A1.p2.3.m3.1.1.2.2.2">𝑖</ci><cn id="A1.p2.3.m3.1.1.2.2.3.cmml" type="integer" xref="A1.p2.3.m3.1.1.2.2.3">1</cn></apply><ci id="A1.p2.3.m3.1.1.2.3.cmml" xref="A1.p2.3.m3.1.1.2.3">𝑘</ci></apply><apply id="A1.p2.3.m3.1.1.3.cmml" xref="A1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.p2.3.m3.1.1.3.1.cmml" xref="A1.p2.3.m3.1.1.3">superscript</csymbol><ci id="A1.p2.3.m3.1.1.3.2.cmml" xref="A1.p2.3.m3.1.1.3.2">𝒳</ci><ci id="A1.p2.3.m3.1.1.3.3.cmml" xref="A1.p2.3.m3.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.3.m3.1c">i_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.3.m3.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="(x_{t})|_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.p2.4.m4.2"><semantics id="A1.p2.4.m4.2a"><msub id="A1.p2.4.m4.2.2.1" xref="A1.p2.4.m4.2.2.2.cmml"><mrow id="A1.p2.4.m4.2.2.1.1" xref="A1.p2.4.m4.2.2.2.cmml"><mrow id="A1.p2.4.m4.2.2.1.1.1.1" xref="A1.p2.4.m4.2.2.1.1.1.1.1.cmml"><mo id="A1.p2.4.m4.2.2.1.1.1.1.2" stretchy="false" xref="A1.p2.4.m4.2.2.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.4.m4.2.2.1.1.1.1.1" xref="A1.p2.4.m4.2.2.1.1.1.1.1.cmml"><mi id="A1.p2.4.m4.2.2.1.1.1.1.1.2" xref="A1.p2.4.m4.2.2.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.4.m4.2.2.1.1.1.1.1.3" xref="A1.p2.4.m4.2.2.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.p2.4.m4.2.2.1.1.1.1.3" stretchy="false" xref="A1.p2.4.m4.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.p2.4.m4.2.2.1.1.2" stretchy="false" xref="A1.p2.4.m4.2.2.2.1.cmml">|</mo></mrow><msubsup id="A1.p2.4.m4.1.1.1" xref="A1.p2.4.m4.1.1.1.cmml"><mi id="A1.p2.4.m4.1.1.1.2.2" xref="A1.p2.4.m4.1.1.1.2.2.cmml">i</mi><mn id="A1.p2.4.m4.1.1.1.2.3" xref="A1.p2.4.m4.1.1.1.2.3.cmml">1</mn><mi id="A1.p2.4.m4.1.1.1.3" xref="A1.p2.4.m4.1.1.1.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.p2.4.m4.2b"><apply id="A1.p2.4.m4.2.2.2.cmml" xref="A1.p2.4.m4.2.2.1"><csymbol cd="latexml" id="A1.p2.4.m4.2.2.2.1.cmml" xref="A1.p2.4.m4.2.2.1.1.2">evaluated-at</csymbol><apply id="A1.p2.4.m4.2.2.1.1.1.1.1.cmml" xref="A1.p2.4.m4.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.4.m4.2.2.1.1.1.1.1.1.cmml" xref="A1.p2.4.m4.2.2.1.1.1.1">subscript</csymbol><ci id="A1.p2.4.m4.2.2.1.1.1.1.1.2.cmml" xref="A1.p2.4.m4.2.2.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.4.m4.2.2.1.1.1.1.1.3.cmml" xref="A1.p2.4.m4.2.2.1.1.1.1.1.3">𝑡</ci></apply><apply id="A1.p2.4.m4.1.1.1.cmml" xref="A1.p2.4.m4.1.1.1"><csymbol cd="ambiguous" id="A1.p2.4.m4.1.1.1.1.cmml" xref="A1.p2.4.m4.1.1.1">superscript</csymbol><apply id="A1.p2.4.m4.1.1.1.2.cmml" xref="A1.p2.4.m4.1.1.1"><csymbol cd="ambiguous" id="A1.p2.4.m4.1.1.1.2.1.cmml" xref="A1.p2.4.m4.1.1.1">subscript</csymbol><ci id="A1.p2.4.m4.1.1.1.2.2.cmml" xref="A1.p2.4.m4.1.1.1.2.2">𝑖</ci><cn id="A1.p2.4.m4.1.1.1.2.3.cmml" type="integer" xref="A1.p2.4.m4.1.1.1.2.3">1</cn></apply><ci id="A1.p2.4.m4.1.1.1.3.cmml" xref="A1.p2.4.m4.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.4.m4.2c">(x_{t})|_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.4.m4.2d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be the subsequence of tokens preceded by <math alttext="i_{1}^{k}" class="ltx_Math" display="inline" id="A1.p2.5.m5.1"><semantics id="A1.p2.5.m5.1a"><msubsup id="A1.p2.5.m5.1.1" xref="A1.p2.5.m5.1.1.cmml"><mi id="A1.p2.5.m5.1.1.2.2" xref="A1.p2.5.m5.1.1.2.2.cmml">i</mi><mn id="A1.p2.5.m5.1.1.2.3" xref="A1.p2.5.m5.1.1.2.3.cmml">1</mn><mi id="A1.p2.5.m5.1.1.3" xref="A1.p2.5.m5.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.p2.5.m5.1b"><apply id="A1.p2.5.m5.1.1.cmml" xref="A1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A1.p2.5.m5.1.1.1.cmml" xref="A1.p2.5.m5.1.1">superscript</csymbol><apply id="A1.p2.5.m5.1.1.2.cmml" xref="A1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A1.p2.5.m5.1.1.2.1.cmml" xref="A1.p2.5.m5.1.1">subscript</csymbol><ci id="A1.p2.5.m5.1.1.2.2.cmml" xref="A1.p2.5.m5.1.1.2.2">𝑖</ci><cn id="A1.p2.5.m5.1.1.2.3.cmml" type="integer" xref="A1.p2.5.m5.1.1.2.3">1</cn></apply><ci id="A1.p2.5.m5.1.1.3.cmml" xref="A1.p2.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.5.m5.1c">i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.5.m5.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Note that, since each sequence <math alttext="(x_{t})" class="ltx_Math" display="inline" id="A1.p2.6.m6.1"><semantics id="A1.p2.6.m6.1a"><mrow id="A1.p2.6.m6.1.1.1" xref="A1.p2.6.m6.1.1.1.1.cmml"><mo id="A1.p2.6.m6.1.1.1.2" stretchy="false" xref="A1.p2.6.m6.1.1.1.1.cmml">(</mo><msub id="A1.p2.6.m6.1.1.1.1" xref="A1.p2.6.m6.1.1.1.1.cmml"><mi id="A1.p2.6.m6.1.1.1.1.2" xref="A1.p2.6.m6.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.6.m6.1.1.1.1.3" xref="A1.p2.6.m6.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.p2.6.m6.1.1.1.3" stretchy="false" xref="A1.p2.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.6.m6.1b"><apply id="A1.p2.6.m6.1.1.1.1.cmml" xref="A1.p2.6.m6.1.1.1"><csymbol cd="ambiguous" id="A1.p2.6.m6.1.1.1.1.1.cmml" xref="A1.p2.6.m6.1.1.1">subscript</csymbol><ci id="A1.p2.6.m6.1.1.1.1.2.cmml" xref="A1.p2.6.m6.1.1.1.1.2">𝑥</ci><ci id="A1.p2.6.m6.1.1.1.1.3.cmml" xref="A1.p2.6.m6.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.6.m6.1c">(x_{t})</annotation><annotation encoding="application/x-llamapun" id="A1.p2.6.m6.1d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> is generated by a <math alttext="k" class="ltx_Math" display="inline" id="A1.p2.7.m7.1"><semantics id="A1.p2.7.m7.1a"><mi id="A1.p2.7.m7.1.1" xref="A1.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.p2.7.m7.1b"><ci id="A1.p2.7.m7.1.1.cmml" xref="A1.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.p2.7.m7.1d">italic_k</annotation></semantics></math>-th order Markov chain, all the tokens in the sequence with the same length-<math alttext="k" class="ltx_Math" display="inline" id="A1.p2.8.m8.1"><semantics id="A1.p2.8.m8.1a"><mi id="A1.p2.8.m8.1.1" xref="A1.p2.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.p2.8.m8.1b"><ci id="A1.p2.8.m8.1.1.cmml" xref="A1.p2.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.p2.8.m8.1d">italic_k</annotation></semantics></math> prefix share the same conditional probability distribution. Furthermore, since each of the conditional distributions of the chain is randomly chosen independently from the others, the subsequence <math alttext="(x_{t})|_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.p2.9.m9.2"><semantics id="A1.p2.9.m9.2a"><msub id="A1.p2.9.m9.2.2.1" xref="A1.p2.9.m9.2.2.2.cmml"><mrow id="A1.p2.9.m9.2.2.1.1" xref="A1.p2.9.m9.2.2.2.cmml"><mrow id="A1.p2.9.m9.2.2.1.1.1.1" xref="A1.p2.9.m9.2.2.1.1.1.1.1.cmml"><mo id="A1.p2.9.m9.2.2.1.1.1.1.2" stretchy="false" xref="A1.p2.9.m9.2.2.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.9.m9.2.2.1.1.1.1.1" xref="A1.p2.9.m9.2.2.1.1.1.1.1.cmml"><mi id="A1.p2.9.m9.2.2.1.1.1.1.1.2" xref="A1.p2.9.m9.2.2.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.9.m9.2.2.1.1.1.1.1.3" xref="A1.p2.9.m9.2.2.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.p2.9.m9.2.2.1.1.1.1.3" stretchy="false" xref="A1.p2.9.m9.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.p2.9.m9.2.2.1.1.2" stretchy="false" xref="A1.p2.9.m9.2.2.2.1.cmml">|</mo></mrow><msubsup id="A1.p2.9.m9.1.1.1" xref="A1.p2.9.m9.1.1.1.cmml"><mi id="A1.p2.9.m9.1.1.1.2.2" xref="A1.p2.9.m9.1.1.1.2.2.cmml">i</mi><mn id="A1.p2.9.m9.1.1.1.2.3" xref="A1.p2.9.m9.1.1.1.2.3.cmml">1</mn><mi id="A1.p2.9.m9.1.1.1.3" xref="A1.p2.9.m9.1.1.1.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.p2.9.m9.2b"><apply id="A1.p2.9.m9.2.2.2.cmml" xref="A1.p2.9.m9.2.2.1"><csymbol cd="latexml" id="A1.p2.9.m9.2.2.2.1.cmml" xref="A1.p2.9.m9.2.2.1.1.2">evaluated-at</csymbol><apply id="A1.p2.9.m9.2.2.1.1.1.1.1.cmml" xref="A1.p2.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.9.m9.2.2.1.1.1.1.1.1.cmml" xref="A1.p2.9.m9.2.2.1.1.1.1">subscript</csymbol><ci id="A1.p2.9.m9.2.2.1.1.1.1.1.2.cmml" xref="A1.p2.9.m9.2.2.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.9.m9.2.2.1.1.1.1.1.3.cmml" xref="A1.p2.9.m9.2.2.1.1.1.1.1.3">𝑡</ci></apply><apply id="A1.p2.9.m9.1.1.1.cmml" xref="A1.p2.9.m9.1.1.1"><csymbol cd="ambiguous" id="A1.p2.9.m9.1.1.1.1.cmml" xref="A1.p2.9.m9.1.1.1">superscript</csymbol><apply id="A1.p2.9.m9.1.1.1.2.cmml" xref="A1.p2.9.m9.1.1.1"><csymbol cd="ambiguous" id="A1.p2.9.m9.1.1.1.2.1.cmml" xref="A1.p2.9.m9.1.1.1">subscript</csymbol><ci id="A1.p2.9.m9.1.1.1.2.2.cmml" xref="A1.p2.9.m9.1.1.1.2.2">𝑖</ci><cn id="A1.p2.9.m9.1.1.1.2.3.cmml" type="integer" xref="A1.p2.9.m9.1.1.1.2.3">1</cn></apply><ci id="A1.p2.9.m9.1.1.1.3.cmml" xref="A1.p2.9.m9.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.9.m9.2c">(x_{t})|_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.9.m9.2d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a sufficient statistic to estimate the probability distribution of all the tokens with the same prefix <math alttext="i_{1}^{k}" class="ltx_Math" display="inline" id="A1.p2.10.m10.1"><semantics id="A1.p2.10.m10.1a"><msubsup id="A1.p2.10.m10.1.1" xref="A1.p2.10.m10.1.1.cmml"><mi id="A1.p2.10.m10.1.1.2.2" xref="A1.p2.10.m10.1.1.2.2.cmml">i</mi><mn id="A1.p2.10.m10.1.1.2.3" xref="A1.p2.10.m10.1.1.2.3.cmml">1</mn><mi id="A1.p2.10.m10.1.1.3" xref="A1.p2.10.m10.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.p2.10.m10.1b"><apply id="A1.p2.10.m10.1.1.cmml" xref="A1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="A1.p2.10.m10.1.1.1.cmml" xref="A1.p2.10.m10.1.1">superscript</csymbol><apply id="A1.p2.10.m10.1.1.2.cmml" xref="A1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="A1.p2.10.m10.1.1.2.1.cmml" xref="A1.p2.10.m10.1.1">subscript</csymbol><ci id="A1.p2.10.m10.1.1.2.2.cmml" xref="A1.p2.10.m10.1.1.2.2">𝑖</ci><cn id="A1.p2.10.m10.1.1.2.3.cmml" type="integer" xref="A1.p2.10.m10.1.1.2.3">1</cn></apply><ci id="A1.p2.10.m10.1.1.3.cmml" xref="A1.p2.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.10.m10.1c">i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.10.m10.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, the optimal prediction for a sequence <math alttext="(x_{t})_{t=1}^{T}" class="ltx_Math" display="inline" id="A1.p2.11.m11.1"><semantics id="A1.p2.11.m11.1a"><msubsup id="A1.p2.11.m11.1.1" xref="A1.p2.11.m11.1.1.cmml"><mrow id="A1.p2.11.m11.1.1.1.1.1" xref="A1.p2.11.m11.1.1.1.1.1.1.cmml"><mo id="A1.p2.11.m11.1.1.1.1.1.2" stretchy="false" xref="A1.p2.11.m11.1.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.11.m11.1.1.1.1.1.1" xref="A1.p2.11.m11.1.1.1.1.1.1.cmml"><mi id="A1.p2.11.m11.1.1.1.1.1.1.2" xref="A1.p2.11.m11.1.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.11.m11.1.1.1.1.1.1.3" xref="A1.p2.11.m11.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.p2.11.m11.1.1.1.1.1.3" stretchy="false" xref="A1.p2.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.p2.11.m11.1.1.1.3" xref="A1.p2.11.m11.1.1.1.3.cmml"><mi id="A1.p2.11.m11.1.1.1.3.2" xref="A1.p2.11.m11.1.1.1.3.2.cmml">t</mi><mo id="A1.p2.11.m11.1.1.1.3.1" xref="A1.p2.11.m11.1.1.1.3.1.cmml">=</mo><mn id="A1.p2.11.m11.1.1.1.3.3" xref="A1.p2.11.m11.1.1.1.3.3.cmml">1</mn></mrow><mi id="A1.p2.11.m11.1.1.3" xref="A1.p2.11.m11.1.1.3.cmml">T</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.p2.11.m11.1b"><apply id="A1.p2.11.m11.1.1.cmml" xref="A1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="A1.p2.11.m11.1.1.2.cmml" xref="A1.p2.11.m11.1.1">superscript</csymbol><apply id="A1.p2.11.m11.1.1.1.cmml" xref="A1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="A1.p2.11.m11.1.1.1.2.cmml" xref="A1.p2.11.m11.1.1">subscript</csymbol><apply id="A1.p2.11.m11.1.1.1.1.1.1.cmml" xref="A1.p2.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.11.m11.1.1.1.1.1.1.1.cmml" xref="A1.p2.11.m11.1.1.1.1.1">subscript</csymbol><ci id="A1.p2.11.m11.1.1.1.1.1.1.2.cmml" xref="A1.p2.11.m11.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.11.m11.1.1.1.1.1.1.3.cmml" xref="A1.p2.11.m11.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="A1.p2.11.m11.1.1.1.3.cmml" xref="A1.p2.11.m11.1.1.1.3"><eq id="A1.p2.11.m11.1.1.1.3.1.cmml" xref="A1.p2.11.m11.1.1.1.3.1"></eq><ci id="A1.p2.11.m11.1.1.1.3.2.cmml" xref="A1.p2.11.m11.1.1.1.3.2">𝑡</ci><cn id="A1.p2.11.m11.1.1.1.3.3.cmml" type="integer" xref="A1.p2.11.m11.1.1.1.3.3">1</cn></apply></apply><ci id="A1.p2.11.m11.1.1.3.cmml" xref="A1.p2.11.m11.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.11.m11.1c">(x_{t})_{t=1}^{T}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.11.m11.1d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> is given by employing the optimal predictor for each i.i.d. subsequence <math alttext="(x_{t})|_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.p2.12.m12.2"><semantics id="A1.p2.12.m12.2a"><msub id="A1.p2.12.m12.2.2.1" xref="A1.p2.12.m12.2.2.2.cmml"><mrow id="A1.p2.12.m12.2.2.1.1" xref="A1.p2.12.m12.2.2.2.cmml"><mrow id="A1.p2.12.m12.2.2.1.1.1.1" xref="A1.p2.12.m12.2.2.1.1.1.1.1.cmml"><mo id="A1.p2.12.m12.2.2.1.1.1.1.2" stretchy="false" xref="A1.p2.12.m12.2.2.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.12.m12.2.2.1.1.1.1.1" xref="A1.p2.12.m12.2.2.1.1.1.1.1.cmml"><mi id="A1.p2.12.m12.2.2.1.1.1.1.1.2" xref="A1.p2.12.m12.2.2.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.12.m12.2.2.1.1.1.1.1.3" xref="A1.p2.12.m12.2.2.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.p2.12.m12.2.2.1.1.1.1.3" stretchy="false" xref="A1.p2.12.m12.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.p2.12.m12.2.2.1.1.2" stretchy="false" xref="A1.p2.12.m12.2.2.2.1.cmml">|</mo></mrow><msubsup id="A1.p2.12.m12.1.1.1" xref="A1.p2.12.m12.1.1.1.cmml"><mi id="A1.p2.12.m12.1.1.1.2.2" xref="A1.p2.12.m12.1.1.1.2.2.cmml">i</mi><mn id="A1.p2.12.m12.1.1.1.2.3" xref="A1.p2.12.m12.1.1.1.2.3.cmml">1</mn><mi id="A1.p2.12.m12.1.1.1.3" xref="A1.p2.12.m12.1.1.1.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.p2.12.m12.2b"><apply id="A1.p2.12.m12.2.2.2.cmml" xref="A1.p2.12.m12.2.2.1"><csymbol cd="latexml" id="A1.p2.12.m12.2.2.2.1.cmml" xref="A1.p2.12.m12.2.2.1.1.2">evaluated-at</csymbol><apply id="A1.p2.12.m12.2.2.1.1.1.1.1.cmml" xref="A1.p2.12.m12.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.12.m12.2.2.1.1.1.1.1.1.cmml" xref="A1.p2.12.m12.2.2.1.1.1.1">subscript</csymbol><ci id="A1.p2.12.m12.2.2.1.1.1.1.1.2.cmml" xref="A1.p2.12.m12.2.2.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.12.m12.2.2.1.1.1.1.1.3.cmml" xref="A1.p2.12.m12.2.2.1.1.1.1.1.3">𝑡</ci></apply><apply id="A1.p2.12.m12.1.1.1.cmml" xref="A1.p2.12.m12.1.1.1"><csymbol cd="ambiguous" id="A1.p2.12.m12.1.1.1.1.cmml" xref="A1.p2.12.m12.1.1.1">superscript</csymbol><apply id="A1.p2.12.m12.1.1.1.2.cmml" xref="A1.p2.12.m12.1.1.1"><csymbol cd="ambiguous" id="A1.p2.12.m12.1.1.1.2.1.cmml" xref="A1.p2.12.m12.1.1.1">subscript</csymbol><ci id="A1.p2.12.m12.1.1.1.2.2.cmml" xref="A1.p2.12.m12.1.1.1.2.2">𝑖</ci><cn id="A1.p2.12.m12.1.1.1.2.3.cmml" type="integer" xref="A1.p2.12.m12.1.1.1.2.3">1</cn></apply><ci id="A1.p2.12.m12.1.1.1.3.cmml" xref="A1.p2.12.m12.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.12.m12.2c">(x_{t})|_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.12.m12.2d">( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, for every <math alttext="i_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A1.p2.13.m13.1"><semantics id="A1.p2.13.m13.1a"><mrow id="A1.p2.13.m13.1.1" xref="A1.p2.13.m13.1.1.cmml"><msubsup id="A1.p2.13.m13.1.1.2" xref="A1.p2.13.m13.1.1.2.cmml"><mi id="A1.p2.13.m13.1.1.2.2.2" xref="A1.p2.13.m13.1.1.2.2.2.cmml">i</mi><mn id="A1.p2.13.m13.1.1.2.2.3" xref="A1.p2.13.m13.1.1.2.2.3.cmml">1</mn><mi id="A1.p2.13.m13.1.1.2.3" xref="A1.p2.13.m13.1.1.2.3.cmml">k</mi></msubsup><mo id="A1.p2.13.m13.1.1.1" xref="A1.p2.13.m13.1.1.1.cmml">∈</mo><msup id="A1.p2.13.m13.1.1.3" xref="A1.p2.13.m13.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p2.13.m13.1.1.3.2" xref="A1.p2.13.m13.1.1.3.2.cmml">𝒳</mi><mi id="A1.p2.13.m13.1.1.3.3" xref="A1.p2.13.m13.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.13.m13.1b"><apply id="A1.p2.13.m13.1.1.cmml" xref="A1.p2.13.m13.1.1"><in id="A1.p2.13.m13.1.1.1.cmml" xref="A1.p2.13.m13.1.1.1"></in><apply id="A1.p2.13.m13.1.1.2.cmml" xref="A1.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="A1.p2.13.m13.1.1.2.1.cmml" xref="A1.p2.13.m13.1.1.2">superscript</csymbol><apply id="A1.p2.13.m13.1.1.2.2.cmml" xref="A1.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="A1.p2.13.m13.1.1.2.2.1.cmml" xref="A1.p2.13.m13.1.1.2">subscript</csymbol><ci id="A1.p2.13.m13.1.1.2.2.2.cmml" xref="A1.p2.13.m13.1.1.2.2.2">𝑖</ci><cn id="A1.p2.13.m13.1.1.2.2.3.cmml" type="integer" xref="A1.p2.13.m13.1.1.2.2.3">1</cn></apply><ci id="A1.p2.13.m13.1.1.2.3.cmml" xref="A1.p2.13.m13.1.1.2.3">𝑘</ci></apply><apply id="A1.p2.13.m13.1.1.3.cmml" xref="A1.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="A1.p2.13.m13.1.1.3.1.cmml" xref="A1.p2.13.m13.1.1.3">superscript</csymbol><ci id="A1.p2.13.m13.1.1.3.2.cmml" xref="A1.p2.13.m13.1.1.3.2">𝒳</ci><ci id="A1.p2.13.m13.1.1.3.3.cmml" xref="A1.p2.13.m13.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.13.m13.1c">i_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.13.m13.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Since each conditional distribution is sampled from a Dirichlet distribution with parameter <math alttext="\beta" class="ltx_Math" display="inline" id="A1.p2.14.m14.1"><semantics id="A1.p2.14.m14.1a"><mi id="A1.p2.14.m14.1.1" xref="A1.p2.14.m14.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A1.p2.14.m14.1b"><ci id="A1.p2.14.m14.1.1.cmml" xref="A1.p2.14.m14.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.14.m14.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A1.p2.14.m14.1d">italic_β</annotation></semantics></math>, it is well known that the optimal predictor for such subsequences is the add-constant estimator, with constant equal to <math alttext="\beta" class="ltx_Math" display="inline" id="A1.p2.15.m15.1"><semantics id="A1.p2.15.m15.1a"><mi id="A1.p2.15.m15.1.1" xref="A1.p2.15.m15.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A1.p2.15.m15.1b"><ci id="A1.p2.15.m15.1.1.cmml" xref="A1.p2.15.m15.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.15.m15.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A1.p2.15.m15.1d">italic_β</annotation></semantics></math>. More specifically, if <math alttext="x_{t-k}^{t-1}=i_{1}^{k}" class="ltx_Math" display="inline" id="A1.p2.16.m16.1"><semantics id="A1.p2.16.m16.1a"><mrow id="A1.p2.16.m16.1.1" xref="A1.p2.16.m16.1.1.cmml"><msubsup id="A1.p2.16.m16.1.1.2" xref="A1.p2.16.m16.1.1.2.cmml"><mi id="A1.p2.16.m16.1.1.2.2.2" xref="A1.p2.16.m16.1.1.2.2.2.cmml">x</mi><mrow id="A1.p2.16.m16.1.1.2.2.3" xref="A1.p2.16.m16.1.1.2.2.3.cmml"><mi id="A1.p2.16.m16.1.1.2.2.3.2" xref="A1.p2.16.m16.1.1.2.2.3.2.cmml">t</mi><mo id="A1.p2.16.m16.1.1.2.2.3.1" xref="A1.p2.16.m16.1.1.2.2.3.1.cmml">−</mo><mi id="A1.p2.16.m16.1.1.2.2.3.3" xref="A1.p2.16.m16.1.1.2.2.3.3.cmml">k</mi></mrow><mrow id="A1.p2.16.m16.1.1.2.3" xref="A1.p2.16.m16.1.1.2.3.cmml"><mi id="A1.p2.16.m16.1.1.2.3.2" xref="A1.p2.16.m16.1.1.2.3.2.cmml">t</mi><mo id="A1.p2.16.m16.1.1.2.3.1" xref="A1.p2.16.m16.1.1.2.3.1.cmml">−</mo><mn id="A1.p2.16.m16.1.1.2.3.3" xref="A1.p2.16.m16.1.1.2.3.3.cmml">1</mn></mrow></msubsup><mo id="A1.p2.16.m16.1.1.1" xref="A1.p2.16.m16.1.1.1.cmml">=</mo><msubsup id="A1.p2.16.m16.1.1.3" xref="A1.p2.16.m16.1.1.3.cmml"><mi id="A1.p2.16.m16.1.1.3.2.2" xref="A1.p2.16.m16.1.1.3.2.2.cmml">i</mi><mn id="A1.p2.16.m16.1.1.3.2.3" xref="A1.p2.16.m16.1.1.3.2.3.cmml">1</mn><mi id="A1.p2.16.m16.1.1.3.3" xref="A1.p2.16.m16.1.1.3.3.cmml">k</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.16.m16.1b"><apply id="A1.p2.16.m16.1.1.cmml" xref="A1.p2.16.m16.1.1"><eq id="A1.p2.16.m16.1.1.1.cmml" xref="A1.p2.16.m16.1.1.1"></eq><apply id="A1.p2.16.m16.1.1.2.cmml" xref="A1.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="A1.p2.16.m16.1.1.2.1.cmml" xref="A1.p2.16.m16.1.1.2">superscript</csymbol><apply id="A1.p2.16.m16.1.1.2.2.cmml" xref="A1.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="A1.p2.16.m16.1.1.2.2.1.cmml" xref="A1.p2.16.m16.1.1.2">subscript</csymbol><ci id="A1.p2.16.m16.1.1.2.2.2.cmml" xref="A1.p2.16.m16.1.1.2.2.2">𝑥</ci><apply id="A1.p2.16.m16.1.1.2.2.3.cmml" xref="A1.p2.16.m16.1.1.2.2.3"><minus id="A1.p2.16.m16.1.1.2.2.3.1.cmml" xref="A1.p2.16.m16.1.1.2.2.3.1"></minus><ci id="A1.p2.16.m16.1.1.2.2.3.2.cmml" xref="A1.p2.16.m16.1.1.2.2.3.2">𝑡</ci><ci id="A1.p2.16.m16.1.1.2.2.3.3.cmml" xref="A1.p2.16.m16.1.1.2.2.3.3">𝑘</ci></apply></apply><apply id="A1.p2.16.m16.1.1.2.3.cmml" xref="A1.p2.16.m16.1.1.2.3"><minus id="A1.p2.16.m16.1.1.2.3.1.cmml" xref="A1.p2.16.m16.1.1.2.3.1"></minus><ci id="A1.p2.16.m16.1.1.2.3.2.cmml" xref="A1.p2.16.m16.1.1.2.3.2">𝑡</ci><cn id="A1.p2.16.m16.1.1.2.3.3.cmml" type="integer" xref="A1.p2.16.m16.1.1.2.3.3">1</cn></apply></apply><apply id="A1.p2.16.m16.1.1.3.cmml" xref="A1.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="A1.p2.16.m16.1.1.3.1.cmml" xref="A1.p2.16.m16.1.1.3">superscript</csymbol><apply id="A1.p2.16.m16.1.1.3.2.cmml" xref="A1.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="A1.p2.16.m16.1.1.3.2.1.cmml" xref="A1.p2.16.m16.1.1.3">subscript</csymbol><ci id="A1.p2.16.m16.1.1.3.2.2.cmml" xref="A1.p2.16.m16.1.1.3.2.2">𝑖</ci><cn id="A1.p2.16.m16.1.1.3.2.3.cmml" type="integer" xref="A1.p2.16.m16.1.1.3.2.3">1</cn></apply><ci id="A1.p2.16.m16.1.1.3.3.cmml" xref="A1.p2.16.m16.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.16.m16.1c">x_{t-k}^{t-1}=i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.16.m16.1d">italic_x start_POSTSUBSCRIPT italic_t - italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT = italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, then the optimal estimation for <math alttext="x_{t}" class="ltx_Math" display="inline" id="A1.p2.17.m17.1"><semantics id="A1.p2.17.m17.1a"><msub id="A1.p2.17.m17.1.1" xref="A1.p2.17.m17.1.1.cmml"><mi id="A1.p2.17.m17.1.1.2" xref="A1.p2.17.m17.1.1.2.cmml">x</mi><mi id="A1.p2.17.m17.1.1.3" xref="A1.p2.17.m17.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" 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start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = italic_j ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = divide start_ARG italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n + 2 italic_β end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.21">where <math alttext="n_{j}" class="ltx_Math" display="inline" id="A1.p2.18.m1.1"><semantics id="A1.p2.18.m1.1a"><msub id="A1.p2.18.m1.1.1" xref="A1.p2.18.m1.1.1.cmml"><mi id="A1.p2.18.m1.1.1.2" xref="A1.p2.18.m1.1.1.2.cmml">n</mi><mi id="A1.p2.18.m1.1.1.3" xref="A1.p2.18.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p2.18.m1.1b"><apply id="A1.p2.18.m1.1.1.cmml" 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id="A1.p2.19.m2.1d">italic_j</annotation></semantics></math> appears in the subsequence <math alttext="x_{1}^{t}|_{i_{1}^{k}}=(x_{\ell}\in x_{1}^{t}:x_{\ell-k}^{\ell-1}=i_{1}^{k})" class="ltx_math_unparsed" display="inline" id="A1.p2.20.m3.1"><semantics id="A1.p2.20.m3.1a"><mrow id="A1.p2.20.m3.1b"><msubsup id="A1.p2.20.m3.1.2"><mi id="A1.p2.20.m3.1.2.2.2">x</mi><mn id="A1.p2.20.m3.1.2.2.3">1</mn><mi id="A1.p2.20.m3.1.2.3">t</mi></msubsup><msub id="A1.p2.20.m3.1.3"><mo fence="false" id="A1.p2.20.m3.1.3.2" stretchy="false">|</mo><msubsup id="A1.p2.20.m3.1.1.1"><mi id="A1.p2.20.m3.1.1.1.2.2">i</mi><mn id="A1.p2.20.m3.1.1.1.2.3">1</mn><mi id="A1.p2.20.m3.1.1.1.3">k</mi></msubsup></msub><mo id="A1.p2.20.m3.1.4" lspace="0.167em">=</mo><mrow id="A1.p2.20.m3.1.5"><mo id="A1.p2.20.m3.1.5.1" stretchy="false">(</mo><msub id="A1.p2.20.m3.1.5.2"><mi id="A1.p2.20.m3.1.5.2.2">x</mi><mi id="A1.p2.20.m3.1.5.2.3" mathvariant="normal">ℓ</mi></msub><mo id="A1.p2.20.m3.1.5.3">∈</mo><msubsup id="A1.p2.20.m3.1.5.4"><mi id="A1.p2.20.m3.1.5.4.2.2">x</mi><mn id="A1.p2.20.m3.1.5.4.2.3">1</mn><mi id="A1.p2.20.m3.1.5.4.3">t</mi></msubsup><mo id="A1.p2.20.m3.1.5.5" lspace="0.278em" rspace="0.278em">:</mo><msubsup id="A1.p2.20.m3.1.5.6"><mi id="A1.p2.20.m3.1.5.6.2.2">x</mi><mrow id="A1.p2.20.m3.1.5.6.2.3"><mi id="A1.p2.20.m3.1.5.6.2.3.2" mathvariant="normal">ℓ</mi><mo id="A1.p2.20.m3.1.5.6.2.3.1">−</mo><mi id="A1.p2.20.m3.1.5.6.2.3.3">k</mi></mrow><mrow id="A1.p2.20.m3.1.5.6.3"><mi id="A1.p2.20.m3.1.5.6.3.2" mathvariant="normal">ℓ</mi><mo id="A1.p2.20.m3.1.5.6.3.1">−</mo><mn id="A1.p2.20.m3.1.5.6.3.3">1</mn></mrow></msubsup><mo id="A1.p2.20.m3.1.5.7">=</mo><msubsup id="A1.p2.20.m3.1.5.8"><mi id="A1.p2.20.m3.1.5.8.2.2">i</mi><mn id="A1.p2.20.m3.1.5.8.2.3">1</mn><mi id="A1.p2.20.m3.1.5.8.3">k</mi></msubsup><mo id="A1.p2.20.m3.1.5.9" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.p2.20.m3.1c">x_{1}^{t}|_{i_{1}^{k}}=(x_{\ell}\in x_{1}^{t}:x_{\ell-k}^{\ell-1}=i_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A1.p2.20.m3.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∈ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT : italic_x start_POSTSUBSCRIPT roman_ℓ - italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ - 1 end_POSTSUPERSCRIPT = italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>, and <math alttext="n" class="ltx_Math" display="inline" id="A1.p2.21.m4.1"><semantics id="A1.p2.21.m4.1a"><mi id="A1.p2.21.m4.1.1" xref="A1.p2.21.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="A1.p2.21.m4.1b"><ci id="A1.p2.21.m4.1.1.cmml" xref="A1.p2.21.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.21.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="A1.p2.21.m4.1d">italic_n</annotation></semantics></math> is the length of the subsequence.</p> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.1">We now provide a formal proof of this fact.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem3.1.1.1">Theorem 3</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem3.p1"> <p class="ltx_p" id="Thmtheorem3.p1.10"><span class="ltx_text ltx_font_italic" id="Thmtheorem3.p1.10.10">Consider the class of all <math alttext="k" class="ltx_Math" display="inline" id="Thmtheorem3.p1.1.1.m1.1"><semantics id="Thmtheorem3.p1.1.1.m1.1a"><mi id="Thmtheorem3.p1.1.1.m1.1.1" xref="Thmtheorem3.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.1.1.m1.1b"><ci id="Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="Thmtheorem3.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-th order Markov kernels <math alttext="P=(P_{i_{1}^{k}})_{i_{1}^{k}\in\mathcal{X}^{k}}" class="ltx_Math" display="inline" id="Thmtheorem3.p1.2.2.m2.1"><semantics id="Thmtheorem3.p1.2.2.m2.1a"><mrow id="Thmtheorem3.p1.2.2.m2.1.1" xref="Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem3.p1.2.2.m2.1.1.3" xref="Thmtheorem3.p1.2.2.m2.1.1.3.cmml">P</mi><mo id="Thmtheorem3.p1.2.2.m2.1.1.2" xref="Thmtheorem3.p1.2.2.m2.1.1.2.cmml">=</mo><msub id="Thmtheorem3.p1.2.2.m2.1.1.1" xref="Thmtheorem3.p1.2.2.m2.1.1.1.cmml"><mrow id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.2" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.2.cmml">P</mi><msubsup id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.cmml"><mi id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.2" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.2.cmml">i</mi><mn id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></msub><mo id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="Thmtheorem3.p1.2.2.m2.1.1.1.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.cmml"><msubsup id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.cmml"><mi id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.2" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.2.cmml">i</mi><mn id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.3.cmml">k</mi></msubsup><mo id="Thmtheorem3.p1.2.2.m2.1.1.1.3.1" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.1.cmml">∈</mo><msup id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.2" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.2.cmml">𝒳</mi><mi id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.3" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.3.cmml">k</mi></msup></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.2.2.m2.1b"><apply id="Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1"><eq id="Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.2"></eq><ci id="Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.3">𝑃</ci><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1">subscript</csymbol><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.2">𝑃</ci><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3">superscript</csymbol><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.2">𝑖</ci><cn id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.2.3">1</cn></apply><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3"><in id="Thmtheorem3.p1.2.2.m2.1.1.1.3.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.1"></in><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2">superscript</csymbol><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2">subscript</csymbol><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.2">𝑖</ci><cn id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.3.cmml" type="integer" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.2.3">1</cn></apply><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.2.3">𝑘</ci></apply><apply id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3">superscript</csymbol><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.2">𝒳</ci><ci id="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1.1.3.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.2.2.m2.1c">P=(P_{i_{1}^{k}})_{i_{1}^{k}\in\mathcal{X}^{k}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.2.2.m2.1d">italic_P = ( italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, where each <math alttext="P_{i_{1}^{k}}=\mathbb{P}(\cdot\mid i_{1}^{k})" class="ltx_math_unparsed" display="inline" id="Thmtheorem3.p1.3.3.m3.1"><semantics id="Thmtheorem3.p1.3.3.m3.1a"><mrow id="Thmtheorem3.p1.3.3.m3.1b"><msub id="Thmtheorem3.p1.3.3.m3.1.1"><mi id="Thmtheorem3.p1.3.3.m3.1.1.2">P</mi><msubsup id="Thmtheorem3.p1.3.3.m3.1.1.3"><mi id="Thmtheorem3.p1.3.3.m3.1.1.3.2.2">i</mi><mn id="Thmtheorem3.p1.3.3.m3.1.1.3.2.3">1</mn><mi id="Thmtheorem3.p1.3.3.m3.1.1.3.3">k</mi></msubsup></msub><mo id="Thmtheorem3.p1.3.3.m3.1.2">=</mo><mi id="Thmtheorem3.p1.3.3.m3.1.3">ℙ</mi><mrow id="Thmtheorem3.p1.3.3.m3.1.4"><mo id="Thmtheorem3.p1.3.3.m3.1.4.1" stretchy="false">(</mo><mo id="Thmtheorem3.p1.3.3.m3.1.4.2" lspace="0em" rspace="0em">⋅</mo><mo id="Thmtheorem3.p1.3.3.m3.1.4.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="Thmtheorem3.p1.3.3.m3.1.4.4"><mi id="Thmtheorem3.p1.3.3.m3.1.4.4.2.2">i</mi><mn id="Thmtheorem3.p1.3.3.m3.1.4.4.2.3">1</mn><mi id="Thmtheorem3.p1.3.3.m3.1.4.4.3">k</mi></msubsup><mo id="Thmtheorem3.p1.3.3.m3.1.4.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmtheorem3.p1.3.3.m3.1c">P_{i_{1}^{k}}=\mathbb{P}(\cdot\mid i_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.3.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = blackboard_P ( ⋅ ∣ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> is a probability distribution on <math alttext="\mathcal{X}=\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem3.p1.4.4.m4.2"><semantics id="Thmtheorem3.p1.4.4.m4.2a"><mrow id="Thmtheorem3.p1.4.4.m4.2.3" xref="Thmtheorem3.p1.4.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem3.p1.4.4.m4.2.3.2" xref="Thmtheorem3.p1.4.4.m4.2.3.2.cmml">𝒳</mi><mo id="Thmtheorem3.p1.4.4.m4.2.3.1" xref="Thmtheorem3.p1.4.4.m4.2.3.1.cmml">=</mo><mrow id="Thmtheorem3.p1.4.4.m4.2.3.3.2" xref="Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml"><mo id="Thmtheorem3.p1.4.4.m4.2.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">{</mo><mn id="Thmtheorem3.p1.4.4.m4.1.1" xref="Thmtheorem3.p1.4.4.m4.1.1.cmml">0</mn><mo id="Thmtheorem3.p1.4.4.m4.2.3.3.2.2" xref="Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">,</mo><mn id="Thmtheorem3.p1.4.4.m4.2.2" xref="Thmtheorem3.p1.4.4.m4.2.2.cmml">1</mn><mo id="Thmtheorem3.p1.4.4.m4.2.3.3.2.3" stretchy="false" xref="Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.4.4.m4.2b"><apply id="Thmtheorem3.p1.4.4.m4.2.3.cmml" xref="Thmtheorem3.p1.4.4.m4.2.3"><eq id="Thmtheorem3.p1.4.4.m4.2.3.1.cmml" xref="Thmtheorem3.p1.4.4.m4.2.3.1"></eq><ci id="Thmtheorem3.p1.4.4.m4.2.3.2.cmml" xref="Thmtheorem3.p1.4.4.m4.2.3.2">𝒳</ci><set id="Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml" xref="Thmtheorem3.p1.4.4.m4.2.3.3.2"><cn id="Thmtheorem3.p1.4.4.m4.1.1.cmml" type="integer" xref="Thmtheorem3.p1.4.4.m4.1.1">0</cn><cn id="Thmtheorem3.p1.4.4.m4.2.2.cmml" type="integer" xref="Thmtheorem3.p1.4.4.m4.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.4.4.m4.2c">\mathcal{X}=\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.4.4.m4.2d">caligraphic_X = { 0 , 1 }</annotation></semantics></math>. Let each <math alttext="P_{i_{1}^{k}}" class="ltx_Math" display="inline" id="Thmtheorem3.p1.5.5.m5.1"><semantics id="Thmtheorem3.p1.5.5.m5.1a"><msub id="Thmtheorem3.p1.5.5.m5.1.1" xref="Thmtheorem3.p1.5.5.m5.1.1.cmml"><mi id="Thmtheorem3.p1.5.5.m5.1.1.2" xref="Thmtheorem3.p1.5.5.m5.1.1.2.cmml">P</mi><msubsup id="Thmtheorem3.p1.5.5.m5.1.1.3" xref="Thmtheorem3.p1.5.5.m5.1.1.3.cmml"><mi id="Thmtheorem3.p1.5.5.m5.1.1.3.2.2" xref="Thmtheorem3.p1.5.5.m5.1.1.3.2.2.cmml">i</mi><mn id="Thmtheorem3.p1.5.5.m5.1.1.3.2.3" xref="Thmtheorem3.p1.5.5.m5.1.1.3.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.5.5.m5.1.1.3.3" xref="Thmtheorem3.p1.5.5.m5.1.1.3.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.5.5.m5.1b"><apply id="Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1">subscript</csymbol><ci id="Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.2">𝑃</ci><apply id="Thmtheorem3.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.5.5.m5.1.1.3.1.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.3">superscript</csymbol><apply id="Thmtheorem3.p1.5.5.m5.1.1.3.2.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.5.5.m5.1.1.3.2.1.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="Thmtheorem3.p1.5.5.m5.1.1.3.2.2.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.3.2.2">𝑖</ci><cn id="Thmtheorem3.p1.5.5.m5.1.1.3.2.3.cmml" type="integer" xref="Thmtheorem3.p1.5.5.m5.1.1.3.2.3">1</cn></apply><ci id="Thmtheorem3.p1.5.5.m5.1.1.3.3.cmml" xref="Thmtheorem3.p1.5.5.m5.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.5.5.m5.1c">P_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.5.5.m5.1d">italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be sampled i.i.d. from <math alttext="\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="Thmtheorem3.p1.6.6.m6.1"><semantics id="Thmtheorem3.p1.6.6.m6.1a"><mrow id="Thmtheorem3.p1.6.6.m6.1.1" xref="Thmtheorem3.p1.6.6.m6.1.1.cmml"><mi id="Thmtheorem3.p1.6.6.m6.1.1.3" xref="Thmtheorem3.p1.6.6.m6.1.1.3.cmml">Dir</mi><mo id="Thmtheorem3.p1.6.6.m6.1.1.2" xref="Thmtheorem3.p1.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem3.p1.6.6.m6.1.1.1.1" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml"><mo id="Thmtheorem3.p1.6.6.m6.1.1.1.1.2" stretchy="false" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml"><mi id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.2" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.2.cmml">β</mi><mo id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.cmml">⋅</mo><mn id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="Thmtheorem3.p1.6.6.m6.1.1.1.1.3" stretchy="false" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.6.6.m6.1b"><apply id="Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="Thmtheorem3.p1.6.6.m6.1.1"><times id="Thmtheorem3.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem3.p1.6.6.m6.1.1.2"></times><ci id="Thmtheorem3.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem3.p1.6.6.m6.1.1.3">Dir</ci><apply id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1"><ci id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1">⋅</ci><ci id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.2.cmml" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.2">𝛽</ci><cn id="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.6.6.m6.1c">\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.6.6.m6.1d">roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math>, and let <math alttext="x_{1}^{k}\sim\mathrm{Unif}(\mathcal{X}^{k})" class="ltx_Math" display="inline" id="Thmtheorem3.p1.7.7.m7.1"><semantics id="Thmtheorem3.p1.7.7.m7.1a"><mrow id="Thmtheorem3.p1.7.7.m7.1.1" xref="Thmtheorem3.p1.7.7.m7.1.1.cmml"><msubsup id="Thmtheorem3.p1.7.7.m7.1.1.3" xref="Thmtheorem3.p1.7.7.m7.1.1.3.cmml"><mi id="Thmtheorem3.p1.7.7.m7.1.1.3.2.2" xref="Thmtheorem3.p1.7.7.m7.1.1.3.2.2.cmml">x</mi><mn id="Thmtheorem3.p1.7.7.m7.1.1.3.2.3" xref="Thmtheorem3.p1.7.7.m7.1.1.3.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.7.7.m7.1.1.3.3" xref="Thmtheorem3.p1.7.7.m7.1.1.3.3.cmml">k</mi></msubsup><mo id="Thmtheorem3.p1.7.7.m7.1.1.2" xref="Thmtheorem3.p1.7.7.m7.1.1.2.cmml">∼</mo><mrow id="Thmtheorem3.p1.7.7.m7.1.1.1" xref="Thmtheorem3.p1.7.7.m7.1.1.1.cmml"><mi id="Thmtheorem3.p1.7.7.m7.1.1.1.3" xref="Thmtheorem3.p1.7.7.m7.1.1.1.3.cmml">Unif</mi><mo id="Thmtheorem3.p1.7.7.m7.1.1.1.2" xref="Thmtheorem3.p1.7.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.cmml"><mo id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.cmml">(</mo><msup id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.2" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.2.cmml">𝒳</mi><mi id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.3" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.3.cmml">k</mi></msup><mo id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.7.7.m7.1b"><apply id="Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1"><csymbol cd="latexml" id="Thmtheorem3.p1.7.7.m7.1.1.2.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.2">similar-to</csymbol><apply id="Thmtheorem3.p1.7.7.m7.1.1.3.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.7.7.m7.1.1.3.1.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.3">superscript</csymbol><apply id="Thmtheorem3.p1.7.7.m7.1.1.3.2.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.7.7.m7.1.1.3.2.1.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="Thmtheorem3.p1.7.7.m7.1.1.3.2.2.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.3.2.2">𝑥</ci><cn id="Thmtheorem3.p1.7.7.m7.1.1.3.2.3.cmml" type="integer" xref="Thmtheorem3.p1.7.7.m7.1.1.3.2.3">1</cn></apply><ci id="Thmtheorem3.p1.7.7.m7.1.1.3.3.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.3.3">𝑘</ci></apply><apply id="Thmtheorem3.p1.7.7.m7.1.1.1.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1"><times id="Thmtheorem3.p1.7.7.m7.1.1.1.2.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1.2"></times><ci id="Thmtheorem3.p1.7.7.m7.1.1.1.3.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1.3">Unif</ci><apply id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1">superscript</csymbol><ci id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.2.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.2">𝒳</ci><ci id="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.3.cmml" xref="Thmtheorem3.p1.7.7.m7.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.7.7.m7.1c">x_{1}^{k}\sim\mathrm{Unif}(\mathcal{X}^{k})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.7.7.m7.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∼ roman_Unif ( caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="x_{t+1}|x_{1}^{t}\sim P_{x_{t-k+1}^{t}}" class="ltx_Math" display="inline" id="Thmtheorem3.p1.8.8.m8.1"><semantics id="Thmtheorem3.p1.8.8.m8.1a"><mrow id="Thmtheorem3.p1.8.8.m8.1.1" xref="Thmtheorem3.p1.8.8.m8.1.1.cmml"><mrow id="Thmtheorem3.p1.8.8.m8.1.1.2" xref="Thmtheorem3.p1.8.8.m8.1.1.2.cmml"><msub id="Thmtheorem3.p1.8.8.m8.1.1.2.2" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.cmml"><mi id="Thmtheorem3.p1.8.8.m8.1.1.2.2.2" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.2.cmml">x</mi><mrow id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.cmml"><mi id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.2" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.2.cmml">t</mi><mo id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.1" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.1.cmml">+</mo><mn id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.3" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.3.cmml">1</mn></mrow></msub><mo fence="false" id="Thmtheorem3.p1.8.8.m8.1.1.2.1" xref="Thmtheorem3.p1.8.8.m8.1.1.2.1.cmml">|</mo><msubsup id="Thmtheorem3.p1.8.8.m8.1.1.2.3" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.cmml"><mi id="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.2" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.2.cmml">x</mi><mn id="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.3" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.8.8.m8.1.1.2.3.3" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.3.cmml">t</mi></msubsup></mrow><mo id="Thmtheorem3.p1.8.8.m8.1.1.1" xref="Thmtheorem3.p1.8.8.m8.1.1.1.cmml">∼</mo><msub id="Thmtheorem3.p1.8.8.m8.1.1.3" xref="Thmtheorem3.p1.8.8.m8.1.1.3.cmml"><mi id="Thmtheorem3.p1.8.8.m8.1.1.3.2" xref="Thmtheorem3.p1.8.8.m8.1.1.3.2.cmml">P</mi><msubsup id="Thmtheorem3.p1.8.8.m8.1.1.3.3" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.cmml"><mi id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.2" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.2.cmml">x</mi><mrow id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.cmml"><mrow id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.cmml"><mi id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.2" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.2.cmml">t</mi><mo id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.1" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.1.cmml">−</mo><mi id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.3" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.3.cmml">k</mi></mrow><mo id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.1" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.1.cmml">+</mo><mn id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.3" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.3.cmml">1</mn></mrow><mi id="Thmtheorem3.p1.8.8.m8.1.1.3.3.3" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.3.cmml">t</mi></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.8.8.m8.1b"><apply id="Thmtheorem3.p1.8.8.m8.1.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1"><csymbol cd="latexml" id="Thmtheorem3.p1.8.8.m8.1.1.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.1">similar-to</csymbol><apply id="Thmtheorem3.p1.8.8.m8.1.1.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2"><csymbol cd="latexml" id="Thmtheorem3.p1.8.8.m8.1.1.2.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.1">conditional</csymbol><apply id="Thmtheorem3.p1.8.8.m8.1.1.2.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.8.m8.1.1.2.2.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2">subscript</csymbol><ci id="Thmtheorem3.p1.8.8.m8.1.1.2.2.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.2">𝑥</ci><apply id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3"><plus id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.1"></plus><ci id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.2">𝑡</ci><cn id="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.3.cmml" type="integer" xref="Thmtheorem3.p1.8.8.m8.1.1.2.2.3.3">1</cn></apply></apply><apply id="Thmtheorem3.p1.8.8.m8.1.1.2.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.8.m8.1.1.2.3.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3">superscript</csymbol><apply id="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3">subscript</csymbol><ci id="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.2">𝑥</ci><cn id="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.3.cmml" type="integer" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.2.3">1</cn></apply><ci id="Thmtheorem3.p1.8.8.m8.1.1.2.3.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.2.3.3">𝑡</ci></apply></apply><apply id="Thmtheorem3.p1.8.8.m8.1.1.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.8.m8.1.1.3.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="Thmtheorem3.p1.8.8.m8.1.1.3.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.2">𝑃</ci><apply id="Thmtheorem3.p1.8.8.m8.1.1.3.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.8.m8.1.1.3.3.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3">superscript</csymbol><apply id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3">subscript</csymbol><ci id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.2">𝑥</ci><apply id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3"><plus id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.1"></plus><apply id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2"><minus id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.1.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.1"></minus><ci id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.2.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.2">𝑡</ci><ci id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.2.3">𝑘</ci></apply><cn id="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.3.cmml" type="integer" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.2.3.3">1</cn></apply></apply><ci id="Thmtheorem3.p1.8.8.m8.1.1.3.3.3.cmml" xref="Thmtheorem3.p1.8.8.m8.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.8.8.m8.1c">x_{t+1}|x_{1}^{t}\sim P_{x_{t-k+1}^{t}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.8.8.m8.1d">italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ italic_P start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Then, the predictor <math alttext="f^{(j)}(x_{1}^{t})=\hat{\mathbb{P}}(x_{t+1}=j\mid x_{1}^{t})" class="ltx_Math" display="inline" id="Thmtheorem3.p1.9.9.m9.3"><semantics id="Thmtheorem3.p1.9.9.m9.3a"><mrow id="Thmtheorem3.p1.9.9.m9.3.3" xref="Thmtheorem3.p1.9.9.m9.3.3.cmml"><mrow id="Thmtheorem3.p1.9.9.m9.2.2.1" xref="Thmtheorem3.p1.9.9.m9.2.2.1.cmml"><msup id="Thmtheorem3.p1.9.9.m9.2.2.1.3" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml"><mi id="Thmtheorem3.p1.9.9.m9.2.2.1.3.2" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3.2.cmml">f</mi><mrow id="Thmtheorem3.p1.9.9.m9.1.1.1.3" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml"><mo id="Thmtheorem3.p1.9.9.m9.1.1.1.3.1" stretchy="false" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml">(</mo><mi id="Thmtheorem3.p1.9.9.m9.1.1.1.1" xref="Thmtheorem3.p1.9.9.m9.1.1.1.1.cmml">j</mi><mo id="Thmtheorem3.p1.9.9.m9.1.1.1.3.2" stretchy="false" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml">)</mo></mrow></msup><mo id="Thmtheorem3.p1.9.9.m9.2.2.1.2" xref="Thmtheorem3.p1.9.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.cmml"><mo id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.2" stretchy="false" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.cmml">(</mo><msubsup id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.cmml"><mi id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.2" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.2.cmml">x</mi><mn id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.3" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.3" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.3.cmml">t</mi></msubsup><mo id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.3" stretchy="false" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem3.p1.9.9.m9.3.3.3" xref="Thmtheorem3.p1.9.9.m9.3.3.3.cmml">=</mo><mrow id="Thmtheorem3.p1.9.9.m9.3.3.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.cmml"><mover accent="true" id="Thmtheorem3.p1.9.9.m9.3.3.2.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.3.cmml"><mi id="Thmtheorem3.p1.9.9.m9.3.3.2.3.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.3.2.cmml">ℙ</mi><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.3.1" xref="Thmtheorem3.p1.9.9.m9.3.3.2.3.1.cmml">^</mo></mover><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.2.cmml">⁢</mo><mrow id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml"><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.2" stretchy="false" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml">(</mo><mrow id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml"><msub id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.cmml"><mi id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.2.cmml">x</mi><mrow id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.cmml"><mi id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.2.cmml">t</mi><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.1" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.1.cmml">+</mo><mn id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.1" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.1.cmml">=</mo><mrow id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.cmml"><mi id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.2.cmml">j</mi><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.1" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.1.cmml">∣</mo><msubsup id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.cmml"><mi id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.2" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.2.cmml">x</mi><mn id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.3.cmml">1</mn><mi id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.3" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.3.cmml">t</mi></msubsup></mrow></mrow><mo id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.3" stretchy="false" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.9.9.m9.3b"><apply id="Thmtheorem3.p1.9.9.m9.3.3.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3"><eq id="Thmtheorem3.p1.9.9.m9.3.3.3.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.3"></eq><apply id="Thmtheorem3.p1.9.9.m9.2.2.1.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1"><times id="Thmtheorem3.p1.9.9.m9.2.2.1.2.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.2"></times><apply id="Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.9.9.m9.2.2.1.3.1.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3">superscript</csymbol><ci id="Thmtheorem3.p1.9.9.m9.2.2.1.3.2.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.3.2">𝑓</ci><ci id="Thmtheorem3.p1.9.9.m9.1.1.1.1.cmml" xref="Thmtheorem3.p1.9.9.m9.1.1.1.1">𝑗</ci></apply><apply id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.1.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1">superscript</csymbol><apply id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.1.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1">subscript</csymbol><ci id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.2.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.2">𝑥</ci><cn id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.3.cmml" type="integer" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.3">1</cn></apply><ci id="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.3.cmml" xref="Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.3">𝑡</ci></apply></apply><apply id="Thmtheorem3.p1.9.9.m9.3.3.2.cmml" 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xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3"><plus id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.1.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.1"></plus><ci id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.2.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.2">𝑡</ci><cn id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.3.cmml" type="integer" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.3.3">1</cn></apply></apply><apply id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3"><csymbol cd="latexml" id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.1.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.1">conditional</csymbol><ci id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.2.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.2">𝑗</ci><apply id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.1.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3">superscript</csymbol><apply id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.1.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3">subscript</csymbol><ci id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.2.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.2">𝑥</ci><cn id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.2.3">1</cn></apply><ci id="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.3.cmml" xref="Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.9.9.m9.3c">f^{(j)}(x_{1}^{t})=\hat{\mathbb{P}}(x_{t+1}=j\mid x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.9.9.m9.3d">italic_f start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = over^ start_ARG blackboard_P end_ARG ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = italic_j ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>, for <math alttext="j\in\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem3.p1.10.10.m10.2"><semantics id="Thmtheorem3.p1.10.10.m10.2a"><mrow id="Thmtheorem3.p1.10.10.m10.2.3" xref="Thmtheorem3.p1.10.10.m10.2.3.cmml"><mi id="Thmtheorem3.p1.10.10.m10.2.3.2" xref="Thmtheorem3.p1.10.10.m10.2.3.2.cmml">j</mi><mo id="Thmtheorem3.p1.10.10.m10.2.3.1" xref="Thmtheorem3.p1.10.10.m10.2.3.1.cmml">∈</mo><mrow id="Thmtheorem3.p1.10.10.m10.2.3.3.2" xref="Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml"><mo id="Thmtheorem3.p1.10.10.m10.2.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml">{</mo><mn id="Thmtheorem3.p1.10.10.m10.1.1" xref="Thmtheorem3.p1.10.10.m10.1.1.cmml">0</mn><mo id="Thmtheorem3.p1.10.10.m10.2.3.3.2.2" xref="Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml">,</mo><mn id="Thmtheorem3.p1.10.10.m10.2.2" xref="Thmtheorem3.p1.10.10.m10.2.2.cmml">1</mn><mo id="Thmtheorem3.p1.10.10.m10.2.3.3.2.3" stretchy="false" xref="Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.10.10.m10.2b"><apply id="Thmtheorem3.p1.10.10.m10.2.3.cmml" xref="Thmtheorem3.p1.10.10.m10.2.3"><in id="Thmtheorem3.p1.10.10.m10.2.3.1.cmml" xref="Thmtheorem3.p1.10.10.m10.2.3.1"></in><ci id="Thmtheorem3.p1.10.10.m10.2.3.2.cmml" xref="Thmtheorem3.p1.10.10.m10.2.3.2">𝑗</ci><set id="Thmtheorem3.p1.10.10.m10.2.3.3.1.cmml" xref="Thmtheorem3.p1.10.10.m10.2.3.3.2"><cn id="Thmtheorem3.p1.10.10.m10.1.1.cmml" type="integer" xref="Thmtheorem3.p1.10.10.m10.1.1">0</cn><cn id="Thmtheorem3.p1.10.10.m10.2.2.cmml" type="integer" xref="Thmtheorem3.p1.10.10.m10.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.10.10.m10.2c">j\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.10.10.m10.2d">italic_j ∈ { 0 , 1 }</annotation></semantics></math>, that minimizes the loss</span></p> <table class="ltx_equation ltx_eqn_table" id="A1.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="L\triangleq-\frac{1}{T}\sum_{t\in[T]}\mathbb{E}_{P}\mathbb{E}_{x_{1}^{t+1}\sim P% }\big{[}x_{t+1}\cdot\log f^{(1)}(x_{1}^{t})+(1-x_{t+1})\cdot\log f^{(0)}(x_{1}% ^{t})\big{]}" class="ltx_Math" display="block" id="A1.E6.m1.4"><semantics id="A1.E6.m1.4a"><mrow id="A1.E6.m1.4.4" xref="A1.E6.m1.4.4.cmml"><mi id="A1.E6.m1.4.4.3" xref="A1.E6.m1.4.4.3.cmml">L</mi><mo id="A1.E6.m1.4.4.2" xref="A1.E6.m1.4.4.2.cmml">≜</mo><mrow id="A1.E6.m1.4.4.1" xref="A1.E6.m1.4.4.1.cmml"><mo id="A1.E6.m1.4.4.1a" xref="A1.E6.m1.4.4.1.cmml">−</mo><mrow id="A1.E6.m1.4.4.1.1" xref="A1.E6.m1.4.4.1.1.cmml"><mfrac id="A1.E6.m1.4.4.1.1.3" xref="A1.E6.m1.4.4.1.1.3.cmml"><mn id="A1.E6.m1.4.4.1.1.3.2" xref="A1.E6.m1.4.4.1.1.3.2.cmml">1</mn><mi id="A1.E6.m1.4.4.1.1.3.3" xref="A1.E6.m1.4.4.1.1.3.3.cmml">T</mi></mfrac><mo id="A1.E6.m1.4.4.1.1.2" xref="A1.E6.m1.4.4.1.1.2.cmml">⁢</mo><mrow id="A1.E6.m1.4.4.1.1.1" xref="A1.E6.m1.4.4.1.1.1.cmml"><munder id="A1.E6.m1.4.4.1.1.1.2" xref="A1.E6.m1.4.4.1.1.1.2.cmml"><mo id="A1.E6.m1.4.4.1.1.1.2.2" movablelimits="false" xref="A1.E6.m1.4.4.1.1.1.2.2.cmml">∑</mo><mrow id="A1.E6.m1.1.1.1" xref="A1.E6.m1.1.1.1.cmml"><mi id="A1.E6.m1.1.1.1.3" xref="A1.E6.m1.1.1.1.3.cmml">t</mi><mo id="A1.E6.m1.1.1.1.2" xref="A1.E6.m1.1.1.1.2.cmml">∈</mo><mrow id="A1.E6.m1.1.1.1.4.2" xref="A1.E6.m1.1.1.1.4.1.cmml"><mo id="A1.E6.m1.1.1.1.4.2.1" stretchy="false" xref="A1.E6.m1.1.1.1.4.1.1.cmml">[</mo><mi id="A1.E6.m1.1.1.1.1" xref="A1.E6.m1.1.1.1.1.cmml">T</mi><mo id="A1.E6.m1.1.1.1.4.2.2" stretchy="false" xref="A1.E6.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder><mrow id="A1.E6.m1.4.4.1.1.1.1" xref="A1.E6.m1.4.4.1.1.1.1.cmml"><msub id="A1.E6.m1.4.4.1.1.1.1.3" xref="A1.E6.m1.4.4.1.1.1.1.3.cmml"><mi id="A1.E6.m1.4.4.1.1.1.1.3.2" xref="A1.E6.m1.4.4.1.1.1.1.3.2.cmml">𝔼</mi><mi id="A1.E6.m1.4.4.1.1.1.1.3.3" xref="A1.E6.m1.4.4.1.1.1.1.3.3.cmml">P</mi></msub><mo id="A1.E6.m1.4.4.1.1.1.1.2" xref="A1.E6.m1.4.4.1.1.1.1.2.cmml">⁢</mo><msub id="A1.E6.m1.4.4.1.1.1.1.4" xref="A1.E6.m1.4.4.1.1.1.1.4.cmml"><mi id="A1.E6.m1.4.4.1.1.1.1.4.2" xref="A1.E6.m1.4.4.1.1.1.1.4.2.cmml">𝔼</mi><mrow id="A1.E6.m1.4.4.1.1.1.1.4.3" xref="A1.E6.m1.4.4.1.1.1.1.4.3.cmml"><msubsup id="A1.E6.m1.4.4.1.1.1.1.4.3.2" xref="A1.E6.m1.4.4.1.1.1.1.4.3.2.cmml"><mi id="A1.E6.m1.4.4.1.1.1.1.4.3.2.2.2" xref="A1.E6.m1.4.4.1.1.1.1.4.3.2.2.2.cmml">x</mi><mn id="A1.E6.m1.4.4.1.1.1.1.4.3.2.2.3" xref="A1.E6.m1.4.4.1.1.1.1.4.3.2.2.3.cmml">1</mn><mrow id="A1.E6.m1.4.4.1.1.1.1.4.3.2.3" xref="A1.E6.m1.4.4.1.1.1.1.4.3.2.3.cmml"><mi 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italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t ∈ [ italic_T ] end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ∼ italic_P end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) + ( 1 - italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem3.p1.13"><span class="ltx_text ltx_font_italic" id="Thmtheorem3.p1.13.3">is the add-<math alttext="\beta" class="ltx_Math" display="inline" id="Thmtheorem3.p1.11.1.m1.1"><semantics id="Thmtheorem3.p1.11.1.m1.1a"><mi id="Thmtheorem3.p1.11.1.m1.1.1" xref="Thmtheorem3.p1.11.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.11.1.m1.1b"><ci id="Thmtheorem3.p1.11.1.m1.1.1.cmml" xref="Thmtheorem3.p1.11.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.11.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.11.1.m1.1d">italic_β</annotation></semantics></math> estimator in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A1.E5" title="Equation 5 ‣ Appendix A Preliminaries on Laplacian smoothing ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">5</span></a>), i.e. the minimizer <math alttext="f_{*}^{(j)}(x_{1}^{t})=\mathbb{P}_{\beta}^{(k)}\left(x_{t+1}=j\mid x_{1}^{t}\right)" class="ltx_Math" display="inline" id="Thmtheorem3.p1.12.2.m2.4"><semantics id="Thmtheorem3.p1.12.2.m2.4a"><mrow id="Thmtheorem3.p1.12.2.m2.4.4" xref="Thmtheorem3.p1.12.2.m2.4.4.cmml"><mrow id="Thmtheorem3.p1.12.2.m2.3.3.1" xref="Thmtheorem3.p1.12.2.m2.3.3.1.cmml"><msubsup id="Thmtheorem3.p1.12.2.m2.3.3.1.3" xref="Thmtheorem3.p1.12.2.m2.3.3.1.3.cmml"><mi id="Thmtheorem3.p1.12.2.m2.3.3.1.3.2.2" xref="Thmtheorem3.p1.12.2.m2.3.3.1.3.2.2.cmml">f</mi><mo id="Thmtheorem3.p1.12.2.m2.3.3.1.3.2.3" xref="Thmtheorem3.p1.12.2.m2.3.3.1.3.2.3.cmml">∗</mo><mrow id="Thmtheorem3.p1.12.2.m2.1.1.1.3" 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id="Thmtheorem3.p1.12.2.m2.4.4.2.1.1.1.3.3.2.2.cmml" xref="Thmtheorem3.p1.12.2.m2.4.4.2.1.1.1.3.3.2.2">𝑥</ci><cn id="Thmtheorem3.p1.12.2.m2.4.4.2.1.1.1.3.3.2.3.cmml" type="integer" xref="Thmtheorem3.p1.12.2.m2.4.4.2.1.1.1.3.3.2.3">1</cn></apply><ci id="Thmtheorem3.p1.12.2.m2.4.4.2.1.1.1.3.3.3.cmml" xref="Thmtheorem3.p1.12.2.m2.4.4.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.12.2.m2.4c">f_{*}^{(j)}(x_{1}^{t})=\mathbb{P}_{\beta}^{(k)}\left(x_{t+1}=j\mid x_{1}^{t}\right)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.12.2.m2.4d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = italic_j ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>, for all <math alttext="t\geq k" class="ltx_Math" display="inline" id="Thmtheorem3.p1.13.3.m3.1"><semantics id="Thmtheorem3.p1.13.3.m3.1a"><mrow id="Thmtheorem3.p1.13.3.m3.1.1" xref="Thmtheorem3.p1.13.3.m3.1.1.cmml"><mi id="Thmtheorem3.p1.13.3.m3.1.1.2" xref="Thmtheorem3.p1.13.3.m3.1.1.2.cmml">t</mi><mo id="Thmtheorem3.p1.13.3.m3.1.1.1" xref="Thmtheorem3.p1.13.3.m3.1.1.1.cmml">≥</mo><mi id="Thmtheorem3.p1.13.3.m3.1.1.3" xref="Thmtheorem3.p1.13.3.m3.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.13.3.m3.1b"><apply id="Thmtheorem3.p1.13.3.m3.1.1.cmml" xref="Thmtheorem3.p1.13.3.m3.1.1"><geq id="Thmtheorem3.p1.13.3.m3.1.1.1.cmml" xref="Thmtheorem3.p1.13.3.m3.1.1.1"></geq><ci id="Thmtheorem3.p1.13.3.m3.1.1.2.cmml" xref="Thmtheorem3.p1.13.3.m3.1.1.2">𝑡</ci><ci id="Thmtheorem3.p1.13.3.m3.1.1.3.cmml" xref="Thmtheorem3.p1.13.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.13.3.m3.1c">t\geq k</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.13.3.m3.1d">italic_t ≥ italic_k</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="A1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.1.p1"> <p class="ltx_p" id="A1.1.p1.20">First note that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx8"> <tbody id="A1.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle L" class="ltx_Math" display="inline" id="A1.Ex21.m1.1"><semantics id="A1.Ex21.m1.1a"><mi id="A1.Ex21.m1.1.1" xref="A1.Ex21.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A1.Ex21.m1.1b"><ci id="A1.Ex21.m1.1.1.cmml" xref="A1.Ex21.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex21.m1.1c">\displaystyle L</annotation><annotation encoding="application/x-llamapun" id="A1.Ex21.m1.1d">italic_L</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{T}\sum_{t}\mathbb{E}_{P}\mathbb{E}_{x_{1}^{t+1}\sim P}% \big{[}x_{t+1}\cdot\log f^{(1)}(x_{1}^{t})+(1-x_{t+1})\cdot\log f^{(0)}(x_{1}^% {t})\big{]}" class="ltx_Math" display="inline" id="A1.Ex21.m2.3"><semantics id="A1.Ex21.m2.3a"><mrow id="A1.Ex21.m2.3.3" xref="A1.Ex21.m2.3.3.cmml"><mi id="A1.Ex21.m2.3.3.3" xref="A1.Ex21.m2.3.3.3.cmml"></mi><mo id="A1.Ex21.m2.3.3.2" xref="A1.Ex21.m2.3.3.2.cmml">=</mo><mrow id="A1.Ex21.m2.3.3.1" xref="A1.Ex21.m2.3.3.1.cmml"><mo id="A1.Ex21.m2.3.3.1a" xref="A1.Ex21.m2.3.3.1.cmml">−</mo><mrow id="A1.Ex21.m2.3.3.1.1" xref="A1.Ex21.m2.3.3.1.1.cmml"><mstyle displaystyle="true" id="A1.Ex21.m2.3.3.1.1.3" xref="A1.Ex21.m2.3.3.1.1.3.cmml"><mfrac id="A1.Ex21.m2.3.3.1.1.3a" xref="A1.Ex21.m2.3.3.1.1.3.cmml"><mn id="A1.Ex21.m2.3.3.1.1.3.2" xref="A1.Ex21.m2.3.3.1.1.3.2.cmml">1</mn><mi id="A1.Ex21.m2.3.3.1.1.3.3" xref="A1.Ex21.m2.3.3.1.1.3.3.cmml">T</mi></mfrac></mstyle><mo id="A1.Ex21.m2.3.3.1.1.2" xref="A1.Ex21.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="A1.Ex21.m2.3.3.1.1.1" xref="A1.Ex21.m2.3.3.1.1.1.cmml"><mstyle displaystyle="true" id="A1.Ex21.m2.3.3.1.1.1.2" xref="A1.Ex21.m2.3.3.1.1.1.2.cmml"><munder id="A1.Ex21.m2.3.3.1.1.1.2a" xref="A1.Ex21.m2.3.3.1.1.1.2.cmml"><mo id="A1.Ex21.m2.3.3.1.1.1.2.2" movablelimits="false" xref="A1.Ex21.m2.3.3.1.1.1.2.2.cmml">∑</mo><mi id="A1.Ex21.m2.3.3.1.1.1.2.3" xref="A1.Ex21.m2.3.3.1.1.1.2.3.cmml">t</mi></munder></mstyle><mrow id="A1.Ex21.m2.3.3.1.1.1.1" xref="A1.Ex21.m2.3.3.1.1.1.1.cmml"><msub id="A1.Ex21.m2.3.3.1.1.1.1.3" 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xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.3" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.4" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.4.cmml">+</mo><mrow id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.3" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.3.cmml"><mrow id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.2.1" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.2.1.cmml"><mrow id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.2.1.1.1" xref="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.2.1.1.1.1.cmml"><mo id="A1.Ex21.m2.3.3.1.1.1.1.1.1.1.2.1.1.1.2" stretchy="false" 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id="A1.Ex21.m2.3c">\displaystyle=-\frac{1}{T}\sum_{t}\mathbb{E}_{P}\mathbb{E}_{x_{1}^{t+1}\sim P}% \big{[}x_{t+1}\cdot\log f^{(1)}(x_{1}^{t})+(1-x_{t+1})\cdot\log f^{(0)}(x_{1}^% {t})\big{]}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex21.m2.3d">= - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT ∼ italic_P end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) + ( 1 - italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{T}\sum_{t}\mathbb{E}_{x_{1}^{t}}\mathbb{E}_{x_{t+1}|x_% {1}^{t}}\big{[}x_{t+1}\cdot\log f^{(1)}(x_{1}^{t})+(1-x_{t+1})\cdot\log f^{(0)% }(x_{1}^{t})\big{]}" class="ltx_Math" display="inline" id="A1.Ex22.m1.3"><semantics id="A1.Ex22.m1.3a"><mrow id="A1.Ex22.m1.3.3" xref="A1.Ex22.m1.3.3.cmml"><mi id="A1.Ex22.m1.3.3.3" xref="A1.Ex22.m1.3.3.3.cmml"></mi><mo id="A1.Ex22.m1.3.3.2" xref="A1.Ex22.m1.3.3.2.cmml">=</mo><mrow id="A1.Ex22.m1.3.3.1" xref="A1.Ex22.m1.3.3.1.cmml"><mo id="A1.Ex22.m1.3.3.1a" xref="A1.Ex22.m1.3.3.1.cmml">−</mo><mrow 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id="A1.Ex22.m1.3.3.1.1.1.1.1.1.1.3.2.1.1.2.3.cmml" type="integer" xref="A1.Ex22.m1.3.3.1.1.1.1.1.1.1.3.2.1.1.2.3">1</cn></apply><ci id="A1.Ex22.m1.3.3.1.1.1.1.1.1.1.3.2.1.1.3.cmml" xref="A1.Ex22.m1.3.3.1.1.1.1.1.1.1.3.2.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex22.m1.3c">\displaystyle=-\frac{1}{T}\sum_{t}\mathbb{E}_{x_{1}^{t}}\mathbb{E}_{x_{t+1}|x_% {1}^{t}}\big{[}x_{t+1}\cdot\log f^{(1)}(x_{1}^{t})+(1-x_{t+1})\cdot\log f^{(0)% }(x_{1}^{t})\big{]}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex22.m1.3d">= - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) + ( 1 - italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{T}\sum_{t}\mathbb{E}_{x_{1}^{t}}\big{[}\mathbb{E}_{x_{% t+1}|x_{1}^{t}}[x_{t+1}]\cdot\log f^{(1)}(x_{1}^{t})+(1-\mathbb{E}_{x_{t+1}|x_% {1}^{t}}[x_{t+1}])\cdot\log f^{(0)}(x_{1}^{t})\big{]}." class="ltx_Math" display="inline" id="A1.Ex23.m1.3"><semantics id="A1.Ex23.m1.3a"><mrow id="A1.Ex23.m1.3.3.1" xref="A1.Ex23.m1.3.3.1.1.cmml"><mrow id="A1.Ex23.m1.3.3.1.1" xref="A1.Ex23.m1.3.3.1.1.cmml"><mi id="A1.Ex23.m1.3.3.1.1.3" xref="A1.Ex23.m1.3.3.1.1.3.cmml"></mi><mo id="A1.Ex23.m1.3.3.1.1.2" xref="A1.Ex23.m1.3.3.1.1.2.cmml">=</mo><mrow id="A1.Ex23.m1.3.3.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.cmml"><mo id="A1.Ex23.m1.3.3.1.1.1a" xref="A1.Ex23.m1.3.3.1.1.1.cmml">−</mo><mrow id="A1.Ex23.m1.3.3.1.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.1.cmml"><mstyle displaystyle="true" id="A1.Ex23.m1.3.3.1.1.1.1.3" xref="A1.Ex23.m1.3.3.1.1.1.1.3.cmml"><mfrac id="A1.Ex23.m1.3.3.1.1.1.1.3a" xref="A1.Ex23.m1.3.3.1.1.1.1.3.cmml"><mn id="A1.Ex23.m1.3.3.1.1.1.1.3.2" 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id="A1.Ex23.m1.3c">\displaystyle=-\frac{1}{T}\sum_{t}\mathbb{E}_{x_{1}^{t}}\big{[}\mathbb{E}_{x_{% t+1}|x_{1}^{t}}[x_{t+1}]\cdot\log f^{(1)}(x_{1}^{t})+(1-\mathbb{E}_{x_{t+1}|x_% {1}^{t}}[x_{t+1}])\cdot\log f^{(0)}(x_{1}^{t})\big{]}.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex23.m1.3d">= - divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ] ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) + ( 1 - blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ] ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.2">Let us define the distribution <math alttext="f_{*}^{(1)}(x_{1}^{t})\triangleq\mathbb{E}_{x_{t+1}|x_{1}^{t}}[x_{t+1}]" class="ltx_Math" display="inline" id="A1.1.p1.1.m1.3"><semantics id="A1.1.p1.1.m1.3a"><mrow id="A1.1.p1.1.m1.3.3" xref="A1.1.p1.1.m1.3.3.cmml"><mrow id="A1.1.p1.1.m1.2.2.1" xref="A1.1.p1.1.m1.2.2.1.cmml"><msubsup id="A1.1.p1.1.m1.2.2.1.3" 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encoding="application/x-tex" id="A1.1.p1.2.m2.4c">f_{*}^{(0)}(x_{1}^{t})\triangleq 1-f_{*}^{(1)}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.2.m2.4d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ≜ 1 - italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then, we can rewrite the loss as</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="L=\frac{1}{T}\sum_{t}\mathbb{E}_{x_{1}^{t}}\big{[}-f_{*}^{(1)}(x_{1}^{t})\cdot% \log f^{(1)}(x_{1}^{t})-f_{*}^{(0)}(x_{1}^{t})\cdot\log f^{(0)}(x_{1}^{t})\big% {]}." class="ltx_Math" display="block" id="A1.Ex24.m1.5"><semantics id="A1.Ex24.m1.5a"><mrow id="A1.Ex24.m1.5.5.1" xref="A1.Ex24.m1.5.5.1.1.cmml"><mrow id="A1.Ex24.m1.5.5.1.1" xref="A1.Ex24.m1.5.5.1.1.cmml"><mi id="A1.Ex24.m1.5.5.1.1.3" xref="A1.Ex24.m1.5.5.1.1.3.cmml">L</mi><mo id="A1.Ex24.m1.5.5.1.1.2" xref="A1.Ex24.m1.5.5.1.1.2.cmml">=</mo><mrow id="A1.Ex24.m1.5.5.1.1.1" xref="A1.Ex24.m1.5.5.1.1.1.cmml"><mfrac id="A1.Ex24.m1.5.5.1.1.1.3" xref="A1.Ex24.m1.5.5.1.1.1.3.cmml"><mn id="A1.Ex24.m1.5.5.1.1.1.3.2" xref="A1.Ex24.m1.5.5.1.1.1.3.2.cmml">1</mn><mi 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xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.3.1.3.2">superscript</csymbol><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.3.1.3.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.3.1.3.2.2">𝑓</ci><cn id="A1.Ex24.m1.4.4.1.1.cmml" type="integer" xref="A1.Ex24.m1.4.4.1.1">0</cn></apply></apply></apply><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1"><csymbol cd="ambiguous" id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1">superscript</csymbol><apply id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1"><csymbol cd="ambiguous" id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.2.1.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1">subscript</csymbol><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.2.2.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.2.2">𝑥</ci><cn id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.2.3.cmml" type="integer" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.2.3">1</cn></apply><ci id="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.3.cmml" xref="A1.Ex24.m1.5.5.1.1.1.1.1.1.1.1.4.2.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex24.m1.5c">L=\frac{1}{T}\sum_{t}\mathbb{E}_{x_{1}^{t}}\big{[}-f_{*}^{(1)}(x_{1}^{t})\cdot% \log f^{(1)}(x_{1}^{t})-f_{*}^{(0)}(x_{1}^{t})\cdot\log f^{(0)}(x_{1}^{t})\big% {]}.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex24.m1.5d">italic_L = divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ - italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) - italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.5">For every <math alttext="t\in[T]" class="ltx_Math" display="inline" id="A1.1.p1.3.m1.1"><semantics id="A1.1.p1.3.m1.1a"><mrow id="A1.1.p1.3.m1.1.2" xref="A1.1.p1.3.m1.1.2.cmml"><mi id="A1.1.p1.3.m1.1.2.2" xref="A1.1.p1.3.m1.1.2.2.cmml">t</mi><mo id="A1.1.p1.3.m1.1.2.1" xref="A1.1.p1.3.m1.1.2.1.cmml">∈</mo><mrow id="A1.1.p1.3.m1.1.2.3.2" xref="A1.1.p1.3.m1.1.2.3.1.cmml"><mo id="A1.1.p1.3.m1.1.2.3.2.1" stretchy="false" xref="A1.1.p1.3.m1.1.2.3.1.1.cmml">[</mo><mi id="A1.1.p1.3.m1.1.1" xref="A1.1.p1.3.m1.1.1.cmml">T</mi><mo id="A1.1.p1.3.m1.1.2.3.2.2" stretchy="false" xref="A1.1.p1.3.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.3.m1.1b"><apply id="A1.1.p1.3.m1.1.2.cmml" xref="A1.1.p1.3.m1.1.2"><in id="A1.1.p1.3.m1.1.2.1.cmml" xref="A1.1.p1.3.m1.1.2.1"></in><ci id="A1.1.p1.3.m1.1.2.2.cmml" xref="A1.1.p1.3.m1.1.2.2">𝑡</ci><apply id="A1.1.p1.3.m1.1.2.3.1.cmml" xref="A1.1.p1.3.m1.1.2.3.2"><csymbol cd="latexml" id="A1.1.p1.3.m1.1.2.3.1.1.cmml" xref="A1.1.p1.3.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="A1.1.p1.3.m1.1.1.cmml" xref="A1.1.p1.3.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.3.m1.1c">t\in[T]</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.3.m1.1d">italic_t ∈ [ italic_T ]</annotation></semantics></math> and every <math alttext="x_{1}^{t}\in\mathcal{X}^{t}" class="ltx_Math" display="inline" id="A1.1.p1.4.m2.1"><semantics id="A1.1.p1.4.m2.1a"><mrow id="A1.1.p1.4.m2.1.1" xref="A1.1.p1.4.m2.1.1.cmml"><msubsup id="A1.1.p1.4.m2.1.1.2" xref="A1.1.p1.4.m2.1.1.2.cmml"><mi id="A1.1.p1.4.m2.1.1.2.2.2" xref="A1.1.p1.4.m2.1.1.2.2.2.cmml">x</mi><mn id="A1.1.p1.4.m2.1.1.2.2.3" xref="A1.1.p1.4.m2.1.1.2.2.3.cmml">1</mn><mi id="A1.1.p1.4.m2.1.1.2.3" xref="A1.1.p1.4.m2.1.1.2.3.cmml">t</mi></msubsup><mo id="A1.1.p1.4.m2.1.1.1" xref="A1.1.p1.4.m2.1.1.1.cmml">∈</mo><msup id="A1.1.p1.4.m2.1.1.3" xref="A1.1.p1.4.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.1.p1.4.m2.1.1.3.2" xref="A1.1.p1.4.m2.1.1.3.2.cmml">𝒳</mi><mi id="A1.1.p1.4.m2.1.1.3.3" xref="A1.1.p1.4.m2.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.4.m2.1b"><apply id="A1.1.p1.4.m2.1.1.cmml" xref="A1.1.p1.4.m2.1.1"><in id="A1.1.p1.4.m2.1.1.1.cmml" xref="A1.1.p1.4.m2.1.1.1"></in><apply id="A1.1.p1.4.m2.1.1.2.cmml" xref="A1.1.p1.4.m2.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.4.m2.1.1.2.1.cmml" xref="A1.1.p1.4.m2.1.1.2">superscript</csymbol><apply id="A1.1.p1.4.m2.1.1.2.2.cmml" xref="A1.1.p1.4.m2.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.4.m2.1.1.2.2.1.cmml" xref="A1.1.p1.4.m2.1.1.2">subscript</csymbol><ci id="A1.1.p1.4.m2.1.1.2.2.2.cmml" xref="A1.1.p1.4.m2.1.1.2.2.2">𝑥</ci><cn id="A1.1.p1.4.m2.1.1.2.2.3.cmml" type="integer" xref="A1.1.p1.4.m2.1.1.2.2.3">1</cn></apply><ci id="A1.1.p1.4.m2.1.1.2.3.cmml" xref="A1.1.p1.4.m2.1.1.2.3">𝑡</ci></apply><apply id="A1.1.p1.4.m2.1.1.3.cmml" xref="A1.1.p1.4.m2.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.4.m2.1.1.3.1.cmml" xref="A1.1.p1.4.m2.1.1.3">superscript</csymbol><ci id="A1.1.p1.4.m2.1.1.3.2.cmml" xref="A1.1.p1.4.m2.1.1.3.2">𝒳</ci><ci id="A1.1.p1.4.m2.1.1.3.3.cmml" xref="A1.1.p1.4.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.4.m2.1c">x_{1}^{t}\in\mathcal{X}^{t}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.4.m2.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, the term inside the expectation is minimized by picking <math alttext="f^{(1)}(x_{1}^{t})=f^{(1)}_{*}(x_{1}^{t})" class="ltx_Math" display="inline" id="A1.1.p1.5.m3.4"><semantics id="A1.1.p1.5.m3.4a"><mrow id="A1.1.p1.5.m3.4.4" xref="A1.1.p1.5.m3.4.4.cmml"><mrow id="A1.1.p1.5.m3.3.3.1" xref="A1.1.p1.5.m3.3.3.1.cmml"><msup id="A1.1.p1.5.m3.3.3.1.3" xref="A1.1.p1.5.m3.3.3.1.3.cmml"><mi id="A1.1.p1.5.m3.3.3.1.3.2" xref="A1.1.p1.5.m3.3.3.1.3.2.cmml">f</mi><mrow id="A1.1.p1.5.m3.1.1.1.3" xref="A1.1.p1.5.m3.3.3.1.3.cmml"><mo id="A1.1.p1.5.m3.1.1.1.3.1" stretchy="false" xref="A1.1.p1.5.m3.3.3.1.3.cmml">(</mo><mn id="A1.1.p1.5.m3.1.1.1.1" xref="A1.1.p1.5.m3.1.1.1.1.cmml">1</mn><mo id="A1.1.p1.5.m3.1.1.1.3.2" stretchy="false" xref="A1.1.p1.5.m3.3.3.1.3.cmml">)</mo></mrow></msup><mo id="A1.1.p1.5.m3.3.3.1.2" xref="A1.1.p1.5.m3.3.3.1.2.cmml">⁢</mo><mrow id="A1.1.p1.5.m3.3.3.1.1.1" xref="A1.1.p1.5.m3.3.3.1.1.1.1.cmml"><mo id="A1.1.p1.5.m3.3.3.1.1.1.2" stretchy="false" xref="A1.1.p1.5.m3.3.3.1.1.1.1.cmml">(</mo><msubsup id="A1.1.p1.5.m3.3.3.1.1.1.1" xref="A1.1.p1.5.m3.3.3.1.1.1.1.cmml"><mi id="A1.1.p1.5.m3.3.3.1.1.1.1.2.2" xref="A1.1.p1.5.m3.3.3.1.1.1.1.2.2.cmml">x</mi><mn id="A1.1.p1.5.m3.3.3.1.1.1.1.2.3" xref="A1.1.p1.5.m3.3.3.1.1.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.5.m3.3.3.1.1.1.1.3" xref="A1.1.p1.5.m3.3.3.1.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.1.p1.5.m3.3.3.1.1.1.3" stretchy="false" xref="A1.1.p1.5.m3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.5.m3.4.4.3" xref="A1.1.p1.5.m3.4.4.3.cmml">=</mo><mrow id="A1.1.p1.5.m3.4.4.2" 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xref="A1.1.p1.5.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.5.m3.3.3.1.1.1.1.1.cmml" xref="A1.1.p1.5.m3.3.3.1.1.1">superscript</csymbol><apply id="A1.1.p1.5.m3.3.3.1.1.1.1.2.cmml" xref="A1.1.p1.5.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.5.m3.3.3.1.1.1.1.2.1.cmml" xref="A1.1.p1.5.m3.3.3.1.1.1">subscript</csymbol><ci id="A1.1.p1.5.m3.3.3.1.1.1.1.2.2.cmml" xref="A1.1.p1.5.m3.3.3.1.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.5.m3.3.3.1.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.5.m3.3.3.1.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.5.m3.3.3.1.1.1.1.3.cmml" xref="A1.1.p1.5.m3.3.3.1.1.1.1.3">𝑡</ci></apply></apply><apply id="A1.1.p1.5.m3.4.4.2.cmml" xref="A1.1.p1.5.m3.4.4.2"><times id="A1.1.p1.5.m3.4.4.2.2.cmml" xref="A1.1.p1.5.m3.4.4.2.2"></times><apply id="A1.1.p1.5.m3.4.4.2.3.cmml" xref="A1.1.p1.5.m3.4.4.2.3"><csymbol cd="ambiguous" id="A1.1.p1.5.m3.4.4.2.3.1.cmml" xref="A1.1.p1.5.m3.4.4.2.3">subscript</csymbol><apply id="A1.1.p1.5.m3.4.4.2.3.2.cmml" xref="A1.1.p1.5.m3.4.4.2.3"><csymbol cd="ambiguous" id="A1.1.p1.5.m3.4.4.2.3.2.1.cmml" xref="A1.1.p1.5.m3.4.4.2.3">superscript</csymbol><ci id="A1.1.p1.5.m3.4.4.2.3.2.2.cmml" xref="A1.1.p1.5.m3.4.4.2.3.2.2">𝑓</ci><cn id="A1.1.p1.5.m3.2.2.1.1.cmml" type="integer" xref="A1.1.p1.5.m3.2.2.1.1">1</cn></apply><times id="A1.1.p1.5.m3.4.4.2.3.3.cmml" xref="A1.1.p1.5.m3.4.4.2.3.3"></times></apply><apply id="A1.1.p1.5.m3.4.4.2.1.1.1.cmml" xref="A1.1.p1.5.m3.4.4.2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.5.m3.4.4.2.1.1.1.1.cmml" xref="A1.1.p1.5.m3.4.4.2.1.1">superscript</csymbol><apply id="A1.1.p1.5.m3.4.4.2.1.1.1.2.cmml" xref="A1.1.p1.5.m3.4.4.2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.5.m3.4.4.2.1.1.1.2.1.cmml" xref="A1.1.p1.5.m3.4.4.2.1.1">subscript</csymbol><ci id="A1.1.p1.5.m3.4.4.2.1.1.1.2.2.cmml" xref="A1.1.p1.5.m3.4.4.2.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.5.m3.4.4.2.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.5.m3.4.4.2.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.5.m3.4.4.2.1.1.1.3.cmml" xref="A1.1.p1.5.m3.4.4.2.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.5.m3.4c">f^{(1)}(x_{1}^{t})=f^{(1)}_{*}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.5.m3.4d">italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>. In fact, note that it can be rewritten as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx9"> <tbody id="A1.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle-f_{*}^{(1)}" class="ltx_Math" display="inline" id="A1.Ex25.m1.1"><semantics id="A1.Ex25.m1.1a"><mrow id="A1.Ex25.m1.1.2" xref="A1.Ex25.m1.1.2.cmml"><mo id="A1.Ex25.m1.1.2a" xref="A1.Ex25.m1.1.2.cmml">−</mo><msubsup id="A1.Ex25.m1.1.2.2" xref="A1.Ex25.m1.1.2.2.cmml"><mi id="A1.Ex25.m1.1.2.2.2.2" xref="A1.Ex25.m1.1.2.2.2.2.cmml">f</mi><mo id="A1.Ex25.m1.1.2.2.2.3" xref="A1.Ex25.m1.1.2.2.2.3.cmml">∗</mo><mrow id="A1.Ex25.m1.1.1.1.3" xref="A1.Ex25.m1.1.2.2.cmml"><mo id="A1.Ex25.m1.1.1.1.3.1" stretchy="false" xref="A1.Ex25.m1.1.2.2.cmml">(</mo><mn id="A1.Ex25.m1.1.1.1.1" xref="A1.Ex25.m1.1.1.1.1.cmml">1</mn><mo id="A1.Ex25.m1.1.1.1.3.2" stretchy="false" xref="A1.Ex25.m1.1.2.2.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex25.m1.1b"><apply id="A1.Ex25.m1.1.2.cmml" xref="A1.Ex25.m1.1.2"><minus id="A1.Ex25.m1.1.2.1.cmml" xref="A1.Ex25.m1.1.2"></minus><apply id="A1.Ex25.m1.1.2.2.cmml" xref="A1.Ex25.m1.1.2.2"><csymbol cd="ambiguous" id="A1.Ex25.m1.1.2.2.1.cmml" xref="A1.Ex25.m1.1.2.2">superscript</csymbol><apply id="A1.Ex25.m1.1.2.2.2.cmml" xref="A1.Ex25.m1.1.2.2"><csymbol cd="ambiguous" id="A1.Ex25.m1.1.2.2.2.1.cmml" xref="A1.Ex25.m1.1.2.2">subscript</csymbol><ci id="A1.Ex25.m1.1.2.2.2.2.cmml" xref="A1.Ex25.m1.1.2.2.2.2">𝑓</ci><times id="A1.Ex25.m1.1.2.2.2.3.cmml" xref="A1.Ex25.m1.1.2.2.2.3"></times></apply><cn id="A1.Ex25.m1.1.1.1.1.cmml" type="integer" xref="A1.Ex25.m1.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex25.m1.1c">\displaystyle-f_{*}^{(1)}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex25.m1.1d">- italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(x_{1}^{t})\cdot\log f^{(1)}(x_{1}^{t})-f_{*}^{(0)}(x_{1}^{t})% \cdot\log f^{(0)}(x_{1}^{t})" class="ltx_Math" display="inline" id="A1.Ex25.m2.7"><semantics id="A1.Ex25.m2.7a"><mrow id="A1.Ex25.m2.7.7" xref="A1.Ex25.m2.7.7.cmml"><mrow id="A1.Ex25.m2.5.5.2" xref="A1.Ex25.m2.5.5.2.cmml"><mrow id="A1.Ex25.m2.4.4.1.1" xref="A1.Ex25.m2.4.4.1.1.cmml"><mrow id="A1.Ex25.m2.4.4.1.1.1.1" xref="A1.Ex25.m2.4.4.1.1.1.1.1.cmml"><mo id="A1.Ex25.m2.4.4.1.1.1.1.2" stretchy="false" xref="A1.Ex25.m2.4.4.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex25.m2.4.4.1.1.1.1.1" xref="A1.Ex25.m2.4.4.1.1.1.1.1.cmml"><mi id="A1.Ex25.m2.4.4.1.1.1.1.1.2.2" xref="A1.Ex25.m2.4.4.1.1.1.1.1.2.2.cmml">x</mi><mn id="A1.Ex25.m2.4.4.1.1.1.1.1.2.3" xref="A1.Ex25.m2.4.4.1.1.1.1.1.2.3.cmml">1</mn><mi id="A1.Ex25.m2.4.4.1.1.1.1.1.3" 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f^{(0)}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.Ex25.m2.7d">( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) - italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ⋅ roman_log italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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id="A1.Ex26.m1.15.15.1.1.1.1.3.2.2" xref="A1.Ex26.m1.15.15.1.1.1.1.3.2.2.cmml">f</mi><mo id="A1.Ex26.m1.15.15.1.1.1.1.3.2.3" xref="A1.Ex26.m1.15.15.1.1.1.1.3.2.3.cmml">∗</mo><mrow id="A1.Ex26.m1.1.1.1.3" xref="A1.Ex26.m1.15.15.1.1.1.1.3.cmml"><mo id="A1.Ex26.m1.1.1.1.3.1" stretchy="false" xref="A1.Ex26.m1.15.15.1.1.1.1.3.cmml">(</mo><mn id="A1.Ex26.m1.1.1.1.1" xref="A1.Ex26.m1.1.1.1.1.cmml">1</mn><mo id="A1.Ex26.m1.1.1.1.3.2" stretchy="false" xref="A1.Ex26.m1.15.15.1.1.1.1.3.cmml">)</mo></mrow></msubsup><mo id="A1.Ex26.m1.15.15.1.1.1.1.2" xref="A1.Ex26.m1.15.15.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex26.m1.15.15.1.1.1.1.1.1" xref="A1.Ex26.m1.15.15.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex26.m1.15.15.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex26.m1.15.15.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex26.m1.15.15.1.1.1.1.1.1.1" xref="A1.Ex26.m1.15.15.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex26.m1.15.15.1.1.1.1.1.1.1.2.2" xref="A1.Ex26.m1.15.15.1.1.1.1.1.1.1.2.2.cmml">x</mi><mn 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xref="A1.Ex26.m1.3.3.2.4.2.2.cmml">f</mi><mo id="A1.Ex26.m1.3.3.2.4.2.3" xref="A1.Ex26.m1.3.3.2.4.2.3.cmml">∗</mo><mrow id="A1.Ex26.m1.2.2.1.1.1.3" xref="A1.Ex26.m1.3.3.2.4.cmml"><mo id="A1.Ex26.m1.2.2.1.1.1.3.1" stretchy="false" xref="A1.Ex26.m1.3.3.2.4.cmml">(</mo><mn id="A1.Ex26.m1.2.2.1.1.1.1" xref="A1.Ex26.m1.2.2.1.1.1.1.cmml">1</mn><mo id="A1.Ex26.m1.2.2.1.1.1.3.2" stretchy="false" xref="A1.Ex26.m1.3.3.2.4.cmml">)</mo></mrow></msubsup><mo id="A1.Ex26.m1.3.3.2.3" xref="A1.Ex26.m1.3.3.2.3.cmml">⁢</mo><mrow id="A1.Ex26.m1.3.3.2.2.1" xref="A1.Ex26.m1.3.3.2.2.1.1.cmml"><mo id="A1.Ex26.m1.3.3.2.2.1.2" stretchy="false" xref="A1.Ex26.m1.3.3.2.2.1.1.cmml">(</mo><msubsup id="A1.Ex26.m1.3.3.2.2.1.1" xref="A1.Ex26.m1.3.3.2.2.1.1.cmml"><mi id="A1.Ex26.m1.3.3.2.2.1.1.2.2" xref="A1.Ex26.m1.3.3.2.2.1.1.2.2.cmml">x</mi><mn id="A1.Ex26.m1.3.3.2.2.1.1.2.3" xref="A1.Ex26.m1.3.3.2.2.1.1.2.3.cmml">1</mn><mi id="A1.Ex26.m1.3.3.2.2.1.1.3" xref="A1.Ex26.m1.3.3.2.2.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex26.m1.3.3.2.2.1.3" stretchy="false" xref="A1.Ex26.m1.3.3.2.2.1.1.cmml">)</mo></mrow></mrow><mrow id="A1.Ex26.m1.5.5.4" xref="A1.Ex26.m1.5.5.4.cmml"><msup id="A1.Ex26.m1.5.5.4.4" xref="A1.Ex26.m1.5.5.4.4.cmml"><mi id="A1.Ex26.m1.5.5.4.4.2" xref="A1.Ex26.m1.5.5.4.4.2.cmml">f</mi><mrow id="A1.Ex26.m1.4.4.3.1.1.3" xref="A1.Ex26.m1.5.5.4.4.cmml"><mo id="A1.Ex26.m1.4.4.3.1.1.3.1" stretchy="false" xref="A1.Ex26.m1.5.5.4.4.cmml">(</mo><mn id="A1.Ex26.m1.4.4.3.1.1.1" xref="A1.Ex26.m1.4.4.3.1.1.1.cmml">1</mn><mo id="A1.Ex26.m1.4.4.3.1.1.3.2" stretchy="false" xref="A1.Ex26.m1.5.5.4.4.cmml">)</mo></mrow></msup><mo id="A1.Ex26.m1.5.5.4.3" xref="A1.Ex26.m1.5.5.4.3.cmml">⁢</mo><mrow id="A1.Ex26.m1.5.5.4.2.1" xref="A1.Ex26.m1.5.5.4.2.1.1.cmml"><mo id="A1.Ex26.m1.5.5.4.2.1.2" stretchy="false" xref="A1.Ex26.m1.5.5.4.2.1.1.cmml">(</mo><msubsup id="A1.Ex26.m1.5.5.4.2.1.1" xref="A1.Ex26.m1.5.5.4.2.1.1.cmml"><mi id="A1.Ex26.m1.5.5.4.2.1.1.2.2" xref="A1.Ex26.m1.5.5.4.2.1.1.2.2.cmml">x</mi><mn id="A1.Ex26.m1.5.5.4.2.1.1.2.3" xref="A1.Ex26.m1.5.5.4.2.1.1.2.3.cmml">1</mn><mi id="A1.Ex26.m1.5.5.4.2.1.1.3" xref="A1.Ex26.m1.5.5.4.2.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex26.m1.5.5.4.2.1.3" stretchy="false" xref="A1.Ex26.m1.5.5.4.2.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><mo id="A1.Ex26.m1.16.16.2.2.3" xref="A1.Ex26.m1.16.16.2.2.3.cmml">+</mo><mrow id="A1.Ex26.m1.16.16.2.2.2" xref="A1.Ex26.m1.16.16.2.2.2.cmml"><mrow id="A1.Ex26.m1.16.16.2.2.2.1" xref="A1.Ex26.m1.16.16.2.2.2.1.cmml"><msubsup id="A1.Ex26.m1.16.16.2.2.2.1.3" xref="A1.Ex26.m1.16.16.2.2.2.1.3.cmml"><mi id="A1.Ex26.m1.16.16.2.2.2.1.3.2.2" xref="A1.Ex26.m1.16.16.2.2.2.1.3.2.2.cmml">f</mi><mo id="A1.Ex26.m1.16.16.2.2.2.1.3.2.3" xref="A1.Ex26.m1.16.16.2.2.2.1.3.2.3.cmml">∗</mo><mrow id="A1.Ex26.m1.6.6.1.3" xref="A1.Ex26.m1.16.16.2.2.2.1.3.cmml"><mo id="A1.Ex26.m1.6.6.1.3.1" stretchy="false" xref="A1.Ex26.m1.16.16.2.2.2.1.3.cmml">(</mo><mn id="A1.Ex26.m1.6.6.1.1" xref="A1.Ex26.m1.6.6.1.1.cmml">0</mn><mo id="A1.Ex26.m1.6.6.1.3.2" stretchy="false" xref="A1.Ex26.m1.16.16.2.2.2.1.3.cmml">)</mo></mrow></msubsup><mo id="A1.Ex26.m1.16.16.2.2.2.1.2" xref="A1.Ex26.m1.16.16.2.2.2.1.2.cmml">⁢</mo><mrow id="A1.Ex26.m1.16.16.2.2.2.1.1.1" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.cmml"><mo id="A1.Ex26.m1.16.16.2.2.2.1.1.1.2" stretchy="false" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex26.m1.16.16.2.2.2.1.1.1.1" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.cmml"><mi id="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.2.2" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.2.2.cmml">x</mi><mn id="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.2.3" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.2.3.cmml">1</mn><mi id="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.3" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex26.m1.16.16.2.2.2.1.1.1.3" rspace="0.055em" stretchy="false" xref="A1.Ex26.m1.16.16.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex26.m1.16.16.2.2.2.2" rspace="0.222em" xref="A1.Ex26.m1.16.16.2.2.2.2.cmml">⋅</mo><mrow id="A1.Ex26.m1.16.16.2.2.2.3" xref="A1.Ex26.m1.16.16.2.2.2.3.cmml"><mi id="A1.Ex26.m1.16.16.2.2.2.3.1" xref="A1.Ex26.m1.16.16.2.2.2.3.1.cmml">log</mi><mo id="A1.Ex26.m1.16.16.2.2.2.3a" lspace="0.167em" xref="A1.Ex26.m1.16.16.2.2.2.3.cmml">⁡</mo><mstyle displaystyle="true" id="A1.Ex26.m1.10.10" xref="A1.Ex26.m1.10.10.cmml"><mfrac id="A1.Ex26.m1.10.10a" xref="A1.Ex26.m1.10.10.cmml"><mrow id="A1.Ex26.m1.8.8.2" xref="A1.Ex26.m1.8.8.2.cmml"><msubsup id="A1.Ex26.m1.8.8.2.4" xref="A1.Ex26.m1.8.8.2.4.cmml"><mi id="A1.Ex26.m1.8.8.2.4.2.2" xref="A1.Ex26.m1.8.8.2.4.2.2.cmml">f</mi><mo id="A1.Ex26.m1.8.8.2.4.2.3" xref="A1.Ex26.m1.8.8.2.4.2.3.cmml">∗</mo><mrow id="A1.Ex26.m1.7.7.1.1.1.3" xref="A1.Ex26.m1.8.8.2.4.cmml"><mo id="A1.Ex26.m1.7.7.1.1.1.3.1" stretchy="false" xref="A1.Ex26.m1.8.8.2.4.cmml">(</mo><mn id="A1.Ex26.m1.7.7.1.1.1.1" xref="A1.Ex26.m1.7.7.1.1.1.1.cmml">0</mn><mo id="A1.Ex26.m1.7.7.1.1.1.3.2" stretchy="false" xref="A1.Ex26.m1.8.8.2.4.cmml">)</mo></mrow></msubsup><mo id="A1.Ex26.m1.8.8.2.3" xref="A1.Ex26.m1.8.8.2.3.cmml">⁢</mo><mrow id="A1.Ex26.m1.8.8.2.2.1" xref="A1.Ex26.m1.8.8.2.2.1.1.cmml"><mo id="A1.Ex26.m1.8.8.2.2.1.2" stretchy="false" xref="A1.Ex26.m1.8.8.2.2.1.1.cmml">(</mo><msubsup id="A1.Ex26.m1.8.8.2.2.1.1" xref="A1.Ex26.m1.8.8.2.2.1.1.cmml"><mi id="A1.Ex26.m1.8.8.2.2.1.1.2.2" xref="A1.Ex26.m1.8.8.2.2.1.1.2.2.cmml">x</mi><mn id="A1.Ex26.m1.8.8.2.2.1.1.2.3" xref="A1.Ex26.m1.8.8.2.2.1.1.2.3.cmml">1</mn><mi id="A1.Ex26.m1.8.8.2.2.1.1.3" xref="A1.Ex26.m1.8.8.2.2.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex26.m1.8.8.2.2.1.3" stretchy="false" xref="A1.Ex26.m1.8.8.2.2.1.1.cmml">)</mo></mrow></mrow><mrow id="A1.Ex26.m1.10.10.4" xref="A1.Ex26.m1.10.10.4.cmml"><msup id="A1.Ex26.m1.10.10.4.4" xref="A1.Ex26.m1.10.10.4.4.cmml"><mi id="A1.Ex26.m1.10.10.4.4.2" xref="A1.Ex26.m1.10.10.4.4.2.cmml">f</mi><mrow id="A1.Ex26.m1.9.9.3.1.1.3" xref="A1.Ex26.m1.10.10.4.4.cmml"><mo id="A1.Ex26.m1.9.9.3.1.1.3.1" stretchy="false" xref="A1.Ex26.m1.10.10.4.4.cmml">(</mo><mn id="A1.Ex26.m1.9.9.3.1.1.1" xref="A1.Ex26.m1.9.9.3.1.1.1.cmml">0</mn><mo id="A1.Ex26.m1.9.9.3.1.1.3.2" stretchy="false" xref="A1.Ex26.m1.10.10.4.4.cmml">)</mo></mrow></msup><mo id="A1.Ex26.m1.10.10.4.3" xref="A1.Ex26.m1.10.10.4.3.cmml">⁢</mo><mrow id="A1.Ex26.m1.10.10.4.2.1" xref="A1.Ex26.m1.10.10.4.2.1.1.cmml"><mo id="A1.Ex26.m1.10.10.4.2.1.2" stretchy="false" xref="A1.Ex26.m1.10.10.4.2.1.1.cmml">(</mo><msubsup id="A1.Ex26.m1.10.10.4.2.1.1" xref="A1.Ex26.m1.10.10.4.2.1.1.cmml"><mi id="A1.Ex26.m1.10.10.4.2.1.1.2.2" xref="A1.Ex26.m1.10.10.4.2.1.1.2.2.cmml">x</mi><mn id="A1.Ex26.m1.10.10.4.2.1.1.2.3" xref="A1.Ex26.m1.10.10.4.2.1.1.2.3.cmml">1</mn><mi id="A1.Ex26.m1.10.10.4.2.1.1.3" xref="A1.Ex26.m1.10.10.4.2.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex26.m1.10.10.4.2.1.3" stretchy="false" xref="A1.Ex26.m1.10.10.4.2.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow></mrow><mo id="A1.Ex26.m1.20.20.6.7" 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xref="A1.Ex26.m1.20.20.6.6.2.1.1.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex26.m1.20c">\displaystyle=f_{*}^{(1)}(x_{1}^{t})\cdot\log\frac{f_{*}^{(1)}(x_{1}^{t})}{f^{% (1)}(x_{1}^{t})}+f_{*}^{(0)}(x_{1}^{t})\cdot\log\frac{f_{*}^{(0)}(x_{1}^{t})}{% f^{(0)}(x_{1}^{t})}-f_{*}^{(1)}(x_{1}^{t})\log f_{*}^{(1)}(x_{1}^{t})-f_{*}^{(% 0)}(x_{1}^{t})\log f_{*}^{(0)}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.Ex26.m1.20d">= italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ⋅ roman_log divide start_ARG italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG + italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ⋅ roman_log divide start_ARG italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_f start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG - italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) roman_log italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) - italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) roman_log italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=D_{\rm KL}\left(f_{*}(x_{1}^{t})\|f(x_{1}^{t})\right)+H(f_{*}(x_% 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xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.3.2">𝑓</ci><times id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.3.3.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.3.3"></times></apply><apply id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1">subscript</csymbol><ci id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.2">𝑥</ci><cn id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.3">1</cn></apply><ci id="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.3.cmml" xref="A1.Ex27.m1.1.1.1.1.2.2.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex27.m1.1c">\displaystyle=D_{\rm KL}\left(f_{*}(x_{1}^{t})\|f(x_{1}^{t})\right)+H(f_{*}(x_% {1}^{t})),</annotation><annotation encoding="application/x-llamapun" id="A1.Ex27.m1.1d">= italic_D start_POSTSUBSCRIPT roman_KL end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∥ italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) + italic_H ( italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.13">which is 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xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A1.1.p1.6.m1.1.1.1.1.1.3" xref="A1.1.p1.6.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.6.m1.1.1.2" xref="A1.1.p1.6.m1.1.1.2.cmml">=</mo><mn id="A1.1.p1.6.m1.1.1.3" xref="A1.1.p1.6.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.6.m1.1b"><apply id="A1.1.p1.6.m1.1.1.cmml" xref="A1.1.p1.6.m1.1.1"><eq id="A1.1.p1.6.m1.1.1.2.cmml" xref="A1.1.p1.6.m1.1.1.2"></eq><apply id="A1.1.p1.6.m1.1.1.1.cmml" xref="A1.1.p1.6.m1.1.1.1"><times id="A1.1.p1.6.m1.1.1.1.2.cmml" xref="A1.1.p1.6.m1.1.1.1.2"></times><apply id="A1.1.p1.6.m1.1.1.1.3.cmml" xref="A1.1.p1.6.m1.1.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.6.m1.1.1.1.3.1.cmml" xref="A1.1.p1.6.m1.1.1.1.3">subscript</csymbol><ci id="A1.1.p1.6.m1.1.1.1.3.2.cmml" xref="A1.1.p1.6.m1.1.1.1.3.2">𝐷</ci><ci id="A1.1.p1.6.m1.1.1.1.3.3.cmml" xref="A1.1.p1.6.m1.1.1.1.3.3">KL</ci></apply><apply id="A1.1.p1.6.m1.1.1.1.1.1.1.cmml" 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xref="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply><apply id="A1.1.p1.6.m1.1.1.1.1.1.1.2.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2"><times id="A1.1.p1.6.m1.1.1.1.1.1.1.2.2.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.2"></times><ci id="A1.1.p1.6.m1.1.1.1.1.1.1.2.3.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.3">𝑓</ci><apply id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.1.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1">superscript</csymbol><apply id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.2.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.2.1.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1">subscript</csymbol><ci id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.2.2.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.3.cmml" xref="A1.1.p1.6.m1.1.1.1.1.1.1.2.1.1.1.3">𝑡</ci></apply></apply></apply></apply><cn id="A1.1.p1.6.m1.1.1.3.cmml" type="integer" xref="A1.1.p1.6.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.6.m1.1c">D_{\rm KL}\left(f_{*}(x_{1}^{t})\|f(x_{1}^{t})\right)=0</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.6.m1.1d">italic_D start_POSTSUBSCRIPT roman_KL end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∥ italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ) = 0</annotation></semantics></math>, i.e., when <math alttext="f(x_{1}^{t})=f_{*}(x_{1}^{t})" class="ltx_Math" display="inline" id="A1.1.p1.7.m2.2"><semantics id="A1.1.p1.7.m2.2a"><mrow id="A1.1.p1.7.m2.2.2" xref="A1.1.p1.7.m2.2.2.cmml"><mrow id="A1.1.p1.7.m2.1.1.1" xref="A1.1.p1.7.m2.1.1.1.cmml"><mi id="A1.1.p1.7.m2.1.1.1.3" xref="A1.1.p1.7.m2.1.1.1.3.cmml">f</mi><mo id="A1.1.p1.7.m2.1.1.1.2" xref="A1.1.p1.7.m2.1.1.1.2.cmml">⁢</mo><mrow id="A1.1.p1.7.m2.1.1.1.1.1" xref="A1.1.p1.7.m2.1.1.1.1.1.1.cmml"><mo id="A1.1.p1.7.m2.1.1.1.1.1.2" stretchy="false" xref="A1.1.p1.7.m2.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.1.p1.7.m2.1.1.1.1.1.1" xref="A1.1.p1.7.m2.1.1.1.1.1.1.cmml"><mi id="A1.1.p1.7.m2.1.1.1.1.1.1.2.2" xref="A1.1.p1.7.m2.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="A1.1.p1.7.m2.1.1.1.1.1.1.2.3" xref="A1.1.p1.7.m2.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.7.m2.1.1.1.1.1.1.3" xref="A1.1.p1.7.m2.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.1.p1.7.m2.1.1.1.1.1.3" stretchy="false" xref="A1.1.p1.7.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.7.m2.2.2.3" xref="A1.1.p1.7.m2.2.2.3.cmml">=</mo><mrow id="A1.1.p1.7.m2.2.2.2" xref="A1.1.p1.7.m2.2.2.2.cmml"><msub id="A1.1.p1.7.m2.2.2.2.3" xref="A1.1.p1.7.m2.2.2.2.3.cmml"><mi id="A1.1.p1.7.m2.2.2.2.3.2" xref="A1.1.p1.7.m2.2.2.2.3.2.cmml">f</mi><mo id="A1.1.p1.7.m2.2.2.2.3.3" xref="A1.1.p1.7.m2.2.2.2.3.3.cmml">∗</mo></msub><mo id="A1.1.p1.7.m2.2.2.2.2" xref="A1.1.p1.7.m2.2.2.2.2.cmml">⁢</mo><mrow id="A1.1.p1.7.m2.2.2.2.1.1" xref="A1.1.p1.7.m2.2.2.2.1.1.1.cmml"><mo id="A1.1.p1.7.m2.2.2.2.1.1.2" stretchy="false" xref="A1.1.p1.7.m2.2.2.2.1.1.1.cmml">(</mo><msubsup id="A1.1.p1.7.m2.2.2.2.1.1.1" xref="A1.1.p1.7.m2.2.2.2.1.1.1.cmml"><mi id="A1.1.p1.7.m2.2.2.2.1.1.1.2.2" xref="A1.1.p1.7.m2.2.2.2.1.1.1.2.2.cmml">x</mi><mn id="A1.1.p1.7.m2.2.2.2.1.1.1.2.3" xref="A1.1.p1.7.m2.2.2.2.1.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.7.m2.2.2.2.1.1.1.3" xref="A1.1.p1.7.m2.2.2.2.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.1.p1.7.m2.2.2.2.1.1.3" stretchy="false" xref="A1.1.p1.7.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.7.m2.2b"><apply id="A1.1.p1.7.m2.2.2.cmml" xref="A1.1.p1.7.m2.2.2"><eq id="A1.1.p1.7.m2.2.2.3.cmml" xref="A1.1.p1.7.m2.2.2.3"></eq><apply id="A1.1.p1.7.m2.1.1.1.cmml" xref="A1.1.p1.7.m2.1.1.1"><times id="A1.1.p1.7.m2.1.1.1.2.cmml" xref="A1.1.p1.7.m2.1.1.1.2"></times><ci id="A1.1.p1.7.m2.1.1.1.3.cmml" xref="A1.1.p1.7.m2.1.1.1.3">𝑓</ci><apply id="A1.1.p1.7.m2.1.1.1.1.1.1.cmml" xref="A1.1.p1.7.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.7.m2.1.1.1.1.1.1.1.cmml" xref="A1.1.p1.7.m2.1.1.1.1.1">superscript</csymbol><apply id="A1.1.p1.7.m2.1.1.1.1.1.1.2.cmml" xref="A1.1.p1.7.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.7.m2.1.1.1.1.1.1.2.1.cmml" xref="A1.1.p1.7.m2.1.1.1.1.1">subscript</csymbol><ci id="A1.1.p1.7.m2.1.1.1.1.1.1.2.2.cmml" xref="A1.1.p1.7.m2.1.1.1.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.7.m2.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.7.m2.1.1.1.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.7.m2.1.1.1.1.1.1.3.cmml" xref="A1.1.p1.7.m2.1.1.1.1.1.1.3">𝑡</ci></apply></apply><apply id="A1.1.p1.7.m2.2.2.2.cmml" xref="A1.1.p1.7.m2.2.2.2"><times id="A1.1.p1.7.m2.2.2.2.2.cmml" xref="A1.1.p1.7.m2.2.2.2.2"></times><apply id="A1.1.p1.7.m2.2.2.2.3.cmml" xref="A1.1.p1.7.m2.2.2.2.3"><csymbol cd="ambiguous" id="A1.1.p1.7.m2.2.2.2.3.1.cmml" xref="A1.1.p1.7.m2.2.2.2.3">subscript</csymbol><ci id="A1.1.p1.7.m2.2.2.2.3.2.cmml" xref="A1.1.p1.7.m2.2.2.2.3.2">𝑓</ci><times id="A1.1.p1.7.m2.2.2.2.3.3.cmml" xref="A1.1.p1.7.m2.2.2.2.3.3"></times></apply><apply id="A1.1.p1.7.m2.2.2.2.1.1.1.cmml" xref="A1.1.p1.7.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.7.m2.2.2.2.1.1.1.1.cmml" xref="A1.1.p1.7.m2.2.2.2.1.1">superscript</csymbol><apply id="A1.1.p1.7.m2.2.2.2.1.1.1.2.cmml" xref="A1.1.p1.7.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="A1.1.p1.7.m2.2.2.2.1.1.1.2.1.cmml" xref="A1.1.p1.7.m2.2.2.2.1.1">subscript</csymbol><ci id="A1.1.p1.7.m2.2.2.2.1.1.1.2.2.cmml" xref="A1.1.p1.7.m2.2.2.2.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.7.m2.2.2.2.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.7.m2.2.2.2.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.7.m2.2.2.2.1.1.1.3.cmml" xref="A1.1.p1.7.m2.2.2.2.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.7.m2.2c">f(x_{1}^{t})=f_{*}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.7.m2.2d">italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>. We will now show that <math alttext="f_{*}(x_{1}^{t})" class="ltx_Math" display="inline" id="A1.1.p1.8.m3.1"><semantics id="A1.1.p1.8.m3.1a"><mrow id="A1.1.p1.8.m3.1.1" xref="A1.1.p1.8.m3.1.1.cmml"><msub id="A1.1.p1.8.m3.1.1.3" xref="A1.1.p1.8.m3.1.1.3.cmml"><mi id="A1.1.p1.8.m3.1.1.3.2" xref="A1.1.p1.8.m3.1.1.3.2.cmml">f</mi><mo id="A1.1.p1.8.m3.1.1.3.3" xref="A1.1.p1.8.m3.1.1.3.3.cmml">∗</mo></msub><mo id="A1.1.p1.8.m3.1.1.2" xref="A1.1.p1.8.m3.1.1.2.cmml">⁢</mo><mrow id="A1.1.p1.8.m3.1.1.1.1" xref="A1.1.p1.8.m3.1.1.1.1.1.cmml"><mo id="A1.1.p1.8.m3.1.1.1.1.2" stretchy="false" xref="A1.1.p1.8.m3.1.1.1.1.1.cmml">(</mo><msubsup id="A1.1.p1.8.m3.1.1.1.1.1" xref="A1.1.p1.8.m3.1.1.1.1.1.cmml"><mi id="A1.1.p1.8.m3.1.1.1.1.1.2.2" xref="A1.1.p1.8.m3.1.1.1.1.1.2.2.cmml">x</mi><mn id="A1.1.p1.8.m3.1.1.1.1.1.2.3" xref="A1.1.p1.8.m3.1.1.1.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.8.m3.1.1.1.1.1.3" xref="A1.1.p1.8.m3.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.1.p1.8.m3.1.1.1.1.3" stretchy="false" xref="A1.1.p1.8.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.8.m3.1b"><apply id="A1.1.p1.8.m3.1.1.cmml" xref="A1.1.p1.8.m3.1.1"><times id="A1.1.p1.8.m3.1.1.2.cmml" xref="A1.1.p1.8.m3.1.1.2"></times><apply id="A1.1.p1.8.m3.1.1.3.cmml" xref="A1.1.p1.8.m3.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.8.m3.1.1.3.1.cmml" xref="A1.1.p1.8.m3.1.1.3">subscript</csymbol><ci id="A1.1.p1.8.m3.1.1.3.2.cmml" xref="A1.1.p1.8.m3.1.1.3.2">𝑓</ci><times id="A1.1.p1.8.m3.1.1.3.3.cmml" xref="A1.1.p1.8.m3.1.1.3.3"></times></apply><apply id="A1.1.p1.8.m3.1.1.1.1.1.cmml" xref="A1.1.p1.8.m3.1.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.8.m3.1.1.1.1.1.1.cmml" xref="A1.1.p1.8.m3.1.1.1.1">superscript</csymbol><apply id="A1.1.p1.8.m3.1.1.1.1.1.2.cmml" xref="A1.1.p1.8.m3.1.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.8.m3.1.1.1.1.1.2.1.cmml" xref="A1.1.p1.8.m3.1.1.1.1">subscript</csymbol><ci id="A1.1.p1.8.m3.1.1.1.1.1.2.2.cmml" xref="A1.1.p1.8.m3.1.1.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.8.m3.1.1.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.8.m3.1.1.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.8.m3.1.1.1.1.1.3.cmml" xref="A1.1.p1.8.m3.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.8.m3.1c">f_{*}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.8.m3.1d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> is precisely the add-<math alttext="\beta" class="ltx_Math" display="inline" id="A1.1.p1.9.m4.1"><semantics id="A1.1.p1.9.m4.1a"><mi id="A1.1.p1.9.m4.1.1" xref="A1.1.p1.9.m4.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A1.1.p1.9.m4.1b"><ci id="A1.1.p1.9.m4.1.1.cmml" xref="A1.1.p1.9.m4.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.9.m4.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.9.m4.1d">italic_β</annotation></semantics></math> estimator. Consider any context <math alttext="i_{1}^{k}" class="ltx_Math" display="inline" id="A1.1.p1.10.m5.1"><semantics id="A1.1.p1.10.m5.1a"><msubsup id="A1.1.p1.10.m5.1.1" xref="A1.1.p1.10.m5.1.1.cmml"><mi id="A1.1.p1.10.m5.1.1.2.2" xref="A1.1.p1.10.m5.1.1.2.2.cmml">i</mi><mn id="A1.1.p1.10.m5.1.1.2.3" xref="A1.1.p1.10.m5.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.10.m5.1.1.3" xref="A1.1.p1.10.m5.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.1.p1.10.m5.1b"><apply id="A1.1.p1.10.m5.1.1.cmml" xref="A1.1.p1.10.m5.1.1"><csymbol cd="ambiguous" id="A1.1.p1.10.m5.1.1.1.cmml" xref="A1.1.p1.10.m5.1.1">superscript</csymbol><apply id="A1.1.p1.10.m5.1.1.2.cmml" xref="A1.1.p1.10.m5.1.1"><csymbol cd="ambiguous" id="A1.1.p1.10.m5.1.1.2.1.cmml" xref="A1.1.p1.10.m5.1.1">subscript</csymbol><ci id="A1.1.p1.10.m5.1.1.2.2.cmml" xref="A1.1.p1.10.m5.1.1.2.2">𝑖</ci><cn id="A1.1.p1.10.m5.1.1.2.3.cmml" type="integer" xref="A1.1.p1.10.m5.1.1.2.3">1</cn></apply><ci id="A1.1.p1.10.m5.1.1.3.cmml" xref="A1.1.p1.10.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.10.m5.1c">i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.10.m5.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> and any sequence <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="A1.1.p1.11.m6.1"><semantics id="A1.1.p1.11.m6.1a"><msubsup id="A1.1.p1.11.m6.1.1" xref="A1.1.p1.11.m6.1.1.cmml"><mi id="A1.1.p1.11.m6.1.1.2.2" xref="A1.1.p1.11.m6.1.1.2.2.cmml">x</mi><mn id="A1.1.p1.11.m6.1.1.2.3" xref="A1.1.p1.11.m6.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.11.m6.1.1.3" xref="A1.1.p1.11.m6.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.1.p1.11.m6.1b"><apply id="A1.1.p1.11.m6.1.1.cmml" xref="A1.1.p1.11.m6.1.1"><csymbol cd="ambiguous" id="A1.1.p1.11.m6.1.1.1.cmml" xref="A1.1.p1.11.m6.1.1">superscript</csymbol><apply id="A1.1.p1.11.m6.1.1.2.cmml" xref="A1.1.p1.11.m6.1.1"><csymbol cd="ambiguous" id="A1.1.p1.11.m6.1.1.2.1.cmml" xref="A1.1.p1.11.m6.1.1">subscript</csymbol><ci id="A1.1.p1.11.m6.1.1.2.2.cmml" xref="A1.1.p1.11.m6.1.1.2.2">𝑥</ci><cn id="A1.1.p1.11.m6.1.1.2.3.cmml" type="integer" xref="A1.1.p1.11.m6.1.1.2.3">1</cn></apply><ci id="A1.1.p1.11.m6.1.1.3.cmml" xref="A1.1.p1.11.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.11.m6.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.11.m6.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="x_{t-k+1}^{t}=i_{1}^{k}" class="ltx_Math" display="inline" id="A1.1.p1.12.m7.1"><semantics id="A1.1.p1.12.m7.1a"><mrow id="A1.1.p1.12.m7.1.1" xref="A1.1.p1.12.m7.1.1.cmml"><msubsup id="A1.1.p1.12.m7.1.1.2" xref="A1.1.p1.12.m7.1.1.2.cmml"><mi id="A1.1.p1.12.m7.1.1.2.2.2" xref="A1.1.p1.12.m7.1.1.2.2.2.cmml">x</mi><mrow id="A1.1.p1.12.m7.1.1.2.2.3" xref="A1.1.p1.12.m7.1.1.2.2.3.cmml"><mrow id="A1.1.p1.12.m7.1.1.2.2.3.2" xref="A1.1.p1.12.m7.1.1.2.2.3.2.cmml"><mi id="A1.1.p1.12.m7.1.1.2.2.3.2.2" xref="A1.1.p1.12.m7.1.1.2.2.3.2.2.cmml">t</mi><mo id="A1.1.p1.12.m7.1.1.2.2.3.2.1" xref="A1.1.p1.12.m7.1.1.2.2.3.2.1.cmml">−</mo><mi id="A1.1.p1.12.m7.1.1.2.2.3.2.3" xref="A1.1.p1.12.m7.1.1.2.2.3.2.3.cmml">k</mi></mrow><mo id="A1.1.p1.12.m7.1.1.2.2.3.1" xref="A1.1.p1.12.m7.1.1.2.2.3.1.cmml">+</mo><mn id="A1.1.p1.12.m7.1.1.2.2.3.3" xref="A1.1.p1.12.m7.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="A1.1.p1.12.m7.1.1.2.3" xref="A1.1.p1.12.m7.1.1.2.3.cmml">t</mi></msubsup><mo id="A1.1.p1.12.m7.1.1.1" xref="A1.1.p1.12.m7.1.1.1.cmml">=</mo><msubsup id="A1.1.p1.12.m7.1.1.3" xref="A1.1.p1.12.m7.1.1.3.cmml"><mi id="A1.1.p1.12.m7.1.1.3.2.2" xref="A1.1.p1.12.m7.1.1.3.2.2.cmml">i</mi><mn id="A1.1.p1.12.m7.1.1.3.2.3" xref="A1.1.p1.12.m7.1.1.3.2.3.cmml">1</mn><mi id="A1.1.p1.12.m7.1.1.3.3" xref="A1.1.p1.12.m7.1.1.3.3.cmml">k</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.12.m7.1b"><apply id="A1.1.p1.12.m7.1.1.cmml" xref="A1.1.p1.12.m7.1.1"><eq id="A1.1.p1.12.m7.1.1.1.cmml" xref="A1.1.p1.12.m7.1.1.1"></eq><apply id="A1.1.p1.12.m7.1.1.2.cmml" xref="A1.1.p1.12.m7.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.12.m7.1.1.2.1.cmml" xref="A1.1.p1.12.m7.1.1.2">superscript</csymbol><apply id="A1.1.p1.12.m7.1.1.2.2.cmml" xref="A1.1.p1.12.m7.1.1.2"><csymbol cd="ambiguous" id="A1.1.p1.12.m7.1.1.2.2.1.cmml" xref="A1.1.p1.12.m7.1.1.2">subscript</csymbol><ci id="A1.1.p1.12.m7.1.1.2.2.2.cmml" xref="A1.1.p1.12.m7.1.1.2.2.2">𝑥</ci><apply id="A1.1.p1.12.m7.1.1.2.2.3.cmml" xref="A1.1.p1.12.m7.1.1.2.2.3"><plus id="A1.1.p1.12.m7.1.1.2.2.3.1.cmml" xref="A1.1.p1.12.m7.1.1.2.2.3.1"></plus><apply id="A1.1.p1.12.m7.1.1.2.2.3.2.cmml" xref="A1.1.p1.12.m7.1.1.2.2.3.2"><minus id="A1.1.p1.12.m7.1.1.2.2.3.2.1.cmml" xref="A1.1.p1.12.m7.1.1.2.2.3.2.1"></minus><ci id="A1.1.p1.12.m7.1.1.2.2.3.2.2.cmml" xref="A1.1.p1.12.m7.1.1.2.2.3.2.2">𝑡</ci><ci id="A1.1.p1.12.m7.1.1.2.2.3.2.3.cmml" xref="A1.1.p1.12.m7.1.1.2.2.3.2.3">𝑘</ci></apply><cn id="A1.1.p1.12.m7.1.1.2.2.3.3.cmml" type="integer" xref="A1.1.p1.12.m7.1.1.2.2.3.3">1</cn></apply></apply><ci id="A1.1.p1.12.m7.1.1.2.3.cmml" xref="A1.1.p1.12.m7.1.1.2.3">𝑡</ci></apply><apply id="A1.1.p1.12.m7.1.1.3.cmml" xref="A1.1.p1.12.m7.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.12.m7.1.1.3.1.cmml" xref="A1.1.p1.12.m7.1.1.3">superscript</csymbol><apply id="A1.1.p1.12.m7.1.1.3.2.cmml" xref="A1.1.p1.12.m7.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.12.m7.1.1.3.2.1.cmml" xref="A1.1.p1.12.m7.1.1.3">subscript</csymbol><ci id="A1.1.p1.12.m7.1.1.3.2.2.cmml" xref="A1.1.p1.12.m7.1.1.3.2.2">𝑖</ci><cn id="A1.1.p1.12.m7.1.1.3.2.3.cmml" type="integer" xref="A1.1.p1.12.m7.1.1.3.2.3">1</cn></apply><ci id="A1.1.p1.12.m7.1.1.3.3.cmml" xref="A1.1.p1.12.m7.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.12.m7.1c">x_{t-k+1}^{t}=i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.12.m7.1d">italic_x start_POSTSUBSCRIPT italic_t - italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT = italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Let also <math alttext="p\triangleq P_{i_{1}^{k}}(1)=\mathbb{P}(1\mid i_{1}^{k})" class="ltx_Math" display="inline" id="A1.1.p1.13.m8.2"><semantics id="A1.1.p1.13.m8.2a"><mrow id="A1.1.p1.13.m8.2.2" xref="A1.1.p1.13.m8.2.2.cmml"><mi id="A1.1.p1.13.m8.2.2.3" xref="A1.1.p1.13.m8.2.2.3.cmml">p</mi><mo id="A1.1.p1.13.m8.2.2.4" xref="A1.1.p1.13.m8.2.2.4.cmml">≜</mo><mrow id="A1.1.p1.13.m8.2.2.5" xref="A1.1.p1.13.m8.2.2.5.cmml"><msub id="A1.1.p1.13.m8.2.2.5.2" xref="A1.1.p1.13.m8.2.2.5.2.cmml"><mi id="A1.1.p1.13.m8.2.2.5.2.2" xref="A1.1.p1.13.m8.2.2.5.2.2.cmml">P</mi><msubsup id="A1.1.p1.13.m8.2.2.5.2.3" xref="A1.1.p1.13.m8.2.2.5.2.3.cmml"><mi id="A1.1.p1.13.m8.2.2.5.2.3.2.2" xref="A1.1.p1.13.m8.2.2.5.2.3.2.2.cmml">i</mi><mn id="A1.1.p1.13.m8.2.2.5.2.3.2.3" xref="A1.1.p1.13.m8.2.2.5.2.3.2.3.cmml">1</mn><mi id="A1.1.p1.13.m8.2.2.5.2.3.3" xref="A1.1.p1.13.m8.2.2.5.2.3.3.cmml">k</mi></msubsup></msub><mo id="A1.1.p1.13.m8.2.2.5.1" xref="A1.1.p1.13.m8.2.2.5.1.cmml">⁢</mo><mrow id="A1.1.p1.13.m8.2.2.5.3.2" xref="A1.1.p1.13.m8.2.2.5.cmml"><mo id="A1.1.p1.13.m8.2.2.5.3.2.1" stretchy="false" xref="A1.1.p1.13.m8.2.2.5.cmml">(</mo><mn id="A1.1.p1.13.m8.1.1" xref="A1.1.p1.13.m8.1.1.cmml">1</mn><mo id="A1.1.p1.13.m8.2.2.5.3.2.2" stretchy="false" xref="A1.1.p1.13.m8.2.2.5.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.13.m8.2.2.6" xref="A1.1.p1.13.m8.2.2.6.cmml">=</mo><mrow id="A1.1.p1.13.m8.2.2.1" xref="A1.1.p1.13.m8.2.2.1.cmml"><mi id="A1.1.p1.13.m8.2.2.1.3" xref="A1.1.p1.13.m8.2.2.1.3.cmml">ℙ</mi><mo id="A1.1.p1.13.m8.2.2.1.2" xref="A1.1.p1.13.m8.2.2.1.2.cmml">⁢</mo><mrow id="A1.1.p1.13.m8.2.2.1.1.1" xref="A1.1.p1.13.m8.2.2.1.1.1.1.cmml"><mo id="A1.1.p1.13.m8.2.2.1.1.1.2" stretchy="false" xref="A1.1.p1.13.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.1.p1.13.m8.2.2.1.1.1.1" xref="A1.1.p1.13.m8.2.2.1.1.1.1.cmml"><mn id="A1.1.p1.13.m8.2.2.1.1.1.1.2" xref="A1.1.p1.13.m8.2.2.1.1.1.1.2.cmml">1</mn><mo id="A1.1.p1.13.m8.2.2.1.1.1.1.1" xref="A1.1.p1.13.m8.2.2.1.1.1.1.1.cmml">∣</mo><msubsup id="A1.1.p1.13.m8.2.2.1.1.1.1.3" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.cmml"><mi id="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.2" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.2.cmml">i</mi><mn id="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.3" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.3.cmml">1</mn><mi id="A1.1.p1.13.m8.2.2.1.1.1.1.3.3" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A1.1.p1.13.m8.2.2.1.1.1.3" stretchy="false" xref="A1.1.p1.13.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.13.m8.2b"><apply id="A1.1.p1.13.m8.2.2.cmml" xref="A1.1.p1.13.m8.2.2"><and id="A1.1.p1.13.m8.2.2a.cmml" xref="A1.1.p1.13.m8.2.2"></and><apply id="A1.1.p1.13.m8.2.2b.cmml" xref="A1.1.p1.13.m8.2.2"><ci id="A1.1.p1.13.m8.2.2.4.cmml" xref="A1.1.p1.13.m8.2.2.4">≜</ci><ci id="A1.1.p1.13.m8.2.2.3.cmml" xref="A1.1.p1.13.m8.2.2.3">𝑝</ci><apply id="A1.1.p1.13.m8.2.2.5.cmml" xref="A1.1.p1.13.m8.2.2.5"><times id="A1.1.p1.13.m8.2.2.5.1.cmml" xref="A1.1.p1.13.m8.2.2.5.1"></times><apply id="A1.1.p1.13.m8.2.2.5.2.cmml" xref="A1.1.p1.13.m8.2.2.5.2"><csymbol cd="ambiguous" id="A1.1.p1.13.m8.2.2.5.2.1.cmml" xref="A1.1.p1.13.m8.2.2.5.2">subscript</csymbol><ci id="A1.1.p1.13.m8.2.2.5.2.2.cmml" xref="A1.1.p1.13.m8.2.2.5.2.2">𝑃</ci><apply id="A1.1.p1.13.m8.2.2.5.2.3.cmml" xref="A1.1.p1.13.m8.2.2.5.2.3"><csymbol cd="ambiguous" id="A1.1.p1.13.m8.2.2.5.2.3.1.cmml" xref="A1.1.p1.13.m8.2.2.5.2.3">superscript</csymbol><apply id="A1.1.p1.13.m8.2.2.5.2.3.2.cmml" xref="A1.1.p1.13.m8.2.2.5.2.3"><csymbol cd="ambiguous" id="A1.1.p1.13.m8.2.2.5.2.3.2.1.cmml" xref="A1.1.p1.13.m8.2.2.5.2.3">subscript</csymbol><ci id="A1.1.p1.13.m8.2.2.5.2.3.2.2.cmml" xref="A1.1.p1.13.m8.2.2.5.2.3.2.2">𝑖</ci><cn id="A1.1.p1.13.m8.2.2.5.2.3.2.3.cmml" type="integer" xref="A1.1.p1.13.m8.2.2.5.2.3.2.3">1</cn></apply><ci id="A1.1.p1.13.m8.2.2.5.2.3.3.cmml" xref="A1.1.p1.13.m8.2.2.5.2.3.3">𝑘</ci></apply></apply><cn id="A1.1.p1.13.m8.1.1.cmml" type="integer" xref="A1.1.p1.13.m8.1.1">1</cn></apply></apply><apply id="A1.1.p1.13.m8.2.2c.cmml" xref="A1.1.p1.13.m8.2.2"><eq id="A1.1.p1.13.m8.2.2.6.cmml" xref="A1.1.p1.13.m8.2.2.6"></eq><share href="https://arxiv.org/html/2502.10178v1#A1.1.p1.13.m8.2.2.5.cmml" id="A1.1.p1.13.m8.2.2d.cmml" xref="A1.1.p1.13.m8.2.2"></share><apply id="A1.1.p1.13.m8.2.2.1.cmml" xref="A1.1.p1.13.m8.2.2.1"><times id="A1.1.p1.13.m8.2.2.1.2.cmml" xref="A1.1.p1.13.m8.2.2.1.2"></times><ci id="A1.1.p1.13.m8.2.2.1.3.cmml" xref="A1.1.p1.13.m8.2.2.1.3">ℙ</ci><apply id="A1.1.p1.13.m8.2.2.1.1.1.1.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1"><csymbol cd="latexml" id="A1.1.p1.13.m8.2.2.1.1.1.1.1.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.1">conditional</csymbol><cn id="A1.1.p1.13.m8.2.2.1.1.1.1.2.cmml" type="integer" xref="A1.1.p1.13.m8.2.2.1.1.1.1.2">1</cn><apply id="A1.1.p1.13.m8.2.2.1.1.1.1.3.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.13.m8.2.2.1.1.1.1.3.1.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3">superscript</csymbol><apply id="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.1.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3">subscript</csymbol><ci id="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.2.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.2">𝑖</ci><cn id="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.3.cmml" type="integer" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.2.3">1</cn></apply><ci id="A1.1.p1.13.m8.2.2.1.1.1.1.3.3.cmml" xref="A1.1.p1.13.m8.2.2.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.13.m8.2c">p\triangleq P_{i_{1}^{k}}(1)=\mathbb{P}(1\mid i_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.13.m8.2d">italic_p ≜ italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 ) = blackboard_P ( 1 ∣ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx10"> <tbody id="A1.Ex28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{*}^{(1)}(x_{1}^{t})" class="ltx_Math" display="inline" id="A1.Ex28.m1.2"><semantics id="A1.Ex28.m1.2a"><mrow id="A1.Ex28.m1.2.2" xref="A1.Ex28.m1.2.2.cmml"><msubsup id="A1.Ex28.m1.2.2.3" xref="A1.Ex28.m1.2.2.3.cmml"><mi id="A1.Ex28.m1.2.2.3.2.2" xref="A1.Ex28.m1.2.2.3.2.2.cmml">f</mi><mo id="A1.Ex28.m1.2.2.3.2.3" xref="A1.Ex28.m1.2.2.3.2.3.cmml">∗</mo><mrow id="A1.Ex28.m1.1.1.1.3" xref="A1.Ex28.m1.2.2.3.cmml"><mo id="A1.Ex28.m1.1.1.1.3.1" stretchy="false" xref="A1.Ex28.m1.2.2.3.cmml">(</mo><mn id="A1.Ex28.m1.1.1.1.1" xref="A1.Ex28.m1.1.1.1.1.cmml">1</mn><mo id="A1.Ex28.m1.1.1.1.3.2" stretchy="false" xref="A1.Ex28.m1.2.2.3.cmml">)</mo></mrow></msubsup><mo id="A1.Ex28.m1.2.2.2" xref="A1.Ex28.m1.2.2.2.cmml">⁢</mo><mrow id="A1.Ex28.m1.2.2.1.1" xref="A1.Ex28.m1.2.2.1.1.1.cmml"><mo id="A1.Ex28.m1.2.2.1.1.2" stretchy="false" xref="A1.Ex28.m1.2.2.1.1.1.cmml">(</mo><msubsup id="A1.Ex28.m1.2.2.1.1.1" xref="A1.Ex28.m1.2.2.1.1.1.cmml"><mi id="A1.Ex28.m1.2.2.1.1.1.2.2" xref="A1.Ex28.m1.2.2.1.1.1.2.2.cmml">x</mi><mn id="A1.Ex28.m1.2.2.1.1.1.2.3" xref="A1.Ex28.m1.2.2.1.1.1.2.3.cmml">1</mn><mi id="A1.Ex28.m1.2.2.1.1.1.3" xref="A1.Ex28.m1.2.2.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex28.m1.2.2.1.1.3" stretchy="false" xref="A1.Ex28.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex28.m1.2b"><apply id="A1.Ex28.m1.2.2.cmml" xref="A1.Ex28.m1.2.2"><times id="A1.Ex28.m1.2.2.2.cmml" xref="A1.Ex28.m1.2.2.2"></times><apply id="A1.Ex28.m1.2.2.3.cmml" xref="A1.Ex28.m1.2.2.3"><csymbol cd="ambiguous" id="A1.Ex28.m1.2.2.3.1.cmml" xref="A1.Ex28.m1.2.2.3">superscript</csymbol><apply id="A1.Ex28.m1.2.2.3.2.cmml" xref="A1.Ex28.m1.2.2.3"><csymbol cd="ambiguous" id="A1.Ex28.m1.2.2.3.2.1.cmml" xref="A1.Ex28.m1.2.2.3">subscript</csymbol><ci id="A1.Ex28.m1.2.2.3.2.2.cmml" xref="A1.Ex28.m1.2.2.3.2.2">𝑓</ci><times id="A1.Ex28.m1.2.2.3.2.3.cmml" xref="A1.Ex28.m1.2.2.3.2.3"></times></apply><cn id="A1.Ex28.m1.1.1.1.1.cmml" type="integer" xref="A1.Ex28.m1.1.1.1.1">1</cn></apply><apply id="A1.Ex28.m1.2.2.1.1.1.cmml" xref="A1.Ex28.m1.2.2.1.1"><csymbol cd="ambiguous" id="A1.Ex28.m1.2.2.1.1.1.1.cmml" xref="A1.Ex28.m1.2.2.1.1">superscript</csymbol><apply id="A1.Ex28.m1.2.2.1.1.1.2.cmml" xref="A1.Ex28.m1.2.2.1.1"><csymbol cd="ambiguous" id="A1.Ex28.m1.2.2.1.1.1.2.1.cmml" xref="A1.Ex28.m1.2.2.1.1">subscript</csymbol><ci id="A1.Ex28.m1.2.2.1.1.1.2.2.cmml" xref="A1.Ex28.m1.2.2.1.1.1.2.2">𝑥</ci><cn id="A1.Ex28.m1.2.2.1.1.1.2.3.cmml" type="integer" xref="A1.Ex28.m1.2.2.1.1.1.2.3">1</cn></apply><ci id="A1.Ex28.m1.2.2.1.1.1.3.cmml" xref="A1.Ex28.m1.2.2.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex28.m1.2c">\displaystyle f_{*}^{(1)}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.Ex28.m1.2d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\triangleq\mathbb{E}_{x_{t+1}|x_{1}^{t}}[x_{t+1}]" class="ltx_Math" display="inline" id="A1.Ex28.m2.1"><semantics id="A1.Ex28.m2.1a"><mrow id="A1.Ex28.m2.1.1" xref="A1.Ex28.m2.1.1.cmml"><mi id="A1.Ex28.m2.1.1.3" xref="A1.Ex28.m2.1.1.3.cmml"></mi><mo id="A1.Ex28.m2.1.1.2" xref="A1.Ex28.m2.1.1.2.cmml">≜</mo><mrow id="A1.Ex28.m2.1.1.1" xref="A1.Ex28.m2.1.1.1.cmml"><msub id="A1.Ex28.m2.1.1.1.3" xref="A1.Ex28.m2.1.1.1.3.cmml"><mi id="A1.Ex28.m2.1.1.1.3.2" xref="A1.Ex28.m2.1.1.1.3.2.cmml">𝔼</mi><mrow id="A1.Ex28.m2.1.1.1.3.3" 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xref="A1.Ex29.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex29.m1.3.3.1.1.1.1.1.cmml" xref="A1.Ex29.m1.3.3.1.1.1.1">subscript</csymbol><ci id="A1.Ex29.m1.3.3.1.1.1.1.2.cmml" xref="A1.Ex29.m1.3.3.1.1.1.1.2">𝑥</ci><apply id="A1.Ex29.m1.3.3.1.1.1.1.3.cmml" xref="A1.Ex29.m1.3.3.1.1.1.1.3"><plus id="A1.Ex29.m1.3.3.1.1.1.1.3.1.cmml" xref="A1.Ex29.m1.3.3.1.1.1.1.3.1"></plus><ci id="A1.Ex29.m1.3.3.1.1.1.1.3.2.cmml" xref="A1.Ex29.m1.3.3.1.1.1.1.3.2">𝑡</ci><cn id="A1.Ex29.m1.3.3.1.1.1.1.3.3.cmml" type="integer" xref="A1.Ex29.m1.3.3.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex29.m1.3c">\displaystyle=\mathbb{E}_{P_{i_{1}^{k}}|x_{1}^{t}}\mathbb{E}_{x_{t+1}|x_{1}^{t% },P_{i_{1}^{k}}}[x_{t+1}]</annotation><annotation encoding="application/x-llamapun" id="A1.Ex29.m1.3d">= blackboard_E start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbb{E}_{P_{i_{1}^{k}}|x_{1}^{t}}[P_{i_{1}^{k}}(1)]" 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id="A1.Ex31.m1.3.3.1.1.1.1.1.1.2.3.2.3.cmml" type="integer" xref="A1.Ex31.m1.3.3.1.1.1.1.1.1.2.3.2.3">1</cn></apply><ci id="A1.Ex31.m1.3.3.1.1.1.1.1.1.2.3.3.cmml" xref="A1.Ex31.m1.3.3.1.1.1.1.1.1.2.3.3">𝑘</ci></apply></apply><cn id="A1.Ex31.m1.2.2.cmml" type="integer" xref="A1.Ex31.m1.2.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex31.m1.3c">\displaystyle=\mathbb{E}_{P_{i_{1}^{k}}|x_{1}^{t}|_{i_{1}^{k}}}[P_{i_{1}^{k}}(% 1)],</annotation><annotation encoding="application/x-llamapun" id="A1.Ex31.m1.3d">= blackboard_E start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.16">where in the last equation we used the fact that, when <math alttext="x_{1}^{k}\sim\mathrm{Unif}(\mathcal{X}^{k})" class="ltx_Math" display="inline" id="A1.1.p1.14.m1.1"><semantics id="A1.1.p1.14.m1.1a"><mrow id="A1.1.p1.14.m1.1.1" xref="A1.1.p1.14.m1.1.1.cmml"><msubsup id="A1.1.p1.14.m1.1.1.3" xref="A1.1.p1.14.m1.1.1.3.cmml"><mi id="A1.1.p1.14.m1.1.1.3.2.2" xref="A1.1.p1.14.m1.1.1.3.2.2.cmml">x</mi><mn id="A1.1.p1.14.m1.1.1.3.2.3" xref="A1.1.p1.14.m1.1.1.3.2.3.cmml">1</mn><mi id="A1.1.p1.14.m1.1.1.3.3" xref="A1.1.p1.14.m1.1.1.3.3.cmml">k</mi></msubsup><mo id="A1.1.p1.14.m1.1.1.2" xref="A1.1.p1.14.m1.1.1.2.cmml">∼</mo><mrow id="A1.1.p1.14.m1.1.1.1" 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id="A1.1.p1.14.m1.1.1.1.1.1.1.2.cmml" xref="A1.1.p1.14.m1.1.1.1.1.1.1.2">𝒳</ci><ci id="A1.1.p1.14.m1.1.1.1.1.1.1.3.cmml" xref="A1.1.p1.14.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.14.m1.1c">x_{1}^{k}\sim\mathrm{Unif}(\mathcal{X}^{k})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.14.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∼ roman_Unif ( caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>, the subsequence <math alttext="x_{1}^{t}|_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.1.p1.15.m2.2"><semantics id="A1.1.p1.15.m2.2a"><msub id="A1.1.p1.15.m2.2.2.1" xref="A1.1.p1.15.m2.2.2.2.cmml"><mrow id="A1.1.p1.15.m2.2.2.1.1" xref="A1.1.p1.15.m2.2.2.2.cmml"><msubsup id="A1.1.p1.15.m2.2.2.1.1.1" xref="A1.1.p1.15.m2.2.2.1.1.1.cmml"><mi id="A1.1.p1.15.m2.2.2.1.1.1.2.2" xref="A1.1.p1.15.m2.2.2.1.1.1.2.2.cmml">x</mi><mn id="A1.1.p1.15.m2.2.2.1.1.1.2.3" xref="A1.1.p1.15.m2.2.2.1.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.15.m2.2.2.1.1.1.3" xref="A1.1.p1.15.m2.2.2.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.1.p1.15.m2.2.2.1.1.2" stretchy="false" xref="A1.1.p1.15.m2.2.2.2.1.cmml">|</mo></mrow><msubsup id="A1.1.p1.15.m2.1.1.1" xref="A1.1.p1.15.m2.1.1.1.cmml"><mi id="A1.1.p1.15.m2.1.1.1.2.2" xref="A1.1.p1.15.m2.1.1.1.2.2.cmml">i</mi><mn id="A1.1.p1.15.m2.1.1.1.2.3" xref="A1.1.p1.15.m2.1.1.1.2.3.cmml">1</mn><mi id="A1.1.p1.15.m2.1.1.1.3" xref="A1.1.p1.15.m2.1.1.1.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.1.p1.15.m2.2b"><apply id="A1.1.p1.15.m2.2.2.2.cmml" xref="A1.1.p1.15.m2.2.2.1"><csymbol cd="latexml" id="A1.1.p1.15.m2.2.2.2.1.cmml" xref="A1.1.p1.15.m2.2.2.1.1.2">evaluated-at</csymbol><apply id="A1.1.p1.15.m2.2.2.1.1.1.cmml" xref="A1.1.p1.15.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.15.m2.2.2.1.1.1.1.cmml" xref="A1.1.p1.15.m2.2.2.1.1.1">superscript</csymbol><apply id="A1.1.p1.15.m2.2.2.1.1.1.2.cmml" xref="A1.1.p1.15.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.15.m2.2.2.1.1.1.2.1.cmml" xref="A1.1.p1.15.m2.2.2.1.1.1">subscript</csymbol><ci id="A1.1.p1.15.m2.2.2.1.1.1.2.2.cmml" xref="A1.1.p1.15.m2.2.2.1.1.1.2.2">𝑥</ci><cn id="A1.1.p1.15.m2.2.2.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.15.m2.2.2.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.15.m2.2.2.1.1.1.3.cmml" xref="A1.1.p1.15.m2.2.2.1.1.1.3">𝑡</ci></apply><apply id="A1.1.p1.15.m2.1.1.1.cmml" xref="A1.1.p1.15.m2.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.15.m2.1.1.1.1.cmml" xref="A1.1.p1.15.m2.1.1.1">superscript</csymbol><apply id="A1.1.p1.15.m2.1.1.1.2.cmml" xref="A1.1.p1.15.m2.1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.15.m2.1.1.1.2.1.cmml" xref="A1.1.p1.15.m2.1.1.1">subscript</csymbol><ci id="A1.1.p1.15.m2.1.1.1.2.2.cmml" xref="A1.1.p1.15.m2.1.1.1.2.2">𝑖</ci><cn id="A1.1.p1.15.m2.1.1.1.2.3.cmml" type="integer" xref="A1.1.p1.15.m2.1.1.1.2.3">1</cn></apply><ci id="A1.1.p1.15.m2.1.1.1.3.cmml" xref="A1.1.p1.15.m2.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.15.m2.2c">x_{1}^{t}|_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.15.m2.2d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a sufficient statistic for <math alttext="P_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.1.p1.16.m3.1"><semantics id="A1.1.p1.16.m3.1a"><msub id="A1.1.p1.16.m3.1.1" xref="A1.1.p1.16.m3.1.1.cmml"><mi id="A1.1.p1.16.m3.1.1.2" xref="A1.1.p1.16.m3.1.1.2.cmml">P</mi><msubsup id="A1.1.p1.16.m3.1.1.3" xref="A1.1.p1.16.m3.1.1.3.cmml"><mi id="A1.1.p1.16.m3.1.1.3.2.2" xref="A1.1.p1.16.m3.1.1.3.2.2.cmml">i</mi><mn id="A1.1.p1.16.m3.1.1.3.2.3" xref="A1.1.p1.16.m3.1.1.3.2.3.cmml">1</mn><mi id="A1.1.p1.16.m3.1.1.3.3" xref="A1.1.p1.16.m3.1.1.3.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.1.p1.16.m3.1b"><apply id="A1.1.p1.16.m3.1.1.cmml" xref="A1.1.p1.16.m3.1.1"><csymbol cd="ambiguous" id="A1.1.p1.16.m3.1.1.1.cmml" xref="A1.1.p1.16.m3.1.1">subscript</csymbol><ci id="A1.1.p1.16.m3.1.1.2.cmml" xref="A1.1.p1.16.m3.1.1.2">𝑃</ci><apply id="A1.1.p1.16.m3.1.1.3.cmml" xref="A1.1.p1.16.m3.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.16.m3.1.1.3.1.cmml" xref="A1.1.p1.16.m3.1.1.3">superscript</csymbol><apply id="A1.1.p1.16.m3.1.1.3.2.cmml" xref="A1.1.p1.16.m3.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.16.m3.1.1.3.2.1.cmml" xref="A1.1.p1.16.m3.1.1.3">subscript</csymbol><ci id="A1.1.p1.16.m3.1.1.3.2.2.cmml" xref="A1.1.p1.16.m3.1.1.3.2.2">𝑖</ci><cn id="A1.1.p1.16.m3.1.1.3.2.3.cmml" type="integer" xref="A1.1.p1.16.m3.1.1.3.2.3">1</cn></apply><ci id="A1.1.p1.16.m3.1.1.3.3.cmml" xref="A1.1.p1.16.m3.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.16.m3.1c">P_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.16.m3.1d">italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Hence,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx11"> <tbody id="A1.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{*}^{(1)}(x_{1}^{t})" class="ltx_Math" display="inline" id="A1.Ex32.m1.2"><semantics id="A1.Ex32.m1.2a"><mrow id="A1.Ex32.m1.2.2" xref="A1.Ex32.m1.2.2.cmml"><msubsup id="A1.Ex32.m1.2.2.3" xref="A1.Ex32.m1.2.2.3.cmml"><mi id="A1.Ex32.m1.2.2.3.2.2" xref="A1.Ex32.m1.2.2.3.2.2.cmml">f</mi><mo id="A1.Ex32.m1.2.2.3.2.3" xref="A1.Ex32.m1.2.2.3.2.3.cmml">∗</mo><mrow id="A1.Ex32.m1.1.1.1.3" xref="A1.Ex32.m1.2.2.3.cmml"><mo id="A1.Ex32.m1.1.1.1.3.1" stretchy="false" xref="A1.Ex32.m1.2.2.3.cmml">(</mo><mn id="A1.Ex32.m1.1.1.1.1" xref="A1.Ex32.m1.1.1.1.1.cmml">1</mn><mo id="A1.Ex32.m1.1.1.1.3.2" stretchy="false" xref="A1.Ex32.m1.2.2.3.cmml">)</mo></mrow></msubsup><mo id="A1.Ex32.m1.2.2.2" xref="A1.Ex32.m1.2.2.2.cmml">⁢</mo><mrow id="A1.Ex32.m1.2.2.1.1" xref="A1.Ex32.m1.2.2.1.1.1.cmml"><mo id="A1.Ex32.m1.2.2.1.1.2" stretchy="false" xref="A1.Ex32.m1.2.2.1.1.1.cmml">(</mo><msubsup id="A1.Ex32.m1.2.2.1.1.1" xref="A1.Ex32.m1.2.2.1.1.1.cmml"><mi id="A1.Ex32.m1.2.2.1.1.1.2.2" xref="A1.Ex32.m1.2.2.1.1.1.2.2.cmml">x</mi><mn id="A1.Ex32.m1.2.2.1.1.1.2.3" xref="A1.Ex32.m1.2.2.1.1.1.2.3.cmml">1</mn><mi id="A1.Ex32.m1.2.2.1.1.1.3" xref="A1.Ex32.m1.2.2.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.Ex32.m1.2.2.1.1.3" stretchy="false" xref="A1.Ex32.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex32.m1.2b"><apply id="A1.Ex32.m1.2.2.cmml" xref="A1.Ex32.m1.2.2"><times id="A1.Ex32.m1.2.2.2.cmml" xref="A1.Ex32.m1.2.2.2"></times><apply id="A1.Ex32.m1.2.2.3.cmml" xref="A1.Ex32.m1.2.2.3"><csymbol cd="ambiguous" id="A1.Ex32.m1.2.2.3.1.cmml" xref="A1.Ex32.m1.2.2.3">superscript</csymbol><apply id="A1.Ex32.m1.2.2.3.2.cmml" xref="A1.Ex32.m1.2.2.3"><csymbol cd="ambiguous" id="A1.Ex32.m1.2.2.3.2.1.cmml" xref="A1.Ex32.m1.2.2.3">subscript</csymbol><ci id="A1.Ex32.m1.2.2.3.2.2.cmml" xref="A1.Ex32.m1.2.2.3.2.2">𝑓</ci><times id="A1.Ex32.m1.2.2.3.2.3.cmml" xref="A1.Ex32.m1.2.2.3.2.3"></times></apply><cn id="A1.Ex32.m1.1.1.1.1.cmml" type="integer" xref="A1.Ex32.m1.1.1.1.1">1</cn></apply><apply id="A1.Ex32.m1.2.2.1.1.1.cmml" xref="A1.Ex32.m1.2.2.1.1"><csymbol cd="ambiguous" id="A1.Ex32.m1.2.2.1.1.1.1.cmml" xref="A1.Ex32.m1.2.2.1.1">superscript</csymbol><apply id="A1.Ex32.m1.2.2.1.1.1.2.cmml" xref="A1.Ex32.m1.2.2.1.1"><csymbol cd="ambiguous" id="A1.Ex32.m1.2.2.1.1.1.2.1.cmml" xref="A1.Ex32.m1.2.2.1.1">subscript</csymbol><ci id="A1.Ex32.m1.2.2.1.1.1.2.2.cmml" xref="A1.Ex32.m1.2.2.1.1.1.2.2">𝑥</ci><cn id="A1.Ex32.m1.2.2.1.1.1.2.3.cmml" type="integer" xref="A1.Ex32.m1.2.2.1.1.1.2.3">1</cn></apply><ci id="A1.Ex32.m1.2.2.1.1.1.3.cmml" xref="A1.Ex32.m1.2.2.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex32.m1.2c">\displaystyle f_{*}^{(1)}(x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A1.Ex32.m1.2d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbb{E}_{P_{i_{1}^{k}}|x_{1}^{t}|_{i_{1}^{k}}}[P_{i_{1}^{k}}(% 1)]" class="ltx_Math" display="inline" id="A1.Ex32.m2.3"><semantics id="A1.Ex32.m2.3a"><mrow id="A1.Ex32.m2.3.3" xref="A1.Ex32.m2.3.3.cmml"><mi id="A1.Ex32.m2.3.3.3" xref="A1.Ex32.m2.3.3.3.cmml"></mi><mo id="A1.Ex32.m2.3.3.2" xref="A1.Ex32.m2.3.3.2.cmml">=</mo><mrow id="A1.Ex32.m2.3.3.1" xref="A1.Ex32.m2.3.3.1.cmml"><msub id="A1.Ex32.m2.3.3.1.3" xref="A1.Ex32.m2.3.3.1.3.cmml"><mi id="A1.Ex32.m2.3.3.1.3.2" xref="A1.Ex32.m2.3.3.1.3.2.cmml">𝔼</mi><mrow id="A1.Ex32.m2.1.1.1" 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xref="A1.Ex32.m2.3.3.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A1.Ex32.m2.3.3.1.1.1.1.2.3.2.1.cmml" xref="A1.Ex32.m2.3.3.1.1.1.1.2.3">subscript</csymbol><ci id="A1.Ex32.m2.3.3.1.1.1.1.2.3.2.2.cmml" xref="A1.Ex32.m2.3.3.1.1.1.1.2.3.2.2">𝑖</ci><cn id="A1.Ex32.m2.3.3.1.1.1.1.2.3.2.3.cmml" type="integer" xref="A1.Ex32.m2.3.3.1.1.1.1.2.3.2.3">1</cn></apply><ci id="A1.Ex32.m2.3.3.1.1.1.1.2.3.3.cmml" xref="A1.Ex32.m2.3.3.1.1.1.1.2.3.3">𝑘</ci></apply></apply><cn id="A1.Ex32.m2.2.2.cmml" type="integer" xref="A1.Ex32.m2.2.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex32.m2.3c">\displaystyle=\mathbb{E}_{P_{i_{1}^{k}}|x_{1}^{t}|_{i_{1}^{k}}}[P_{i_{1}^{k}}(% 1)]</annotation><annotation encoding="application/x-llamapun" id="A1.Ex32.m2.3d">= blackboard_E start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 ) ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\int_{0}^{1}\frac{p^{\beta-1}(1-p)^{\beta-1}p^{n_{1}}(1-p)^{n_{0% }}}{\int_{0}^{1}q^{\beta-1}(1-q)^{\beta-1}q^{n_{1}}(1-q)^{n_{0}}\,dq}\cdot p\,dp" class="ltx_Math" display="inline" id="A1.Ex33.m1.4"><semantics id="A1.Ex33.m1.4a"><mrow 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id="A1.Ex33.m1.4c">\displaystyle=\int_{0}^{1}\frac{p^{\beta-1}(1-p)^{\beta-1}p^{n_{1}}(1-p)^{n_{0% }}}{\int_{0}^{1}q^{\beta-1}(1-q)^{\beta-1}q^{n_{1}}(1-q)^{n_{0}}\,dq}\cdot p\,dp</annotation><annotation encoding="application/x-llamapun" id="A1.Ex33.m1.4d">= ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT divide start_ARG italic_p start_POSTSUPERSCRIPT italic_β - 1 end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_β - 1 end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_β - 1 end_POSTSUPERSCRIPT ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_β - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d italic_q end_ARG ⋅ italic_p italic_d italic_p</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\int_{0}^{1}p^{n_{1}+\beta}(1-p)^{n_{0}+\beta-1}\,dp}{\int% _{0}^{1}q^{n_{1}+\beta-1}(1-q)^{n_{0}+\beta-1}\,dq}" class="ltx_Math" display="inline" id="A1.Ex34.m1.2"><semantics id="A1.Ex34.m1.2a"><mrow id="A1.Ex34.m1.2.3" xref="A1.Ex34.m1.2.3.cmml"><mi id="A1.Ex34.m1.2.3.2" xref="A1.Ex34.m1.2.3.2.cmml"></mi><mo id="A1.Ex34.m1.2.3.1" xref="A1.Ex34.m1.2.3.1.cmml">=</mo><mstyle displaystyle="true" id="A1.Ex34.m1.2.2" 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start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_β end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_β - 1 end_POSTSUPERSCRIPT italic_d italic_p end_ARG start_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_β - 1 end_POSTSUPERSCRIPT ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_β - 1 end_POSTSUPERSCRIPT italic_d italic_q end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_β ) end_ARG start_ARG roman_Γ ( italic_n + 2 italic_β + 1 ) end_ARG ⋅ divide start_ARG roman_Γ ( italic_n + 2 italic_β ) end_ARG start_ARG roman_Γ ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_β ) roman_Γ ( italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_β ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{n_{1}+\beta}{n+2\beta}," class="ltx_Math" display="inline" id="A1.Ex36.m1.1"><semantics id="A1.Ex36.m1.1a"><mrow id="A1.Ex36.m1.1.1.1" xref="A1.Ex36.m1.1.1.1.1.cmml"><mrow id="A1.Ex36.m1.1.1.1.1" xref="A1.Ex36.m1.1.1.1.1.cmml"><mi id="A1.Ex36.m1.1.1.1.1.2" xref="A1.Ex36.m1.1.1.1.1.2.cmml"></mi><mo 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id="A1.Ex36.m1.1.1.1.1.3.3.3.2" xref="A1.Ex36.m1.1.1.1.1.3.3.3.2.cmml">2</mn><mo id="A1.Ex36.m1.1.1.1.1.3.3.3.1" xref="A1.Ex36.m1.1.1.1.1.3.3.3.1.cmml">⁢</mo><mi id="A1.Ex36.m1.1.1.1.1.3.3.3.3" xref="A1.Ex36.m1.1.1.1.1.3.3.3.3.cmml">β</mi></mrow></mrow></mfrac></mstyle></mrow><mo id="A1.Ex36.m1.1.1.1.2" xref="A1.Ex36.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex36.m1.1b"><apply id="A1.Ex36.m1.1.1.1.1.cmml" xref="A1.Ex36.m1.1.1.1"><eq id="A1.Ex36.m1.1.1.1.1.1.cmml" xref="A1.Ex36.m1.1.1.1.1.1"></eq><csymbol cd="latexml" id="A1.Ex36.m1.1.1.1.1.2.cmml" xref="A1.Ex36.m1.1.1.1.1.2">absent</csymbol><apply id="A1.Ex36.m1.1.1.1.1.3.cmml" xref="A1.Ex36.m1.1.1.1.1.3"><divide id="A1.Ex36.m1.1.1.1.1.3.1.cmml" xref="A1.Ex36.m1.1.1.1.1.3"></divide><apply id="A1.Ex36.m1.1.1.1.1.3.2.cmml" xref="A1.Ex36.m1.1.1.1.1.3.2"><plus id="A1.Ex36.m1.1.1.1.1.3.2.1.cmml" xref="A1.Ex36.m1.1.1.1.1.3.2.1"></plus><apply id="A1.Ex36.m1.1.1.1.1.3.2.2.cmml" xref="A1.Ex36.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A1.Ex36.m1.1.1.1.1.3.2.2.1.cmml" xref="A1.Ex36.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="A1.Ex36.m1.1.1.1.1.3.2.2.2.cmml" xref="A1.Ex36.m1.1.1.1.1.3.2.2.2">𝑛</ci><cn id="A1.Ex36.m1.1.1.1.1.3.2.2.3.cmml" type="integer" xref="A1.Ex36.m1.1.1.1.1.3.2.2.3">1</cn></apply><ci id="A1.Ex36.m1.1.1.1.1.3.2.3.cmml" xref="A1.Ex36.m1.1.1.1.1.3.2.3">𝛽</ci></apply><apply id="A1.Ex36.m1.1.1.1.1.3.3.cmml" xref="A1.Ex36.m1.1.1.1.1.3.3"><plus id="A1.Ex36.m1.1.1.1.1.3.3.1.cmml" xref="A1.Ex36.m1.1.1.1.1.3.3.1"></plus><ci id="A1.Ex36.m1.1.1.1.1.3.3.2.cmml" xref="A1.Ex36.m1.1.1.1.1.3.3.2">𝑛</ci><apply id="A1.Ex36.m1.1.1.1.1.3.3.3.cmml" xref="A1.Ex36.m1.1.1.1.1.3.3.3"><times id="A1.Ex36.m1.1.1.1.1.3.3.3.1.cmml" xref="A1.Ex36.m1.1.1.1.1.3.3.3.1"></times><cn id="A1.Ex36.m1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="A1.Ex36.m1.1.1.1.1.3.3.3.2">2</cn><ci id="A1.Ex36.m1.1.1.1.1.3.3.3.3.cmml" xref="A1.Ex36.m1.1.1.1.1.3.3.3.3">𝛽</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex36.m1.1c">\displaystyle=\frac{n_{1}+\beta}{n+2\beta},</annotation><annotation encoding="application/x-llamapun" id="A1.Ex36.m1.1d">= divide start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n + 2 italic_β end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.19">where we used the fact that <math alttext="P_{i_{1}^{k}}\sim\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="A1.1.p1.17.m1.1"><semantics id="A1.1.p1.17.m1.1a"><mrow id="A1.1.p1.17.m1.1.1" xref="A1.1.p1.17.m1.1.1.cmml"><msub id="A1.1.p1.17.m1.1.1.3" xref="A1.1.p1.17.m1.1.1.3.cmml"><mi id="A1.1.p1.17.m1.1.1.3.2" xref="A1.1.p1.17.m1.1.1.3.2.cmml">P</mi><msubsup id="A1.1.p1.17.m1.1.1.3.3" xref="A1.1.p1.17.m1.1.1.3.3.cmml"><mi id="A1.1.p1.17.m1.1.1.3.3.2.2" xref="A1.1.p1.17.m1.1.1.3.3.2.2.cmml">i</mi><mn id="A1.1.p1.17.m1.1.1.3.3.2.3" xref="A1.1.p1.17.m1.1.1.3.3.2.3.cmml">1</mn><mi id="A1.1.p1.17.m1.1.1.3.3.3" xref="A1.1.p1.17.m1.1.1.3.3.3.cmml">k</mi></msubsup></msub><mo id="A1.1.p1.17.m1.1.1.2" xref="A1.1.p1.17.m1.1.1.2.cmml">∼</mo><mrow id="A1.1.p1.17.m1.1.1.1" xref="A1.1.p1.17.m1.1.1.1.cmml"><mi id="A1.1.p1.17.m1.1.1.1.3" xref="A1.1.p1.17.m1.1.1.1.3.cmml">Dir</mi><mo id="A1.1.p1.17.m1.1.1.1.2" xref="A1.1.p1.17.m1.1.1.1.2.cmml">⁢</mo><mrow id="A1.1.p1.17.m1.1.1.1.1.1" xref="A1.1.p1.17.m1.1.1.1.1.1.1.cmml"><mo id="A1.1.p1.17.m1.1.1.1.1.1.2" stretchy="false" xref="A1.1.p1.17.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.1.p1.17.m1.1.1.1.1.1.1" xref="A1.1.p1.17.m1.1.1.1.1.1.1.cmml"><mi id="A1.1.p1.17.m1.1.1.1.1.1.1.2" xref="A1.1.p1.17.m1.1.1.1.1.1.1.2.cmml">β</mi><mo id="A1.1.p1.17.m1.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="A1.1.p1.17.m1.1.1.1.1.1.1.1.cmml">⋅</mo><mn id="A1.1.p1.17.m1.1.1.1.1.1.1.3" xref="A1.1.p1.17.m1.1.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="A1.1.p1.17.m1.1.1.1.1.1.3" stretchy="false" xref="A1.1.p1.17.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.17.m1.1b"><apply id="A1.1.p1.17.m1.1.1.cmml" xref="A1.1.p1.17.m1.1.1"><csymbol cd="latexml" id="A1.1.p1.17.m1.1.1.2.cmml" xref="A1.1.p1.17.m1.1.1.2">similar-to</csymbol><apply id="A1.1.p1.17.m1.1.1.3.cmml" xref="A1.1.p1.17.m1.1.1.3"><csymbol cd="ambiguous" id="A1.1.p1.17.m1.1.1.3.1.cmml" xref="A1.1.p1.17.m1.1.1.3">subscript</csymbol><ci id="A1.1.p1.17.m1.1.1.3.2.cmml" xref="A1.1.p1.17.m1.1.1.3.2">𝑃</ci><apply id="A1.1.p1.17.m1.1.1.3.3.cmml" xref="A1.1.p1.17.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.1.p1.17.m1.1.1.3.3.1.cmml" xref="A1.1.p1.17.m1.1.1.3.3">superscript</csymbol><apply id="A1.1.p1.17.m1.1.1.3.3.2.cmml" xref="A1.1.p1.17.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.1.p1.17.m1.1.1.3.3.2.1.cmml" xref="A1.1.p1.17.m1.1.1.3.3">subscript</csymbol><ci id="A1.1.p1.17.m1.1.1.3.3.2.2.cmml" xref="A1.1.p1.17.m1.1.1.3.3.2.2">𝑖</ci><cn id="A1.1.p1.17.m1.1.1.3.3.2.3.cmml" type="integer" xref="A1.1.p1.17.m1.1.1.3.3.2.3">1</cn></apply><ci id="A1.1.p1.17.m1.1.1.3.3.3.cmml" xref="A1.1.p1.17.m1.1.1.3.3.3">𝑘</ci></apply></apply><apply id="A1.1.p1.17.m1.1.1.1.cmml" xref="A1.1.p1.17.m1.1.1.1"><times id="A1.1.p1.17.m1.1.1.1.2.cmml" xref="A1.1.p1.17.m1.1.1.1.2"></times><ci id="A1.1.p1.17.m1.1.1.1.3.cmml" xref="A1.1.p1.17.m1.1.1.1.3">Dir</ci><apply id="A1.1.p1.17.m1.1.1.1.1.1.1.cmml" xref="A1.1.p1.17.m1.1.1.1.1.1"><ci id="A1.1.p1.17.m1.1.1.1.1.1.1.1.cmml" xref="A1.1.p1.17.m1.1.1.1.1.1.1.1">⋅</ci><ci id="A1.1.p1.17.m1.1.1.1.1.1.1.2.cmml" xref="A1.1.p1.17.m1.1.1.1.1.1.1.2">𝛽</ci><cn id="A1.1.p1.17.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.1.p1.17.m1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.17.m1.1c">P_{i_{1}^{k}}\sim\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation 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id="A1.1.p1.18.m2.4c">\int_{0}^{1}q^{z_{1}-1}(1-q)^{z_{0}-1}=\Gamma(z_{1})\Gamma(z_{0})/\Gamma(z_{1}% +z_{0})</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.18.m2.4d">∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT = roman_Γ ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) roman_Γ ( italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) / roman_Γ ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>, and that <math alttext="\Gamma(z+1)=z\Gamma(z)" class="ltx_Math" display="inline" id="A1.1.p1.19.m3.2"><semantics id="A1.1.p1.19.m3.2a"><mrow id="A1.1.p1.19.m3.2.2" xref="A1.1.p1.19.m3.2.2.cmml"><mrow id="A1.1.p1.19.m3.2.2.1" xref="A1.1.p1.19.m3.2.2.1.cmml"><mi id="A1.1.p1.19.m3.2.2.1.3" mathvariant="normal" xref="A1.1.p1.19.m3.2.2.1.3.cmml">Γ</mi><mo id="A1.1.p1.19.m3.2.2.1.2" xref="A1.1.p1.19.m3.2.2.1.2.cmml">⁢</mo><mrow id="A1.1.p1.19.m3.2.2.1.1.1" xref="A1.1.p1.19.m3.2.2.1.1.1.1.cmml"><mo id="A1.1.p1.19.m3.2.2.1.1.1.2" stretchy="false" xref="A1.1.p1.19.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.1.p1.19.m3.2.2.1.1.1.1" xref="A1.1.p1.19.m3.2.2.1.1.1.1.cmml"><mi id="A1.1.p1.19.m3.2.2.1.1.1.1.2" xref="A1.1.p1.19.m3.2.2.1.1.1.1.2.cmml">z</mi><mo id="A1.1.p1.19.m3.2.2.1.1.1.1.1" xref="A1.1.p1.19.m3.2.2.1.1.1.1.1.cmml">+</mo><mn id="A1.1.p1.19.m3.2.2.1.1.1.1.3" xref="A1.1.p1.19.m3.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.1.p1.19.m3.2.2.1.1.1.3" stretchy="false" xref="A1.1.p1.19.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.19.m3.2.2.2" xref="A1.1.p1.19.m3.2.2.2.cmml">=</mo><mrow id="A1.1.p1.19.m3.2.2.3" xref="A1.1.p1.19.m3.2.2.3.cmml"><mi id="A1.1.p1.19.m3.2.2.3.2" xref="A1.1.p1.19.m3.2.2.3.2.cmml">z</mi><mo id="A1.1.p1.19.m3.2.2.3.1" xref="A1.1.p1.19.m3.2.2.3.1.cmml">⁢</mo><mi id="A1.1.p1.19.m3.2.2.3.3" mathvariant="normal" xref="A1.1.p1.19.m3.2.2.3.3.cmml">Γ</mi><mo id="A1.1.p1.19.m3.2.2.3.1a" xref="A1.1.p1.19.m3.2.2.3.1.cmml">⁢</mo><mrow id="A1.1.p1.19.m3.2.2.3.4.2" xref="A1.1.p1.19.m3.2.2.3.cmml"><mo id="A1.1.p1.19.m3.2.2.3.4.2.1" stretchy="false" xref="A1.1.p1.19.m3.2.2.3.cmml">(</mo><mi id="A1.1.p1.19.m3.1.1" xref="A1.1.p1.19.m3.1.1.cmml">z</mi><mo id="A1.1.p1.19.m3.2.2.3.4.2.2" stretchy="false" xref="A1.1.p1.19.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.19.m3.2b"><apply id="A1.1.p1.19.m3.2.2.cmml" xref="A1.1.p1.19.m3.2.2"><eq id="A1.1.p1.19.m3.2.2.2.cmml" xref="A1.1.p1.19.m3.2.2.2"></eq><apply id="A1.1.p1.19.m3.2.2.1.cmml" xref="A1.1.p1.19.m3.2.2.1"><times id="A1.1.p1.19.m3.2.2.1.2.cmml" xref="A1.1.p1.19.m3.2.2.1.2"></times><ci id="A1.1.p1.19.m3.2.2.1.3.cmml" xref="A1.1.p1.19.m3.2.2.1.3">Γ</ci><apply id="A1.1.p1.19.m3.2.2.1.1.1.1.cmml" xref="A1.1.p1.19.m3.2.2.1.1.1"><plus id="A1.1.p1.19.m3.2.2.1.1.1.1.1.cmml" xref="A1.1.p1.19.m3.2.2.1.1.1.1.1"></plus><ci id="A1.1.p1.19.m3.2.2.1.1.1.1.2.cmml" xref="A1.1.p1.19.m3.2.2.1.1.1.1.2">𝑧</ci><cn id="A1.1.p1.19.m3.2.2.1.1.1.1.3.cmml" type="integer" xref="A1.1.p1.19.m3.2.2.1.1.1.1.3">1</cn></apply></apply><apply id="A1.1.p1.19.m3.2.2.3.cmml" xref="A1.1.p1.19.m3.2.2.3"><times id="A1.1.p1.19.m3.2.2.3.1.cmml" xref="A1.1.p1.19.m3.2.2.3.1"></times><ci id="A1.1.p1.19.m3.2.2.3.2.cmml" xref="A1.1.p1.19.m3.2.2.3.2">𝑧</ci><ci id="A1.1.p1.19.m3.2.2.3.3.cmml" xref="A1.1.p1.19.m3.2.2.3.3">Γ</ci><ci id="A1.1.p1.19.m3.1.1.cmml" xref="A1.1.p1.19.m3.1.1">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.19.m3.2c">\Gamma(z+1)=z\Gamma(z)</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.19.m3.2d">roman_Γ ( italic_z + 1 ) = italic_z roman_Γ ( italic_z )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="A1.p4"> <p class="ltx_p" id="A1.p4.6"><span class="ltx_text ltx_font_bold" id="A1.p4.6.1">Remark.</span> The proof above is for <math alttext="x_{1}^{k}\sim\mathrm{Unif}(\mathcal{X}^{k})" class="ltx_Math" display="inline" id="A1.p4.1.m1.1"><semantics id="A1.p4.1.m1.1a"><mrow id="A1.p4.1.m1.1.1" xref="A1.p4.1.m1.1.1.cmml"><msubsup id="A1.p4.1.m1.1.1.3" xref="A1.p4.1.m1.1.1.3.cmml"><mi id="A1.p4.1.m1.1.1.3.2.2" xref="A1.p4.1.m1.1.1.3.2.2.cmml">x</mi><mn id="A1.p4.1.m1.1.1.3.2.3" xref="A1.p4.1.m1.1.1.3.2.3.cmml">1</mn><mi id="A1.p4.1.m1.1.1.3.3" xref="A1.p4.1.m1.1.1.3.3.cmml">k</mi></msubsup><mo id="A1.p4.1.m1.1.1.2" xref="A1.p4.1.m1.1.1.2.cmml">∼</mo><mrow id="A1.p4.1.m1.1.1.1" xref="A1.p4.1.m1.1.1.1.cmml"><mi id="A1.p4.1.m1.1.1.1.3" xref="A1.p4.1.m1.1.1.1.3.cmml">Unif</mi><mo id="A1.p4.1.m1.1.1.1.2" xref="A1.p4.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="A1.p4.1.m1.1.1.1.1.1" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml"><mo id="A1.p4.1.m1.1.1.1.1.1.2" stretchy="false" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="A1.p4.1.m1.1.1.1.1.1.1" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p4.1.m1.1.1.1.1.1.1.2" xref="A1.p4.1.m1.1.1.1.1.1.1.2.cmml">𝒳</mi><mi id="A1.p4.1.m1.1.1.1.1.1.1.3" xref="A1.p4.1.m1.1.1.1.1.1.1.3.cmml">k</mi></msup><mo id="A1.p4.1.m1.1.1.1.1.1.3" stretchy="false" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p4.1.m1.1b"><apply id="A1.p4.1.m1.1.1.cmml" xref="A1.p4.1.m1.1.1"><csymbol cd="latexml" id="A1.p4.1.m1.1.1.2.cmml" xref="A1.p4.1.m1.1.1.2">similar-to</csymbol><apply id="A1.p4.1.m1.1.1.3.cmml" xref="A1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.p4.1.m1.1.1.3.1.cmml" xref="A1.p4.1.m1.1.1.3">superscript</csymbol><apply id="A1.p4.1.m1.1.1.3.2.cmml" xref="A1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.p4.1.m1.1.1.3.2.1.cmml" xref="A1.p4.1.m1.1.1.3">subscript</csymbol><ci id="A1.p4.1.m1.1.1.3.2.2.cmml" xref="A1.p4.1.m1.1.1.3.2.2">𝑥</ci><cn id="A1.p4.1.m1.1.1.3.2.3.cmml" type="integer" xref="A1.p4.1.m1.1.1.3.2.3">1</cn></apply><ci id="A1.p4.1.m1.1.1.3.3.cmml" xref="A1.p4.1.m1.1.1.3.3">𝑘</ci></apply><apply id="A1.p4.1.m1.1.1.1.cmml" xref="A1.p4.1.m1.1.1.1"><times id="A1.p4.1.m1.1.1.1.2.cmml" xref="A1.p4.1.m1.1.1.1.2"></times><ci id="A1.p4.1.m1.1.1.1.3.cmml" xref="A1.p4.1.m1.1.1.1.3">Unif</ci><apply id="A1.p4.1.m1.1.1.1.1.1.1.cmml" xref="A1.p4.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.p4.1.m1.1.1.1.1.1">superscript</csymbol><ci id="A1.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="A1.p4.1.m1.1.1.1.1.1.1.2">𝒳</ci><ci id="A1.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="A1.p4.1.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.1.m1.1c">x_{1}^{k}\sim\mathrm{Unif}(\mathcal{X}^{k})</annotation><annotation encoding="application/x-llamapun" id="A1.p4.1.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∼ roman_Unif ( caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>. However, note that the same proof would also work for <math alttext="x_{1}^{k}" class="ltx_Math" display="inline" id="A1.p4.2.m2.1"><semantics id="A1.p4.2.m2.1a"><msubsup id="A1.p4.2.m2.1.1" xref="A1.p4.2.m2.1.1.cmml"><mi id="A1.p4.2.m2.1.1.2.2" xref="A1.p4.2.m2.1.1.2.2.cmml">x</mi><mn id="A1.p4.2.m2.1.1.2.3" xref="A1.p4.2.m2.1.1.2.3.cmml">1</mn><mi id="A1.p4.2.m2.1.1.3" xref="A1.p4.2.m2.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.p4.2.m2.1b"><apply id="A1.p4.2.m2.1.1.cmml" xref="A1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="A1.p4.2.m2.1.1.1.cmml" xref="A1.p4.2.m2.1.1">superscript</csymbol><apply id="A1.p4.2.m2.1.1.2.cmml" xref="A1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="A1.p4.2.m2.1.1.2.1.cmml" xref="A1.p4.2.m2.1.1">subscript</csymbol><ci id="A1.p4.2.m2.1.1.2.2.cmml" xref="A1.p4.2.m2.1.1.2.2">𝑥</ci><cn id="A1.p4.2.m2.1.1.2.3.cmml" type="integer" xref="A1.p4.2.m2.1.1.2.3">1</cn></apply><ci id="A1.p4.2.m2.1.1.3.cmml" xref="A1.p4.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.2.m2.1c">x_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A1.p4.2.m2.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> distributed according to any distribution that is independent of the Markov kernel <math alttext="P" class="ltx_Math" display="inline" id="A1.p4.3.m3.1"><semantics id="A1.p4.3.m3.1a"><mi id="A1.p4.3.m3.1.1" xref="A1.p4.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A1.p4.3.m3.1b"><ci id="A1.p4.3.m3.1.1.cmml" xref="A1.p4.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="A1.p4.3.m3.1d">italic_P</annotation></semantics></math>. If instead the distribution depends on <math alttext="P" class="ltx_Math" display="inline" id="A1.p4.4.m4.1"><semantics id="A1.p4.4.m4.1a"><mi id="A1.p4.4.m4.1.1" xref="A1.p4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A1.p4.4.m4.1b"><ci id="A1.p4.4.m4.1.1.cmml" xref="A1.p4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="A1.p4.4.m4.1d">italic_P</annotation></semantics></math> (e.g., the stationary distribution of the Markov chain), then the proof would fail in the step where <math alttext="x_{1}^{t}|_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.p4.5.m5.2"><semantics id="A1.p4.5.m5.2a"><msub id="A1.p4.5.m5.2.2.1" xref="A1.p4.5.m5.2.2.2.cmml"><mrow id="A1.p4.5.m5.2.2.1.1" xref="A1.p4.5.m5.2.2.2.cmml"><msubsup id="A1.p4.5.m5.2.2.1.1.1" xref="A1.p4.5.m5.2.2.1.1.1.cmml"><mi id="A1.p4.5.m5.2.2.1.1.1.2.2" xref="A1.p4.5.m5.2.2.1.1.1.2.2.cmml">x</mi><mn id="A1.p4.5.m5.2.2.1.1.1.2.3" xref="A1.p4.5.m5.2.2.1.1.1.2.3.cmml">1</mn><mi id="A1.p4.5.m5.2.2.1.1.1.3" xref="A1.p4.5.m5.2.2.1.1.1.3.cmml">t</mi></msubsup><mo id="A1.p4.5.m5.2.2.1.1.2" stretchy="false" xref="A1.p4.5.m5.2.2.2.1.cmml">|</mo></mrow><msubsup id="A1.p4.5.m5.1.1.1" xref="A1.p4.5.m5.1.1.1.cmml"><mi id="A1.p4.5.m5.1.1.1.2.2" xref="A1.p4.5.m5.1.1.1.2.2.cmml">i</mi><mn id="A1.p4.5.m5.1.1.1.2.3" xref="A1.p4.5.m5.1.1.1.2.3.cmml">1</mn><mi id="A1.p4.5.m5.1.1.1.3" xref="A1.p4.5.m5.1.1.1.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.p4.5.m5.2b"><apply id="A1.p4.5.m5.2.2.2.cmml" xref="A1.p4.5.m5.2.2.1"><csymbol cd="latexml" id="A1.p4.5.m5.2.2.2.1.cmml" xref="A1.p4.5.m5.2.2.1.1.2">evaluated-at</csymbol><apply id="A1.p4.5.m5.2.2.1.1.1.cmml" xref="A1.p4.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.p4.5.m5.2.2.1.1.1.1.cmml" xref="A1.p4.5.m5.2.2.1.1.1">superscript</csymbol><apply id="A1.p4.5.m5.2.2.1.1.1.2.cmml" xref="A1.p4.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.p4.5.m5.2.2.1.1.1.2.1.cmml" xref="A1.p4.5.m5.2.2.1.1.1">subscript</csymbol><ci id="A1.p4.5.m5.2.2.1.1.1.2.2.cmml" xref="A1.p4.5.m5.2.2.1.1.1.2.2">𝑥</ci><cn id="A1.p4.5.m5.2.2.1.1.1.2.3.cmml" type="integer" xref="A1.p4.5.m5.2.2.1.1.1.2.3">1</cn></apply><ci id="A1.p4.5.m5.2.2.1.1.1.3.cmml" xref="A1.p4.5.m5.2.2.1.1.1.3">𝑡</ci></apply><apply id="A1.p4.5.m5.1.1.1.cmml" xref="A1.p4.5.m5.1.1.1"><csymbol cd="ambiguous" id="A1.p4.5.m5.1.1.1.1.cmml" xref="A1.p4.5.m5.1.1.1">superscript</csymbol><apply id="A1.p4.5.m5.1.1.1.2.cmml" xref="A1.p4.5.m5.1.1.1"><csymbol cd="ambiguous" id="A1.p4.5.m5.1.1.1.2.1.cmml" xref="A1.p4.5.m5.1.1.1">subscript</csymbol><ci id="A1.p4.5.m5.1.1.1.2.2.cmml" xref="A1.p4.5.m5.1.1.1.2.2">𝑖</ci><cn id="A1.p4.5.m5.1.1.1.2.3.cmml" type="integer" xref="A1.p4.5.m5.1.1.1.2.3">1</cn></apply><ci id="A1.p4.5.m5.1.1.1.3.cmml" xref="A1.p4.5.m5.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.5.m5.2c">x_{1}^{t}|_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.p4.5.m5.2d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a sufficient statistic for <math alttext="P_{i_{1}^{k}}" class="ltx_Math" display="inline" id="A1.p4.6.m6.1"><semantics id="A1.p4.6.m6.1a"><msub id="A1.p4.6.m6.1.1" xref="A1.p4.6.m6.1.1.cmml"><mi id="A1.p4.6.m6.1.1.2" xref="A1.p4.6.m6.1.1.2.cmml">P</mi><msubsup id="A1.p4.6.m6.1.1.3" xref="A1.p4.6.m6.1.1.3.cmml"><mi id="A1.p4.6.m6.1.1.3.2.2" xref="A1.p4.6.m6.1.1.3.2.2.cmml">i</mi><mn id="A1.p4.6.m6.1.1.3.2.3" xref="A1.p4.6.m6.1.1.3.2.3.cmml">1</mn><mi id="A1.p4.6.m6.1.1.3.3" xref="A1.p4.6.m6.1.1.3.3.cmml">k</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="A1.p4.6.m6.1b"><apply id="A1.p4.6.m6.1.1.cmml" xref="A1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="A1.p4.6.m6.1.1.1.cmml" xref="A1.p4.6.m6.1.1">subscript</csymbol><ci id="A1.p4.6.m6.1.1.2.cmml" xref="A1.p4.6.m6.1.1.2">𝑃</ci><apply id="A1.p4.6.m6.1.1.3.cmml" xref="A1.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="A1.p4.6.m6.1.1.3.1.cmml" xref="A1.p4.6.m6.1.1.3">superscript</csymbol><apply id="A1.p4.6.m6.1.1.3.2.cmml" xref="A1.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="A1.p4.6.m6.1.1.3.2.1.cmml" xref="A1.p4.6.m6.1.1.3">subscript</csymbol><ci id="A1.p4.6.m6.1.1.3.2.2.cmml" xref="A1.p4.6.m6.1.1.3.2.2">𝑖</ci><cn id="A1.p4.6.m6.1.1.3.2.3.cmml" type="integer" xref="A1.p4.6.m6.1.1.3.2.3">1</cn></apply><ci id="A1.p4.6.m6.1.1.3.3.cmml" xref="A1.p4.6.m6.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.6.m6.1c">P_{i_{1}^{k}}</annotation><annotation encoding="application/x-llamapun" id="A1.p4.6.m6.1d">italic_P start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.p5"> <p class="ltx_p" id="A1.p5.1"><span class="ltx_text ltx_font_bold" id="A1.p5.1.1">Remark.</span> It is important to note that, to be able to implement such a predictor requires in-context capabilities: at inference, in order to optimally predict the next token, the model must be able to look into the previous tokens of the test sequence, and count the tokens with the correct prefix.</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Preliminaries and Proof of Thm. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#Thmtheorem1" title="Theorem 1 (𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈 represents order-1 Laplacian smoothing). ‣ 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1</span></a> </h2> <section class="ltx_subsection" id="A2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.1 </span>Empirical insights</h3> <div class="ltx_para" id="A2.SS1.p1"> <p class="ltx_p" id="A2.SS1.p1.1">Here we expand upon our empirical observations in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.SS2.SSS1" title="4.2.1 Proof sketch ‣ 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">4.2.1</span></a>, which form the basis of our proof.</p> </div> <div class="ltx_para" id="A2.SS1.p2"> <p class="ltx_p" id="A2.SS1.p2.2"><span class="ltx_text ltx_font_bold" id="A2.SS1.p2.2.2">State-to-state transition factor <math alttext="a_{t}\approx 1" class="ltx_Math" display="inline" id="A2.SS1.p2.1.1.m1.1"><semantics id="A2.SS1.p2.1.1.m1.1a"><mrow id="A2.SS1.p2.1.1.m1.1.1" xref="A2.SS1.p2.1.1.m1.1.1.cmml"><msub id="A2.SS1.p2.1.1.m1.1.1.2" xref="A2.SS1.p2.1.1.m1.1.1.2.cmml"><mi id="A2.SS1.p2.1.1.m1.1.1.2.2" xref="A2.SS1.p2.1.1.m1.1.1.2.2.cmml">a</mi><mi id="A2.SS1.p2.1.1.m1.1.1.2.3" xref="A2.SS1.p2.1.1.m1.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS1.p2.1.1.m1.1.1.1" xref="A2.SS1.p2.1.1.m1.1.1.1.cmml">≈</mo><mn id="A2.SS1.p2.1.1.m1.1.1.3" xref="A2.SS1.p2.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p2.1.1.m1.1b"><apply id="A2.SS1.p2.1.1.m1.1.1.cmml" xref="A2.SS1.p2.1.1.m1.1.1"><approx id="A2.SS1.p2.1.1.m1.1.1.1.cmml" xref="A2.SS1.p2.1.1.m1.1.1.1"></approx><apply id="A2.SS1.p2.1.1.m1.1.1.2.cmml" xref="A2.SS1.p2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.p2.1.1.m1.1.1.2.1.cmml" xref="A2.SS1.p2.1.1.m1.1.1.2">subscript</csymbol><ci id="A2.SS1.p2.1.1.m1.1.1.2.2.cmml" xref="A2.SS1.p2.1.1.m1.1.1.2.2">𝑎</ci><ci id="A2.SS1.p2.1.1.m1.1.1.2.3.cmml" xref="A2.SS1.p2.1.1.m1.1.1.2.3">𝑡</ci></apply><cn id="A2.SS1.p2.1.1.m1.1.1.3.cmml" type="integer" xref="A2.SS1.p2.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p2.1.1.m1.1c">a_{t}\approx 1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p2.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≈ 1</annotation></semantics></math> for all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="A2.SS1.p2.2.2.m2.1"><semantics id="A2.SS1.p2.2.2.m2.1a"><mrow id="A2.SS1.p2.2.2.m2.1.1" xref="A2.SS1.p2.2.2.m2.1.1.cmml"><mi id="A2.SS1.p2.2.2.m2.1.1.2" xref="A2.SS1.p2.2.2.m2.1.1.2.cmml">t</mi><mo id="A2.SS1.p2.2.2.m2.1.1.1" xref="A2.SS1.p2.2.2.m2.1.1.1.cmml">≥</mo><mn id="A2.SS1.p2.2.2.m2.1.1.3" xref="A2.SS1.p2.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p2.2.2.m2.1b"><apply id="A2.SS1.p2.2.2.m2.1.1.cmml" xref="A2.SS1.p2.2.2.m2.1.1"><geq id="A2.SS1.p2.2.2.m2.1.1.1.cmml" xref="A2.SS1.p2.2.2.m2.1.1.1"></geq><ci id="A2.SS1.p2.2.2.m2.1.1.2.cmml" xref="A2.SS1.p2.2.2.m2.1.1.2">𝑡</ci><cn id="A2.SS1.p2.2.2.m2.1.1.3.cmml" type="integer" xref="A2.SS1.p2.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p2.2.2.m2.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p2.2.2.m2.1d">italic_t ≥ 1</annotation></semantics></math>.</span> We empirical evidence supporting this observation in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A2.F5" title="Figure 5 ‣ B.1 Empirical insights ‣ Appendix B Preliminaries and Proof of Thm. 1 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> <figure class="ltx_figure" id="A2.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="358" id="A2.F5.g1" src="x7.png" width="831"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="A2.F5.1"><svg height="16.07" overflow="visible" version="1.1" width="67.03"><g transform="translate(0,16.07) scale(1,-1)"><g transform="translate(-492.6,-9.69)"><text fill="black" transform="scale(1, -1)" x="0" y="0">Position <g transform="translate(0,1.383700013837) scale(1, -1)"><foreignobject height="1.383700013837" overflow="visible" width="1.383700013837"><math alttext="t" class="ltx_Math" display="inline" id="A2.F5.1.pic1.1.1.m1.1"><semantics id="A2.F5.1.pic1.1.1.m1.1a"><mi id="A2.F5.1.pic1.1.1.m1.1.1" xref="A2.F5.1.pic1.1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.F5.1.pic1.1.1.m1.1b"><ci id="A2.F5.1.pic1.1.1.m1.1.1.cmml" xref="A2.F5.1.pic1.1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F5.1.pic1.1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.F5.1.pic1.1.1.m1.1d">italic_t</annotation></semantics></math></foreignobject></g></text></g><g transform="translate(-664.18,128.68)"><g transform="translate(0,10.1010101010101) scale(1, -1)"><foreignobject height="16.0509201605092" overflow="visible" width="8.302200083022"> <div class="ltx_inline-block ltx_transformed_outer" id="A2.F5.1.pic1.2.1" style="width:6.0pt;height:7.3pt;vertical-align:4.3pt;"><span class="ltx_transformed_inner" style="width:7.3pt;transform:translate(-0.64pt,-4.08pt) rotate(-90deg) ;"> <p class="ltx_p" id="A2.F5.1.pic1.2.1.1"><math alttext="a_{t}" class="ltx_Math" display="inline" id="A2.F5.1.pic1.2.1.1.m1.1"><semantics id="A2.F5.1.pic1.2.1.1.m1.1a"><msub id="A2.F5.1.pic1.2.1.1.m1.1.1" xref="A2.F5.1.pic1.2.1.1.m1.1.1.cmml"><mi id="A2.F5.1.pic1.2.1.1.m1.1.1.2" xref="A2.F5.1.pic1.2.1.1.m1.1.1.2.cmml">a</mi><mi id="A2.F5.1.pic1.2.1.1.m1.1.1.3" xref="A2.F5.1.pic1.2.1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.F5.1.pic1.2.1.1.m1.1b"><apply id="A2.F5.1.pic1.2.1.1.m1.1.1.cmml" xref="A2.F5.1.pic1.2.1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.F5.1.pic1.2.1.1.m1.1.1.1.cmml" xref="A2.F5.1.pic1.2.1.1.m1.1.1">subscript</csymbol><ci id="A2.F5.1.pic1.2.1.1.m1.1.1.2.cmml" xref="A2.F5.1.pic1.2.1.1.m1.1.1.2">𝑎</ci><ci id="A2.F5.1.pic1.2.1.1.m1.1.1.3.cmml" xref="A2.F5.1.pic1.2.1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F5.1.pic1.2.1.1.m1.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.F5.1.pic1.2.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></p> </span></div></foreignobject></g></g></g></svg></p> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A2.F5.5.2.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="A2.F5.3.1" style="font-size:90%;">Value of <math alttext="a_{t}" class="ltx_Math" display="inline" id="A2.F5.3.1.m1.1"><semantics id="A2.F5.3.1.m1.1b"><msub id="A2.F5.3.1.m1.1.1" xref="A2.F5.3.1.m1.1.1.cmml"><mi id="A2.F5.3.1.m1.1.1.2" xref="A2.F5.3.1.m1.1.1.2.cmml">a</mi><mi id="A2.F5.3.1.m1.1.1.3" xref="A2.F5.3.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.F5.3.1.m1.1c"><apply id="A2.F5.3.1.m1.1.1.cmml" xref="A2.F5.3.1.m1.1.1"><csymbol cd="ambiguous" id="A2.F5.3.1.m1.1.1.1.cmml" xref="A2.F5.3.1.m1.1.1">subscript</csymbol><ci id="A2.F5.3.1.m1.1.1.2.cmml" xref="A2.F5.3.1.m1.1.1.2">𝑎</ci><ci id="A2.F5.3.1.m1.1.1.3.cmml" xref="A2.F5.3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F5.3.1.m1.1d">a_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.F5.3.1.m1.1e">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> across positions at convergence.</span></figcaption> </figure> <div class="ltx_para" id="A2.SS1.p3"> <p class="ltx_p" id="A2.SS1.p3.15"><span class="ltx_text ltx_font_bold" id="A2.SS1.p3.1.1">Convolution window <math alttext="w\geq k+1" class="ltx_Math" display="inline" id="A2.SS1.p3.1.1.m1.1"><semantics id="A2.SS1.p3.1.1.m1.1a"><mrow id="A2.SS1.p3.1.1.m1.1.1" xref="A2.SS1.p3.1.1.m1.1.1.cmml"><mi id="A2.SS1.p3.1.1.m1.1.1.2" xref="A2.SS1.p3.1.1.m1.1.1.2.cmml">w</mi><mo id="A2.SS1.p3.1.1.m1.1.1.1" xref="A2.SS1.p3.1.1.m1.1.1.1.cmml">≥</mo><mrow id="A2.SS1.p3.1.1.m1.1.1.3" xref="A2.SS1.p3.1.1.m1.1.1.3.cmml"><mi id="A2.SS1.p3.1.1.m1.1.1.3.2" xref="A2.SS1.p3.1.1.m1.1.1.3.2.cmml">k</mi><mo id="A2.SS1.p3.1.1.m1.1.1.3.1" xref="A2.SS1.p3.1.1.m1.1.1.3.1.cmml">+</mo><mn id="A2.SS1.p3.1.1.m1.1.1.3.3" xref="A2.SS1.p3.1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.1.1.m1.1b"><apply id="A2.SS1.p3.1.1.m1.1.1.cmml" xref="A2.SS1.p3.1.1.m1.1.1"><geq id="A2.SS1.p3.1.1.m1.1.1.1.cmml" xref="A2.SS1.p3.1.1.m1.1.1.1"></geq><ci id="A2.SS1.p3.1.1.m1.1.1.2.cmml" xref="A2.SS1.p3.1.1.m1.1.1.2">𝑤</ci><apply id="A2.SS1.p3.1.1.m1.1.1.3.cmml" xref="A2.SS1.p3.1.1.m1.1.1.3"><plus id="A2.SS1.p3.1.1.m1.1.1.3.1.cmml" xref="A2.SS1.p3.1.1.m1.1.1.3.1"></plus><ci id="A2.SS1.p3.1.1.m1.1.1.3.2.cmml" xref="A2.SS1.p3.1.1.m1.1.1.3.2">𝑘</ci><cn id="A2.SS1.p3.1.1.m1.1.1.3.3.cmml" type="integer" xref="A2.SS1.p3.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.1.1.m1.1c">w\geq k+1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.1.1.m1.1d">italic_w ≥ italic_k + 1</annotation></semantics></math>.</span> Recalling that <math alttext="k" class="ltx_Math" display="inline" id="A2.SS1.p3.2.m1.1"><semantics id="A2.SS1.p3.2.m1.1a"><mi id="A2.SS1.p3.2.m1.1.1" xref="A2.SS1.p3.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.2.m1.1b"><ci id="A2.SS1.p3.2.m1.1.1.cmml" xref="A2.SS1.p3.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.2.m1.1d">italic_k</annotation></semantics></math> is the Markov order, we empirically observe that the window that <math alttext="w=k+1" class="ltx_Math" display="inline" id="A2.SS1.p3.3.m2.1"><semantics id="A2.SS1.p3.3.m2.1a"><mrow id="A2.SS1.p3.3.m2.1.1" xref="A2.SS1.p3.3.m2.1.1.cmml"><mi id="A2.SS1.p3.3.m2.1.1.2" xref="A2.SS1.p3.3.m2.1.1.2.cmml">w</mi><mo id="A2.SS1.p3.3.m2.1.1.1" xref="A2.SS1.p3.3.m2.1.1.1.cmml">=</mo><mrow id="A2.SS1.p3.3.m2.1.1.3" xref="A2.SS1.p3.3.m2.1.1.3.cmml"><mi id="A2.SS1.p3.3.m2.1.1.3.2" xref="A2.SS1.p3.3.m2.1.1.3.2.cmml">k</mi><mo id="A2.SS1.p3.3.m2.1.1.3.1" xref="A2.SS1.p3.3.m2.1.1.3.1.cmml">+</mo><mn id="A2.SS1.p3.3.m2.1.1.3.3" xref="A2.SS1.p3.3.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.3.m2.1b"><apply id="A2.SS1.p3.3.m2.1.1.cmml" xref="A2.SS1.p3.3.m2.1.1"><eq id="A2.SS1.p3.3.m2.1.1.1.cmml" xref="A2.SS1.p3.3.m2.1.1.1"></eq><ci id="A2.SS1.p3.3.m2.1.1.2.cmml" xref="A2.SS1.p3.3.m2.1.1.2">𝑤</ci><apply id="A2.SS1.p3.3.m2.1.1.3.cmml" xref="A2.SS1.p3.3.m2.1.1.3"><plus id="A2.SS1.p3.3.m2.1.1.3.1.cmml" xref="A2.SS1.p3.3.m2.1.1.3.1"></plus><ci id="A2.SS1.p3.3.m2.1.1.3.2.cmml" xref="A2.SS1.p3.3.m2.1.1.3.2">𝑘</ci><cn id="A2.SS1.p3.3.m2.1.1.3.3.cmml" type="integer" xref="A2.SS1.p3.3.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.3.m2.1c">w=k+1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.3.m2.1d">italic_w = italic_k + 1</annotation></semantics></math> is sufficient for the full Mamba to learn the Laplacian smoothing on <math alttext="k^{\text{th}}" class="ltx_Math" display="inline" id="A2.SS1.p3.4.m3.1"><semantics id="A2.SS1.p3.4.m3.1a"><msup id="A2.SS1.p3.4.m3.1.1" xref="A2.SS1.p3.4.m3.1.1.cmml"><mi id="A2.SS1.p3.4.m3.1.1.2" xref="A2.SS1.p3.4.m3.1.1.2.cmml">k</mi><mtext id="A2.SS1.p3.4.m3.1.1.3" xref="A2.SS1.p3.4.m3.1.1.3a.cmml">th</mtext></msup><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.4.m3.1b"><apply id="A2.SS1.p3.4.m3.1.1.cmml" xref="A2.SS1.p3.4.m3.1.1"><csymbol cd="ambiguous" id="A2.SS1.p3.4.m3.1.1.1.cmml" xref="A2.SS1.p3.4.m3.1.1">superscript</csymbol><ci id="A2.SS1.p3.4.m3.1.1.2.cmml" xref="A2.SS1.p3.4.m3.1.1.2">𝑘</ci><ci id="A2.SS1.p3.4.m3.1.1.3a.cmml" xref="A2.SS1.p3.4.m3.1.1.3"><mtext id="A2.SS1.p3.4.m3.1.1.3.cmml" mathsize="70%" xref="A2.SS1.p3.4.m3.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.4.m3.1c">k^{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.4.m3.1d">italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT</annotation></semantics></math>-order Markov chains. To understand why, note that in the <math alttext="\mathsf{MambaZero}" class="ltx_Math" display="inline" id="A2.SS1.p3.5.m4.1"><semantics id="A2.SS1.p3.5.m4.1a"><mi id="A2.SS1.p3.5.m4.1.1" xref="A2.SS1.p3.5.m4.1.1.cmml">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.5.m4.1b"><ci id="A2.SS1.p3.5.m4.1.1.cmml" xref="A2.SS1.p3.5.m4.1.1">𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.5.m4.1c">\mathsf{MambaZero}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.5.m4.1d">sansserif_MambaZero</annotation></semantics></math> architecture above, apart from the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex13" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a> block, all remaining equations operate on the current token at time <math alttext="t" class="ltx_Math" display="inline" id="A2.SS1.p3.6.m5.1"><semantics id="A2.SS1.p3.6.m5.1a"><mi id="A2.SS1.p3.6.m5.1.1" xref="A2.SS1.p3.6.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.6.m5.1b"><ci id="A2.SS1.p3.6.m5.1.1.cmml" xref="A2.SS1.p3.6.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.6.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.6.m5.1d">italic_t</annotation></semantics></math>. In the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.Ex13" title="Equation MambaZero ‣ 4.1 MambaZero: Simplified model ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">MambaZero</span></a> block, same as the <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex1" title="Equation Mamba ‣ 2.2 Mamba architecture ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Mamba</span></a> block except ReLU, the dependency of the output <math alttext="\bm{y}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p3.7.m6.1"><semantics id="A2.SS1.p3.7.m6.1a"><msub id="A2.SS1.p3.7.m6.1.1" xref="A2.SS1.p3.7.m6.1.1.cmml"><mi id="A2.SS1.p3.7.m6.1.1.2" xref="A2.SS1.p3.7.m6.1.1.2.cmml">𝒚</mi><mi id="A2.SS1.p3.7.m6.1.1.3" xref="A2.SS1.p3.7.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.7.m6.1b"><apply id="A2.SS1.p3.7.m6.1.1.cmml" xref="A2.SS1.p3.7.m6.1.1"><csymbol cd="ambiguous" id="A2.SS1.p3.7.m6.1.1.1.cmml" xref="A2.SS1.p3.7.m6.1.1">subscript</csymbol><ci id="A2.SS1.p3.7.m6.1.1.2.cmml" xref="A2.SS1.p3.7.m6.1.1.2">𝒚</ci><ci id="A2.SS1.p3.7.m6.1.1.3.cmml" xref="A2.SS1.p3.7.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.7.m6.1c">\bm{y}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.7.m6.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> on the previous tokens is due to that of the state <math alttext="H_{t}" class="ltx_Math" display="inline" id="A2.SS1.p3.8.m7.1"><semantics id="A2.SS1.p3.8.m7.1a"><msub id="A2.SS1.p3.8.m7.1.1" xref="A2.SS1.p3.8.m7.1.1.cmml"><mi id="A2.SS1.p3.8.m7.1.1.2" xref="A2.SS1.p3.8.m7.1.1.2.cmml">H</mi><mi id="A2.SS1.p3.8.m7.1.1.3" xref="A2.SS1.p3.8.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.8.m7.1b"><apply id="A2.SS1.p3.8.m7.1.1.cmml" xref="A2.SS1.p3.8.m7.1.1"><csymbol cd="ambiguous" id="A2.SS1.p3.8.m7.1.1.1.cmml" xref="A2.SS1.p3.8.m7.1.1">subscript</csymbol><ci id="A2.SS1.p3.8.m7.1.1.2.cmml" xref="A2.SS1.p3.8.m7.1.1.2">𝐻</ci><ci id="A2.SS1.p3.8.m7.1.1.3.cmml" xref="A2.SS1.p3.8.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.8.m7.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.8.m7.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="(\widetilde{\bm{x}}_{t},\bm{b}_{t})" class="ltx_Math" display="inline" id="A2.SS1.p3.9.m8.2"><semantics id="A2.SS1.p3.9.m8.2a"><mrow id="A2.SS1.p3.9.m8.2.2.2" xref="A2.SS1.p3.9.m8.2.2.3.cmml"><mo id="A2.SS1.p3.9.m8.2.2.2.3" stretchy="false" xref="A2.SS1.p3.9.m8.2.2.3.cmml">(</mo><msub id="A2.SS1.p3.9.m8.1.1.1.1" xref="A2.SS1.p3.9.m8.1.1.1.1.cmml"><mover accent="true" id="A2.SS1.p3.9.m8.1.1.1.1.2" xref="A2.SS1.p3.9.m8.1.1.1.1.2.cmml"><mi id="A2.SS1.p3.9.m8.1.1.1.1.2.2" xref="A2.SS1.p3.9.m8.1.1.1.1.2.2.cmml">𝒙</mi><mo id="A2.SS1.p3.9.m8.1.1.1.1.2.1" xref="A2.SS1.p3.9.m8.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A2.SS1.p3.9.m8.1.1.1.1.3" xref="A2.SS1.p3.9.m8.1.1.1.1.3.cmml">t</mi></msub><mo id="A2.SS1.p3.9.m8.2.2.2.4" xref="A2.SS1.p3.9.m8.2.2.3.cmml">,</mo><msub id="A2.SS1.p3.9.m8.2.2.2.2" xref="A2.SS1.p3.9.m8.2.2.2.2.cmml"><mi id="A2.SS1.p3.9.m8.2.2.2.2.2" xref="A2.SS1.p3.9.m8.2.2.2.2.2.cmml">𝒃</mi><mi id="A2.SS1.p3.9.m8.2.2.2.2.3" xref="A2.SS1.p3.9.m8.2.2.2.2.3.cmml">t</mi></msub><mo id="A2.SS1.p3.9.m8.2.2.2.5" stretchy="false" xref="A2.SS1.p3.9.m8.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.9.m8.2b"><interval closure="open" id="A2.SS1.p3.9.m8.2.2.3.cmml" xref="A2.SS1.p3.9.m8.2.2.2"><apply id="A2.SS1.p3.9.m8.1.1.1.1.cmml" xref="A2.SS1.p3.9.m8.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p3.9.m8.1.1.1.1.1.cmml" xref="A2.SS1.p3.9.m8.1.1.1.1">subscript</csymbol><apply id="A2.SS1.p3.9.m8.1.1.1.1.2.cmml" xref="A2.SS1.p3.9.m8.1.1.1.1.2"><ci id="A2.SS1.p3.9.m8.1.1.1.1.2.1.cmml" xref="A2.SS1.p3.9.m8.1.1.1.1.2.1">~</ci><ci id="A2.SS1.p3.9.m8.1.1.1.1.2.2.cmml" xref="A2.SS1.p3.9.m8.1.1.1.1.2.2">𝒙</ci></apply><ci id="A2.SS1.p3.9.m8.1.1.1.1.3.cmml" xref="A2.SS1.p3.9.m8.1.1.1.1.3">𝑡</ci></apply><apply id="A2.SS1.p3.9.m8.2.2.2.2.cmml" xref="A2.SS1.p3.9.m8.2.2.2.2"><csymbol cd="ambiguous" id="A2.SS1.p3.9.m8.2.2.2.2.1.cmml" xref="A2.SS1.p3.9.m8.2.2.2.2">subscript</csymbol><ci id="A2.SS1.p3.9.m8.2.2.2.2.2.cmml" xref="A2.SS1.p3.9.m8.2.2.2.2.2">𝒃</ci><ci id="A2.SS1.p3.9.m8.2.2.2.2.3.cmml" xref="A2.SS1.p3.9.m8.2.2.2.2.3">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.9.m8.2c">(\widetilde{\bm{x}}_{t},\bm{b}_{t})</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.9.m8.2d">( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> in the update equation, and of <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p3.10.m9.1"><semantics id="A2.SS1.p3.10.m9.1a"><msub id="A2.SS1.p3.10.m9.1.1" xref="A2.SS1.p3.10.m9.1.1.cmml"><mi id="A2.SS1.p3.10.m9.1.1.2" xref="A2.SS1.p3.10.m9.1.1.2.cmml">𝒄</mi><mi id="A2.SS1.p3.10.m9.1.1.3" xref="A2.SS1.p3.10.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.10.m9.1b"><apply id="A2.SS1.p3.10.m9.1.1.cmml" xref="A2.SS1.p3.10.m9.1.1"><csymbol cd="ambiguous" id="A2.SS1.p3.10.m9.1.1.1.cmml" xref="A2.SS1.p3.10.m9.1.1">subscript</csymbol><ci id="A2.SS1.p3.10.m9.1.1.2.cmml" xref="A2.SS1.p3.10.m9.1.1.2">𝒄</ci><ci id="A2.SS1.p3.10.m9.1.1.3.cmml" xref="A2.SS1.p3.10.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.10.m9.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.10.m9.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in the state projection. Since <math alttext="(\widetilde{\bm{x}}_{t},\bm{b}_{t},\bm{c}_{t})" class="ltx_Math" display="inline" id="A2.SS1.p3.11.m10.3"><semantics id="A2.SS1.p3.11.m10.3a"><mrow id="A2.SS1.p3.11.m10.3.3.3" xref="A2.SS1.p3.11.m10.3.3.4.cmml"><mo id="A2.SS1.p3.11.m10.3.3.3.4" stretchy="false" xref="A2.SS1.p3.11.m10.3.3.4.cmml">(</mo><msub id="A2.SS1.p3.11.m10.1.1.1.1" xref="A2.SS1.p3.11.m10.1.1.1.1.cmml"><mover accent="true" id="A2.SS1.p3.11.m10.1.1.1.1.2" xref="A2.SS1.p3.11.m10.1.1.1.1.2.cmml"><mi id="A2.SS1.p3.11.m10.1.1.1.1.2.2" xref="A2.SS1.p3.11.m10.1.1.1.1.2.2.cmml">𝒙</mi><mo id="A2.SS1.p3.11.m10.1.1.1.1.2.1" xref="A2.SS1.p3.11.m10.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A2.SS1.p3.11.m10.1.1.1.1.3" xref="A2.SS1.p3.11.m10.1.1.1.1.3.cmml">t</mi></msub><mo id="A2.SS1.p3.11.m10.3.3.3.5" xref="A2.SS1.p3.11.m10.3.3.4.cmml">,</mo><msub id="A2.SS1.p3.11.m10.2.2.2.2" xref="A2.SS1.p3.11.m10.2.2.2.2.cmml"><mi id="A2.SS1.p3.11.m10.2.2.2.2.2" xref="A2.SS1.p3.11.m10.2.2.2.2.2.cmml">𝒃</mi><mi id="A2.SS1.p3.11.m10.2.2.2.2.3" xref="A2.SS1.p3.11.m10.2.2.2.2.3.cmml">t</mi></msub><mo id="A2.SS1.p3.11.m10.3.3.3.6" xref="A2.SS1.p3.11.m10.3.3.4.cmml">,</mo><msub id="A2.SS1.p3.11.m10.3.3.3.3" xref="A2.SS1.p3.11.m10.3.3.3.3.cmml"><mi id="A2.SS1.p3.11.m10.3.3.3.3.2" xref="A2.SS1.p3.11.m10.3.3.3.3.2.cmml">𝒄</mi><mi id="A2.SS1.p3.11.m10.3.3.3.3.3" xref="A2.SS1.p3.11.m10.3.3.3.3.3.cmml">t</mi></msub><mo id="A2.SS1.p3.11.m10.3.3.3.7" stretchy="false" xref="A2.SS1.p3.11.m10.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.11.m10.3b"><vector id="A2.SS1.p3.11.m10.3.3.4.cmml" xref="A2.SS1.p3.11.m10.3.3.3"><apply id="A2.SS1.p3.11.m10.1.1.1.1.cmml" xref="A2.SS1.p3.11.m10.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p3.11.m10.1.1.1.1.1.cmml" xref="A2.SS1.p3.11.m10.1.1.1.1">subscript</csymbol><apply id="A2.SS1.p3.11.m10.1.1.1.1.2.cmml" xref="A2.SS1.p3.11.m10.1.1.1.1.2"><ci id="A2.SS1.p3.11.m10.1.1.1.1.2.1.cmml" xref="A2.SS1.p3.11.m10.1.1.1.1.2.1">~</ci><ci id="A2.SS1.p3.11.m10.1.1.1.1.2.2.cmml" xref="A2.SS1.p3.11.m10.1.1.1.1.2.2">𝒙</ci></apply><ci id="A2.SS1.p3.11.m10.1.1.1.1.3.cmml" xref="A2.SS1.p3.11.m10.1.1.1.1.3">𝑡</ci></apply><apply id="A2.SS1.p3.11.m10.2.2.2.2.cmml" xref="A2.SS1.p3.11.m10.2.2.2.2"><csymbol cd="ambiguous" id="A2.SS1.p3.11.m10.2.2.2.2.1.cmml" xref="A2.SS1.p3.11.m10.2.2.2.2">subscript</csymbol><ci id="A2.SS1.p3.11.m10.2.2.2.2.2.cmml" xref="A2.SS1.p3.11.m10.2.2.2.2.2">𝒃</ci><ci id="A2.SS1.p3.11.m10.2.2.2.2.3.cmml" xref="A2.SS1.p3.11.m10.2.2.2.2.3">𝑡</ci></apply><apply id="A2.SS1.p3.11.m10.3.3.3.3.cmml" xref="A2.SS1.p3.11.m10.3.3.3.3"><csymbol cd="ambiguous" id="A2.SS1.p3.11.m10.3.3.3.3.1.cmml" xref="A2.SS1.p3.11.m10.3.3.3.3">subscript</csymbol><ci id="A2.SS1.p3.11.m10.3.3.3.3.2.cmml" xref="A2.SS1.p3.11.m10.3.3.3.3.2">𝒄</ci><ci id="A2.SS1.p3.11.m10.3.3.3.3.3.cmml" xref="A2.SS1.p3.11.m10.3.3.3.3.3">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.11.m10.3c">(\widetilde{\bm{x}}_{t},\bm{b}_{t},\bm{c}_{t})</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.11.m10.3d">( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> depend on the past through the convolutions, a window of size <math alttext="k+1" class="ltx_Math" display="inline" id="A2.SS1.p3.12.m11.1"><semantics id="A2.SS1.p3.12.m11.1a"><mrow id="A2.SS1.p3.12.m11.1.1" xref="A2.SS1.p3.12.m11.1.1.cmml"><mi id="A2.SS1.p3.12.m11.1.1.2" xref="A2.SS1.p3.12.m11.1.1.2.cmml">k</mi><mo id="A2.SS1.p3.12.m11.1.1.1" xref="A2.SS1.p3.12.m11.1.1.1.cmml">+</mo><mn id="A2.SS1.p3.12.m11.1.1.3" xref="A2.SS1.p3.12.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.12.m11.1b"><apply id="A2.SS1.p3.12.m11.1.1.cmml" xref="A2.SS1.p3.12.m11.1.1"><plus id="A2.SS1.p3.12.m11.1.1.1.cmml" xref="A2.SS1.p3.12.m11.1.1.1"></plus><ci id="A2.SS1.p3.12.m11.1.1.2.cmml" xref="A2.SS1.p3.12.m11.1.1.2">𝑘</ci><cn id="A2.SS1.p3.12.m11.1.1.3.cmml" type="integer" xref="A2.SS1.p3.12.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.12.m11.1c">k+1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.12.m11.1d">italic_k + 1</annotation></semantics></math> enables them to keep track of the current token as well as its length-<math alttext="k" class="ltx_Math" display="inline" id="A2.SS1.p3.13.m12.1"><semantics id="A2.SS1.p3.13.m12.1a"><mi id="A2.SS1.p3.13.m12.1.1" xref="A2.SS1.p3.13.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.13.m12.1b"><ci id="A2.SS1.p3.13.m12.1.1.cmml" xref="A2.SS1.p3.13.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.13.m12.1c">k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.13.m12.1d">italic_k</annotation></semantics></math> prefix, which is necessary to compute the counts needed in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S2.Ex8" title="Equation Laplacian smoothing ‣ 2.4 Optimal estimator: Laplacian smoothing ‣ 2 Problem setup ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Laplacian smoothing</span></a>. On the other hand, if <math alttext="w\leq k" class="ltx_Math" display="inline" id="A2.SS1.p3.14.m13.1"><semantics id="A2.SS1.p3.14.m13.1a"><mrow id="A2.SS1.p3.14.m13.1.1" xref="A2.SS1.p3.14.m13.1.1.cmml"><mi id="A2.SS1.p3.14.m13.1.1.2" xref="A2.SS1.p3.14.m13.1.1.2.cmml">w</mi><mo id="A2.SS1.p3.14.m13.1.1.1" xref="A2.SS1.p3.14.m13.1.1.1.cmml">≤</mo><mi id="A2.SS1.p3.14.m13.1.1.3" xref="A2.SS1.p3.14.m13.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.14.m13.1b"><apply id="A2.SS1.p3.14.m13.1.1.cmml" xref="A2.SS1.p3.14.m13.1.1"><leq id="A2.SS1.p3.14.m13.1.1.1.cmml" xref="A2.SS1.p3.14.m13.1.1.1"></leq><ci id="A2.SS1.p3.14.m13.1.1.2.cmml" xref="A2.SS1.p3.14.m13.1.1.2">𝑤</ci><ci id="A2.SS1.p3.14.m13.1.1.3.cmml" xref="A2.SS1.p3.14.m13.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.14.m13.1c">w\leq k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.14.m13.1d">italic_w ≤ italic_k</annotation></semantics></math>, then one can find <em class="ltx_emph ltx_font_italic" id="A2.SS1.p3.15.2">confusable</em> sequences, i.e. sequences that share the same number of occurrences of all length-<math alttext="k" class="ltx_Math" display="inline" id="A2.SS1.p3.15.m14.1"><semantics id="A2.SS1.p3.15.m14.1a"><mi id="A2.SS1.p3.15.m14.1.1" xref="A2.SS1.p3.15.m14.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p3.15.m14.1b"><ci id="A2.SS1.p3.15.m14.1.1.cmml" xref="A2.SS1.p3.15.m14.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p3.15.m14.1c">k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p3.15.m14.1d">italic_k</annotation></semantics></math> prefixes, but whose counts of the tokens following each prefix is different.</p> </div> <div class="ltx_para" id="A2.SS1.p4"> <p class="ltx_p" id="A2.SS1.p4.18">For such sequences, the state <math alttext="H_{t}" class="ltx_Math" display="inline" id="A2.SS1.p4.1.m1.1"><semantics id="A2.SS1.p4.1.m1.1a"><msub id="A2.SS1.p4.1.m1.1.1" xref="A2.SS1.p4.1.m1.1.1.cmml"><mi id="A2.SS1.p4.1.m1.1.1.2" xref="A2.SS1.p4.1.m1.1.1.2.cmml">H</mi><mi id="A2.SS1.p4.1.m1.1.1.3" xref="A2.SS1.p4.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.1.m1.1b"><apply id="A2.SS1.p4.1.m1.1.1.cmml" xref="A2.SS1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p4.1.m1.1.1.1.cmml" xref="A2.SS1.p4.1.m1.1.1">subscript</csymbol><ci id="A2.SS1.p4.1.m1.1.1.2.cmml" xref="A2.SS1.p4.1.m1.1.1.2">𝐻</ci><ci id="A2.SS1.p4.1.m1.1.1.3.cmml" xref="A2.SS1.p4.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.1.m1.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.1.m1.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is the same, and so are the predicted probabilities by the Mamba model; however, the optimal estimator, depending on the transition counts, would give very different probability estimates, allowing Mamba’s prediction loss to deviate from that of the optimal. For example, consider <math alttext="k=1" class="ltx_Math" display="inline" id="A2.SS1.p4.2.m2.1"><semantics id="A2.SS1.p4.2.m2.1a"><mrow id="A2.SS1.p4.2.m2.1.1" xref="A2.SS1.p4.2.m2.1.1.cmml"><mi id="A2.SS1.p4.2.m2.1.1.2" xref="A2.SS1.p4.2.m2.1.1.2.cmml">k</mi><mo id="A2.SS1.p4.2.m2.1.1.1" xref="A2.SS1.p4.2.m2.1.1.1.cmml">=</mo><mn id="A2.SS1.p4.2.m2.1.1.3" xref="A2.SS1.p4.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.2.m2.1b"><apply id="A2.SS1.p4.2.m2.1.1.cmml" xref="A2.SS1.p4.2.m2.1.1"><eq id="A2.SS1.p4.2.m2.1.1.1.cmml" xref="A2.SS1.p4.2.m2.1.1.1"></eq><ci id="A2.SS1.p4.2.m2.1.1.2.cmml" xref="A2.SS1.p4.2.m2.1.1.2">𝑘</ci><cn id="A2.SS1.p4.2.m2.1.1.3.cmml" type="integer" xref="A2.SS1.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.2.m2.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.2.m2.1d">italic_k = 1</annotation></semantics></math>. If <math alttext="w=1" class="ltx_Math" display="inline" id="A2.SS1.p4.3.m3.1"><semantics id="A2.SS1.p4.3.m3.1a"><mrow id="A2.SS1.p4.3.m3.1.1" xref="A2.SS1.p4.3.m3.1.1.cmml"><mi id="A2.SS1.p4.3.m3.1.1.2" xref="A2.SS1.p4.3.m3.1.1.2.cmml">w</mi><mo id="A2.SS1.p4.3.m3.1.1.1" xref="A2.SS1.p4.3.m3.1.1.1.cmml">=</mo><mn id="A2.SS1.p4.3.m3.1.1.3" xref="A2.SS1.p4.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.3.m3.1b"><apply id="A2.SS1.p4.3.m3.1.1.cmml" xref="A2.SS1.p4.3.m3.1.1"><eq id="A2.SS1.p4.3.m3.1.1.1.cmml" xref="A2.SS1.p4.3.m3.1.1.1"></eq><ci id="A2.SS1.p4.3.m3.1.1.2.cmml" xref="A2.SS1.p4.3.m3.1.1.2">𝑤</ci><cn id="A2.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="A2.SS1.p4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.3.m3.1c">w=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.3.m3.1d">italic_w = 1</annotation></semantics></math>, then <math alttext="(\widetilde{\bm{x}}_{t},\bm{b}_{t},\bm{c}_{t})" class="ltx_Math" display="inline" id="A2.SS1.p4.4.m4.3"><semantics id="A2.SS1.p4.4.m4.3a"><mrow id="A2.SS1.p4.4.m4.3.3.3" xref="A2.SS1.p4.4.m4.3.3.4.cmml"><mo id="A2.SS1.p4.4.m4.3.3.3.4" stretchy="false" xref="A2.SS1.p4.4.m4.3.3.4.cmml">(</mo><msub id="A2.SS1.p4.4.m4.1.1.1.1" xref="A2.SS1.p4.4.m4.1.1.1.1.cmml"><mover accent="true" id="A2.SS1.p4.4.m4.1.1.1.1.2" xref="A2.SS1.p4.4.m4.1.1.1.1.2.cmml"><mi id="A2.SS1.p4.4.m4.1.1.1.1.2.2" xref="A2.SS1.p4.4.m4.1.1.1.1.2.2.cmml">𝒙</mi><mo id="A2.SS1.p4.4.m4.1.1.1.1.2.1" xref="A2.SS1.p4.4.m4.1.1.1.1.2.1.cmml">~</mo></mover><mi id="A2.SS1.p4.4.m4.1.1.1.1.3" xref="A2.SS1.p4.4.m4.1.1.1.1.3.cmml">t</mi></msub><mo id="A2.SS1.p4.4.m4.3.3.3.5" xref="A2.SS1.p4.4.m4.3.3.4.cmml">,</mo><msub id="A2.SS1.p4.4.m4.2.2.2.2" xref="A2.SS1.p4.4.m4.2.2.2.2.cmml"><mi id="A2.SS1.p4.4.m4.2.2.2.2.2" xref="A2.SS1.p4.4.m4.2.2.2.2.2.cmml">𝒃</mi><mi id="A2.SS1.p4.4.m4.2.2.2.2.3" xref="A2.SS1.p4.4.m4.2.2.2.2.3.cmml">t</mi></msub><mo id="A2.SS1.p4.4.m4.3.3.3.6" xref="A2.SS1.p4.4.m4.3.3.4.cmml">,</mo><msub id="A2.SS1.p4.4.m4.3.3.3.3" xref="A2.SS1.p4.4.m4.3.3.3.3.cmml"><mi id="A2.SS1.p4.4.m4.3.3.3.3.2" xref="A2.SS1.p4.4.m4.3.3.3.3.2.cmml">𝒄</mi><mi id="A2.SS1.p4.4.m4.3.3.3.3.3" xref="A2.SS1.p4.4.m4.3.3.3.3.3.cmml">t</mi></msub><mo id="A2.SS1.p4.4.m4.3.3.3.7" stretchy="false" xref="A2.SS1.p4.4.m4.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.4.m4.3b"><vector id="A2.SS1.p4.4.m4.3.3.4.cmml" xref="A2.SS1.p4.4.m4.3.3.3"><apply id="A2.SS1.p4.4.m4.1.1.1.1.cmml" xref="A2.SS1.p4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p4.4.m4.1.1.1.1.1.cmml" xref="A2.SS1.p4.4.m4.1.1.1.1">subscript</csymbol><apply id="A2.SS1.p4.4.m4.1.1.1.1.2.cmml" xref="A2.SS1.p4.4.m4.1.1.1.1.2"><ci id="A2.SS1.p4.4.m4.1.1.1.1.2.1.cmml" xref="A2.SS1.p4.4.m4.1.1.1.1.2.1">~</ci><ci id="A2.SS1.p4.4.m4.1.1.1.1.2.2.cmml" xref="A2.SS1.p4.4.m4.1.1.1.1.2.2">𝒙</ci></apply><ci id="A2.SS1.p4.4.m4.1.1.1.1.3.cmml" xref="A2.SS1.p4.4.m4.1.1.1.1.3">𝑡</ci></apply><apply id="A2.SS1.p4.4.m4.2.2.2.2.cmml" xref="A2.SS1.p4.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="A2.SS1.p4.4.m4.2.2.2.2.1.cmml" xref="A2.SS1.p4.4.m4.2.2.2.2">subscript</csymbol><ci id="A2.SS1.p4.4.m4.2.2.2.2.2.cmml" xref="A2.SS1.p4.4.m4.2.2.2.2.2">𝒃</ci><ci id="A2.SS1.p4.4.m4.2.2.2.2.3.cmml" xref="A2.SS1.p4.4.m4.2.2.2.2.3">𝑡</ci></apply><apply id="A2.SS1.p4.4.m4.3.3.3.3.cmml" xref="A2.SS1.p4.4.m4.3.3.3.3"><csymbol cd="ambiguous" id="A2.SS1.p4.4.m4.3.3.3.3.1.cmml" xref="A2.SS1.p4.4.m4.3.3.3.3">subscript</csymbol><ci id="A2.SS1.p4.4.m4.3.3.3.3.2.cmml" xref="A2.SS1.p4.4.m4.3.3.3.3.2">𝒄</ci><ci id="A2.SS1.p4.4.m4.3.3.3.3.3.cmml" xref="A2.SS1.p4.4.m4.3.3.3.3.3">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.4.m4.3c">(\widetilde{\bm{x}}_{t},\bm{b}_{t},\bm{c}_{t})</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.4.m4.3d">( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> depend only on the current token <math alttext="x_{t}" class="ltx_Math" display="inline" id="A2.SS1.p4.5.m5.1"><semantics id="A2.SS1.p4.5.m5.1a"><msub id="A2.SS1.p4.5.m5.1.1" xref="A2.SS1.p4.5.m5.1.1.cmml"><mi id="A2.SS1.p4.5.m5.1.1.2" xref="A2.SS1.p4.5.m5.1.1.2.cmml">x</mi><mi id="A2.SS1.p4.5.m5.1.1.3" xref="A2.SS1.p4.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.5.m5.1b"><apply id="A2.SS1.p4.5.m5.1.1.cmml" xref="A2.SS1.p4.5.m5.1.1"><csymbol cd="ambiguous" id="A2.SS1.p4.5.m5.1.1.1.cmml" xref="A2.SS1.p4.5.m5.1.1">subscript</csymbol><ci id="A2.SS1.p4.5.m5.1.1.2.cmml" xref="A2.SS1.p4.5.m5.1.1.2">𝑥</ci><ci id="A2.SS1.p4.5.m5.1.1.3.cmml" xref="A2.SS1.p4.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.5.m5.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.5.m5.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. Then, consider the two sequences <math alttext="x=(0,1,0,1,0,1)" class="ltx_Math" display="inline" id="A2.SS1.p4.6.m6.6"><semantics id="A2.SS1.p4.6.m6.6a"><mrow id="A2.SS1.p4.6.m6.6.7" xref="A2.SS1.p4.6.m6.6.7.cmml"><mi id="A2.SS1.p4.6.m6.6.7.2" xref="A2.SS1.p4.6.m6.6.7.2.cmml">x</mi><mo id="A2.SS1.p4.6.m6.6.7.1" xref="A2.SS1.p4.6.m6.6.7.1.cmml">=</mo><mrow id="A2.SS1.p4.6.m6.6.7.3.2" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml"><mo id="A2.SS1.p4.6.m6.6.7.3.2.1" stretchy="false" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">(</mo><mn id="A2.SS1.p4.6.m6.1.1" xref="A2.SS1.p4.6.m6.1.1.cmml">0</mn><mo id="A2.SS1.p4.6.m6.6.7.3.2.2" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.6.m6.2.2" xref="A2.SS1.p4.6.m6.2.2.cmml">1</mn><mo id="A2.SS1.p4.6.m6.6.7.3.2.3" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.6.m6.3.3" xref="A2.SS1.p4.6.m6.3.3.cmml">0</mn><mo id="A2.SS1.p4.6.m6.6.7.3.2.4" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.6.m6.4.4" xref="A2.SS1.p4.6.m6.4.4.cmml">1</mn><mo id="A2.SS1.p4.6.m6.6.7.3.2.5" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.6.m6.5.5" xref="A2.SS1.p4.6.m6.5.5.cmml">0</mn><mo id="A2.SS1.p4.6.m6.6.7.3.2.6" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.6.m6.6.6" xref="A2.SS1.p4.6.m6.6.6.cmml">1</mn><mo id="A2.SS1.p4.6.m6.6.7.3.2.7" stretchy="false" xref="A2.SS1.p4.6.m6.6.7.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.6.m6.6b"><apply id="A2.SS1.p4.6.m6.6.7.cmml" xref="A2.SS1.p4.6.m6.6.7"><eq id="A2.SS1.p4.6.m6.6.7.1.cmml" xref="A2.SS1.p4.6.m6.6.7.1"></eq><ci id="A2.SS1.p4.6.m6.6.7.2.cmml" xref="A2.SS1.p4.6.m6.6.7.2">𝑥</ci><vector id="A2.SS1.p4.6.m6.6.7.3.1.cmml" xref="A2.SS1.p4.6.m6.6.7.3.2"><cn id="A2.SS1.p4.6.m6.1.1.cmml" type="integer" xref="A2.SS1.p4.6.m6.1.1">0</cn><cn id="A2.SS1.p4.6.m6.2.2.cmml" type="integer" xref="A2.SS1.p4.6.m6.2.2">1</cn><cn id="A2.SS1.p4.6.m6.3.3.cmml" type="integer" xref="A2.SS1.p4.6.m6.3.3">0</cn><cn id="A2.SS1.p4.6.m6.4.4.cmml" type="integer" xref="A2.SS1.p4.6.m6.4.4">1</cn><cn id="A2.SS1.p4.6.m6.5.5.cmml" type="integer" xref="A2.SS1.p4.6.m6.5.5">0</cn><cn id="A2.SS1.p4.6.m6.6.6.cmml" type="integer" xref="A2.SS1.p4.6.m6.6.6">1</cn></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.6.m6.6c">x=(0,1,0,1,0,1)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.6.m6.6d">italic_x = ( 0 , 1 , 0 , 1 , 0 , 1 )</annotation></semantics></math> and <math alttext="\widetilde{x}=(0,0,0,1,1,1)" class="ltx_Math" display="inline" id="A2.SS1.p4.7.m7.6"><semantics id="A2.SS1.p4.7.m7.6a"><mrow id="A2.SS1.p4.7.m7.6.7" xref="A2.SS1.p4.7.m7.6.7.cmml"><mover accent="true" id="A2.SS1.p4.7.m7.6.7.2" xref="A2.SS1.p4.7.m7.6.7.2.cmml"><mi id="A2.SS1.p4.7.m7.6.7.2.2" xref="A2.SS1.p4.7.m7.6.7.2.2.cmml">x</mi><mo id="A2.SS1.p4.7.m7.6.7.2.1" xref="A2.SS1.p4.7.m7.6.7.2.1.cmml">~</mo></mover><mo id="A2.SS1.p4.7.m7.6.7.1" xref="A2.SS1.p4.7.m7.6.7.1.cmml">=</mo><mrow id="A2.SS1.p4.7.m7.6.7.3.2" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml"><mo id="A2.SS1.p4.7.m7.6.7.3.2.1" stretchy="false" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">(</mo><mn id="A2.SS1.p4.7.m7.1.1" xref="A2.SS1.p4.7.m7.1.1.cmml">0</mn><mo id="A2.SS1.p4.7.m7.6.7.3.2.2" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.7.m7.2.2" xref="A2.SS1.p4.7.m7.2.2.cmml">0</mn><mo id="A2.SS1.p4.7.m7.6.7.3.2.3" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.7.m7.3.3" xref="A2.SS1.p4.7.m7.3.3.cmml">0</mn><mo id="A2.SS1.p4.7.m7.6.7.3.2.4" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.7.m7.4.4" xref="A2.SS1.p4.7.m7.4.4.cmml">1</mn><mo id="A2.SS1.p4.7.m7.6.7.3.2.5" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.7.m7.5.5" xref="A2.SS1.p4.7.m7.5.5.cmml">1</mn><mo id="A2.SS1.p4.7.m7.6.7.3.2.6" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">,</mo><mn id="A2.SS1.p4.7.m7.6.6" xref="A2.SS1.p4.7.m7.6.6.cmml">1</mn><mo id="A2.SS1.p4.7.m7.6.7.3.2.7" stretchy="false" xref="A2.SS1.p4.7.m7.6.7.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.7.m7.6b"><apply id="A2.SS1.p4.7.m7.6.7.cmml" xref="A2.SS1.p4.7.m7.6.7"><eq id="A2.SS1.p4.7.m7.6.7.1.cmml" xref="A2.SS1.p4.7.m7.6.7.1"></eq><apply id="A2.SS1.p4.7.m7.6.7.2.cmml" xref="A2.SS1.p4.7.m7.6.7.2"><ci id="A2.SS1.p4.7.m7.6.7.2.1.cmml" xref="A2.SS1.p4.7.m7.6.7.2.1">~</ci><ci id="A2.SS1.p4.7.m7.6.7.2.2.cmml" xref="A2.SS1.p4.7.m7.6.7.2.2">𝑥</ci></apply><vector id="A2.SS1.p4.7.m7.6.7.3.1.cmml" xref="A2.SS1.p4.7.m7.6.7.3.2"><cn id="A2.SS1.p4.7.m7.1.1.cmml" type="integer" xref="A2.SS1.p4.7.m7.1.1">0</cn><cn id="A2.SS1.p4.7.m7.2.2.cmml" type="integer" xref="A2.SS1.p4.7.m7.2.2">0</cn><cn id="A2.SS1.p4.7.m7.3.3.cmml" type="integer" xref="A2.SS1.p4.7.m7.3.3">0</cn><cn id="A2.SS1.p4.7.m7.4.4.cmml" type="integer" xref="A2.SS1.p4.7.m7.4.4">1</cn><cn id="A2.SS1.p4.7.m7.5.5.cmml" type="integer" xref="A2.SS1.p4.7.m7.5.5">1</cn><cn id="A2.SS1.p4.7.m7.6.6.cmml" type="integer" xref="A2.SS1.p4.7.m7.6.6">1</cn></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.7.m7.6c">\widetilde{x}=(0,0,0,1,1,1)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.7.m7.6d">over~ start_ARG italic_x end_ARG = ( 0 , 0 , 0 , 1 , 1 , 1 )</annotation></semantics></math>. At time <math alttext="t=6" class="ltx_Math" display="inline" id="A2.SS1.p4.8.m8.1"><semantics id="A2.SS1.p4.8.m8.1a"><mrow id="A2.SS1.p4.8.m8.1.1" xref="A2.SS1.p4.8.m8.1.1.cmml"><mi id="A2.SS1.p4.8.m8.1.1.2" xref="A2.SS1.p4.8.m8.1.1.2.cmml">t</mi><mo id="A2.SS1.p4.8.m8.1.1.1" xref="A2.SS1.p4.8.m8.1.1.1.cmml">=</mo><mn id="A2.SS1.p4.8.m8.1.1.3" xref="A2.SS1.p4.8.m8.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.8.m8.1b"><apply id="A2.SS1.p4.8.m8.1.1.cmml" xref="A2.SS1.p4.8.m8.1.1"><eq id="A2.SS1.p4.8.m8.1.1.1.cmml" xref="A2.SS1.p4.8.m8.1.1.1"></eq><ci id="A2.SS1.p4.8.m8.1.1.2.cmml" xref="A2.SS1.p4.8.m8.1.1.2">𝑡</ci><cn id="A2.SS1.p4.8.m8.1.1.3.cmml" type="integer" xref="A2.SS1.p4.8.m8.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.8.m8.1c">t=6</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.8.m8.1d">italic_t = 6</annotation></semantics></math>, these two sequences would give the same state <math alttext="H_{t}" class="ltx_Math" display="inline" id="A2.SS1.p4.9.m9.1"><semantics id="A2.SS1.p4.9.m9.1a"><msub id="A2.SS1.p4.9.m9.1.1" xref="A2.SS1.p4.9.m9.1.1.cmml"><mi id="A2.SS1.p4.9.m9.1.1.2" xref="A2.SS1.p4.9.m9.1.1.2.cmml">H</mi><mi id="A2.SS1.p4.9.m9.1.1.3" xref="A2.SS1.p4.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.9.m9.1b"><apply id="A2.SS1.p4.9.m9.1.1.cmml" xref="A2.SS1.p4.9.m9.1.1"><csymbol cd="ambiguous" id="A2.SS1.p4.9.m9.1.1.1.cmml" xref="A2.SS1.p4.9.m9.1.1">subscript</csymbol><ci id="A2.SS1.p4.9.m9.1.1.2.cmml" xref="A2.SS1.p4.9.m9.1.1.2">𝐻</ci><ci id="A2.SS1.p4.9.m9.1.1.3.cmml" xref="A2.SS1.p4.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.9.m9.1c">H_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.9.m9.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and the same output <math alttext="\bm{y}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p4.10.m10.1"><semantics id="A2.SS1.p4.10.m10.1a"><msub id="A2.SS1.p4.10.m10.1.1" xref="A2.SS1.p4.10.m10.1.1.cmml"><mi id="A2.SS1.p4.10.m10.1.1.2" xref="A2.SS1.p4.10.m10.1.1.2.cmml">𝒚</mi><mi id="A2.SS1.p4.10.m10.1.1.3" xref="A2.SS1.p4.10.m10.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.10.m10.1b"><apply id="A2.SS1.p4.10.m10.1.1.cmml" xref="A2.SS1.p4.10.m10.1.1"><csymbol cd="ambiguous" id="A2.SS1.p4.10.m10.1.1.1.cmml" xref="A2.SS1.p4.10.m10.1.1">subscript</csymbol><ci id="A2.SS1.p4.10.m10.1.1.2.cmml" xref="A2.SS1.p4.10.m10.1.1.2">𝒚</ci><ci id="A2.SS1.p4.10.m10.1.1.3.cmml" xref="A2.SS1.p4.10.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.10.m10.1c">\bm{y}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.10.m10.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, since they share the same number of tokens <math alttext="0" class="ltx_Math" display="inline" id="A2.SS1.p4.11.m11.1"><semantics id="A2.SS1.p4.11.m11.1a"><mn id="A2.SS1.p4.11.m11.1.1" xref="A2.SS1.p4.11.m11.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.11.m11.1b"><cn id="A2.SS1.p4.11.m11.1.1.cmml" type="integer" xref="A2.SS1.p4.11.m11.1.1">0</cn></annotation-xml></semantics></math> and <math alttext="1" class="ltx_Math" display="inline" id="A2.SS1.p4.12.m12.1"><semantics id="A2.SS1.p4.12.m12.1a"><mn id="A2.SS1.p4.12.m12.1.1" xref="A2.SS1.p4.12.m12.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.12.m12.1b"><cn id="A2.SS1.p4.12.m12.1.1.cmml" type="integer" xref="A2.SS1.p4.12.m12.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.12.m12.1c">1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.12.m12.1d">1</annotation></semantics></math>. Therefore, the estimated probability given by the model would be the same in both cases. However, the optimal add-constant estimator (with <math alttext="\beta=1" class="ltx_Math" display="inline" id="A2.SS1.p4.13.m13.1"><semantics id="A2.SS1.p4.13.m13.1a"><mrow id="A2.SS1.p4.13.m13.1.1" xref="A2.SS1.p4.13.m13.1.1.cmml"><mi id="A2.SS1.p4.13.m13.1.1.2" xref="A2.SS1.p4.13.m13.1.1.2.cmml">β</mi><mo id="A2.SS1.p4.13.m13.1.1.1" xref="A2.SS1.p4.13.m13.1.1.1.cmml">=</mo><mn id="A2.SS1.p4.13.m13.1.1.3" xref="A2.SS1.p4.13.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.13.m13.1b"><apply id="A2.SS1.p4.13.m13.1.1.cmml" xref="A2.SS1.p4.13.m13.1.1"><eq id="A2.SS1.p4.13.m13.1.1.1.cmml" xref="A2.SS1.p4.13.m13.1.1.1"></eq><ci id="A2.SS1.p4.13.m13.1.1.2.cmml" xref="A2.SS1.p4.13.m13.1.1.2">𝛽</ci><cn id="A2.SS1.p4.13.m13.1.1.3.cmml" type="integer" xref="A2.SS1.p4.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.13.m13.1c">\beta=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.13.m13.1d">italic_β = 1</annotation></semantics></math>) would estimate the probability of <math alttext="x_{t+1}=1" class="ltx_Math" display="inline" id="A2.SS1.p4.14.m14.1"><semantics id="A2.SS1.p4.14.m14.1a"><mrow id="A2.SS1.p4.14.m14.1.1" xref="A2.SS1.p4.14.m14.1.1.cmml"><msub id="A2.SS1.p4.14.m14.1.1.2" xref="A2.SS1.p4.14.m14.1.1.2.cmml"><mi id="A2.SS1.p4.14.m14.1.1.2.2" xref="A2.SS1.p4.14.m14.1.1.2.2.cmml">x</mi><mrow id="A2.SS1.p4.14.m14.1.1.2.3" xref="A2.SS1.p4.14.m14.1.1.2.3.cmml"><mi id="A2.SS1.p4.14.m14.1.1.2.3.2" xref="A2.SS1.p4.14.m14.1.1.2.3.2.cmml">t</mi><mo id="A2.SS1.p4.14.m14.1.1.2.3.1" xref="A2.SS1.p4.14.m14.1.1.2.3.1.cmml">+</mo><mn id="A2.SS1.p4.14.m14.1.1.2.3.3" xref="A2.SS1.p4.14.m14.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="A2.SS1.p4.14.m14.1.1.1" xref="A2.SS1.p4.14.m14.1.1.1.cmml">=</mo><mn id="A2.SS1.p4.14.m14.1.1.3" xref="A2.SS1.p4.14.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.14.m14.1b"><apply id="A2.SS1.p4.14.m14.1.1.cmml" xref="A2.SS1.p4.14.m14.1.1"><eq id="A2.SS1.p4.14.m14.1.1.1.cmml" xref="A2.SS1.p4.14.m14.1.1.1"></eq><apply id="A2.SS1.p4.14.m14.1.1.2.cmml" xref="A2.SS1.p4.14.m14.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.p4.14.m14.1.1.2.1.cmml" xref="A2.SS1.p4.14.m14.1.1.2">subscript</csymbol><ci id="A2.SS1.p4.14.m14.1.1.2.2.cmml" xref="A2.SS1.p4.14.m14.1.1.2.2">𝑥</ci><apply id="A2.SS1.p4.14.m14.1.1.2.3.cmml" xref="A2.SS1.p4.14.m14.1.1.2.3"><plus id="A2.SS1.p4.14.m14.1.1.2.3.1.cmml" xref="A2.SS1.p4.14.m14.1.1.2.3.1"></plus><ci id="A2.SS1.p4.14.m14.1.1.2.3.2.cmml" xref="A2.SS1.p4.14.m14.1.1.2.3.2">𝑡</ci><cn id="A2.SS1.p4.14.m14.1.1.2.3.3.cmml" type="integer" xref="A2.SS1.p4.14.m14.1.1.2.3.3">1</cn></apply></apply><cn id="A2.SS1.p4.14.m14.1.1.3.cmml" type="integer" xref="A2.SS1.p4.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.14.m14.1c">x_{t+1}=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.14.m14.1d">italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> to be <math alttext="1/4" class="ltx_Math" display="inline" id="A2.SS1.p4.15.m15.1"><semantics id="A2.SS1.p4.15.m15.1a"><mrow id="A2.SS1.p4.15.m15.1.1" xref="A2.SS1.p4.15.m15.1.1.cmml"><mn id="A2.SS1.p4.15.m15.1.1.2" xref="A2.SS1.p4.15.m15.1.1.2.cmml">1</mn><mo id="A2.SS1.p4.15.m15.1.1.1" xref="A2.SS1.p4.15.m15.1.1.1.cmml">/</mo><mn id="A2.SS1.p4.15.m15.1.1.3" xref="A2.SS1.p4.15.m15.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.15.m15.1b"><apply id="A2.SS1.p4.15.m15.1.1.cmml" xref="A2.SS1.p4.15.m15.1.1"><divide id="A2.SS1.p4.15.m15.1.1.1.cmml" xref="A2.SS1.p4.15.m15.1.1.1"></divide><cn id="A2.SS1.p4.15.m15.1.1.2.cmml" type="integer" xref="A2.SS1.p4.15.m15.1.1.2">1</cn><cn id="A2.SS1.p4.15.m15.1.1.3.cmml" type="integer" xref="A2.SS1.p4.15.m15.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.15.m15.1c">1/4</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.15.m15.1d">1 / 4</annotation></semantics></math> for <math alttext="\bm{x}" class="ltx_Math" display="inline" id="A2.SS1.p4.16.m16.1"><semantics id="A2.SS1.p4.16.m16.1a"><mi id="A2.SS1.p4.16.m16.1.1" xref="A2.SS1.p4.16.m16.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.16.m16.1b"><ci id="A2.SS1.p4.16.m16.1.1.cmml" xref="A2.SS1.p4.16.m16.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.16.m16.1c">\bm{x}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.16.m16.1d">bold_italic_x</annotation></semantics></math>, and <math alttext="3/4" class="ltx_Math" display="inline" id="A2.SS1.p4.17.m17.1"><semantics id="A2.SS1.p4.17.m17.1a"><mrow id="A2.SS1.p4.17.m17.1.1" xref="A2.SS1.p4.17.m17.1.1.cmml"><mn id="A2.SS1.p4.17.m17.1.1.2" xref="A2.SS1.p4.17.m17.1.1.2.cmml">3</mn><mo id="A2.SS1.p4.17.m17.1.1.1" xref="A2.SS1.p4.17.m17.1.1.1.cmml">/</mo><mn id="A2.SS1.p4.17.m17.1.1.3" xref="A2.SS1.p4.17.m17.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.17.m17.1b"><apply id="A2.SS1.p4.17.m17.1.1.cmml" xref="A2.SS1.p4.17.m17.1.1"><divide id="A2.SS1.p4.17.m17.1.1.1.cmml" xref="A2.SS1.p4.17.m17.1.1.1"></divide><cn id="A2.SS1.p4.17.m17.1.1.2.cmml" type="integer" xref="A2.SS1.p4.17.m17.1.1.2">3</cn><cn id="A2.SS1.p4.17.m17.1.1.3.cmml" type="integer" xref="A2.SS1.p4.17.m17.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.17.m17.1c">3/4</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.17.m17.1d">3 / 4</annotation></semantics></math> for <math alttext="\widetilde{\bm{x}}" class="ltx_Math" display="inline" id="A2.SS1.p4.18.m18.1"><semantics id="A2.SS1.p4.18.m18.1a"><mover accent="true" id="A2.SS1.p4.18.m18.1.1" xref="A2.SS1.p4.18.m18.1.1.cmml"><mi id="A2.SS1.p4.18.m18.1.1.2" xref="A2.SS1.p4.18.m18.1.1.2.cmml">𝒙</mi><mo id="A2.SS1.p4.18.m18.1.1.1" xref="A2.SS1.p4.18.m18.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A2.SS1.p4.18.m18.1b"><apply id="A2.SS1.p4.18.m18.1.1.cmml" xref="A2.SS1.p4.18.m18.1.1"><ci id="A2.SS1.p4.18.m18.1.1.1.cmml" xref="A2.SS1.p4.18.m18.1.1.1">~</ci><ci id="A2.SS1.p4.18.m18.1.1.2.cmml" xref="A2.SS1.p4.18.m18.1.1.2">𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p4.18.m18.1c">\widetilde{\bm{x}}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p4.18.m18.1d">over~ start_ARG bold_italic_x end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.SS1.p5"> <p class="ltx_p" id="A2.SS1.p5.13">Further, <em class="ltx_emph ltx_font_italic" id="A2.SS1.p5.2.2">it is sufficient that the convolution for <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p5.1.1.m1.1"><semantics id="A2.SS1.p5.1.1.m1.1a"><msub id="A2.SS1.p5.1.1.m1.1.1" xref="A2.SS1.p5.1.1.m1.1.1.cmml"><mi id="A2.SS1.p5.1.1.m1.1.1.2" xref="A2.SS1.p5.1.1.m1.1.1.2.cmml">𝐜</mi><mi id="A2.SS1.p5.1.1.m1.1.1.3" xref="A2.SS1.p5.1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.1.1.m1.1b"><apply id="A2.SS1.p5.1.1.m1.1.1.cmml" xref="A2.SS1.p5.1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.1.1.m1.1.1.1.cmml" xref="A2.SS1.p5.1.1.m1.1.1">subscript</csymbol><ci id="A2.SS1.p5.1.1.m1.1.1.2.cmml" xref="A2.SS1.p5.1.1.m1.1.1.2">𝐜</ci><ci id="A2.SS1.p5.1.1.m1.1.1.3.cmml" xref="A2.SS1.p5.1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.1.1.m1.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.1.1.m1.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> has window <math alttext="w_{C}=k" class="ltx_Math" display="inline" id="A2.SS1.p5.2.2.m2.1"><semantics id="A2.SS1.p5.2.2.m2.1a"><mrow id="A2.SS1.p5.2.2.m2.1.1" xref="A2.SS1.p5.2.2.m2.1.1.cmml"><msub id="A2.SS1.p5.2.2.m2.1.1.2" xref="A2.SS1.p5.2.2.m2.1.1.2.cmml"><mi id="A2.SS1.p5.2.2.m2.1.1.2.2" xref="A2.SS1.p5.2.2.m2.1.1.2.2.cmml">w</mi><mi id="A2.SS1.p5.2.2.m2.1.1.2.3" xref="A2.SS1.p5.2.2.m2.1.1.2.3.cmml">C</mi></msub><mo id="A2.SS1.p5.2.2.m2.1.1.1" xref="A2.SS1.p5.2.2.m2.1.1.1.cmml">=</mo><mi id="A2.SS1.p5.2.2.m2.1.1.3" xref="A2.SS1.p5.2.2.m2.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.2.2.m2.1b"><apply id="A2.SS1.p5.2.2.m2.1.1.cmml" xref="A2.SS1.p5.2.2.m2.1.1"><eq id="A2.SS1.p5.2.2.m2.1.1.1.cmml" xref="A2.SS1.p5.2.2.m2.1.1.1"></eq><apply id="A2.SS1.p5.2.2.m2.1.1.2.cmml" xref="A2.SS1.p5.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.p5.2.2.m2.1.1.2.1.cmml" xref="A2.SS1.p5.2.2.m2.1.1.2">subscript</csymbol><ci id="A2.SS1.p5.2.2.m2.1.1.2.2.cmml" xref="A2.SS1.p5.2.2.m2.1.1.2.2">𝑤</ci><ci id="A2.SS1.p5.2.2.m2.1.1.2.3.cmml" xref="A2.SS1.p5.2.2.m2.1.1.2.3">𝐶</ci></apply><ci id="A2.SS1.p5.2.2.m2.1.1.3.cmml" xref="A2.SS1.p5.2.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.2.2.m2.1c">w_{C}=k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.2.2.m2.1d">italic_w start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = italic_k</annotation></semantics></math>.</em> That is, the convolution <math alttext="\mathrm{conv}_{C}" class="ltx_Math" display="inline" id="A2.SS1.p5.3.m1.1"><semantics id="A2.SS1.p5.3.m1.1a"><msub id="A2.SS1.p5.3.m1.1.1" xref="A2.SS1.p5.3.m1.1.1.cmml"><mi id="A2.SS1.p5.3.m1.1.1.2" xref="A2.SS1.p5.3.m1.1.1.2.cmml">conv</mi><mi id="A2.SS1.p5.3.m1.1.1.3" xref="A2.SS1.p5.3.m1.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.3.m1.1b"><apply id="A2.SS1.p5.3.m1.1.1.cmml" xref="A2.SS1.p5.3.m1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.3.m1.1.1.1.cmml" xref="A2.SS1.p5.3.m1.1.1">subscript</csymbol><ci id="A2.SS1.p5.3.m1.1.1.2.cmml" xref="A2.SS1.p5.3.m1.1.1.2">conv</ci><ci id="A2.SS1.p5.3.m1.1.1.3.cmml" xref="A2.SS1.p5.3.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.3.m1.1c">\mathrm{conv}_{C}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.3.m1.1d">roman_conv start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> involved in the computation of <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p5.4.m2.1"><semantics id="A2.SS1.p5.4.m2.1a"><msub id="A2.SS1.p5.4.m2.1.1" xref="A2.SS1.p5.4.m2.1.1.cmml"><mi id="A2.SS1.p5.4.m2.1.1.2" xref="A2.SS1.p5.4.m2.1.1.2.cmml">𝒄</mi><mi id="A2.SS1.p5.4.m2.1.1.3" xref="A2.SS1.p5.4.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.4.m2.1b"><apply id="A2.SS1.p5.4.m2.1.1.cmml" xref="A2.SS1.p5.4.m2.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.4.m2.1.1.1.cmml" xref="A2.SS1.p5.4.m2.1.1">subscript</csymbol><ci id="A2.SS1.p5.4.m2.1.1.2.cmml" xref="A2.SS1.p5.4.m2.1.1.2">𝒄</ci><ci id="A2.SS1.p5.4.m2.1.1.3.cmml" xref="A2.SS1.p5.4.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.4.m2.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.4.m2.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> can have a window size equal to the Markov order <math alttext="k" class="ltx_Math" display="inline" id="A2.SS1.p5.5.m3.1"><semantics id="A2.SS1.p5.5.m3.1a"><mi id="A2.SS1.p5.5.m3.1.1" xref="A2.SS1.p5.5.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.5.m3.1b"><ci id="A2.SS1.p5.5.m3.1.1.cmml" xref="A2.SS1.p5.5.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.5.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.5.m3.1d">italic_k</annotation></semantics></math> (i.e., one less than <math alttext="\mathrm{conv}_{X}" class="ltx_Math" display="inline" id="A2.SS1.p5.6.m4.1"><semantics id="A2.SS1.p5.6.m4.1a"><msub id="A2.SS1.p5.6.m4.1.1" xref="A2.SS1.p5.6.m4.1.1.cmml"><mi id="A2.SS1.p5.6.m4.1.1.2" xref="A2.SS1.p5.6.m4.1.1.2.cmml">conv</mi><mi id="A2.SS1.p5.6.m4.1.1.3" xref="A2.SS1.p5.6.m4.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.6.m4.1b"><apply id="A2.SS1.p5.6.m4.1.1.cmml" xref="A2.SS1.p5.6.m4.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.6.m4.1.1.1.cmml" xref="A2.SS1.p5.6.m4.1.1">subscript</csymbol><ci id="A2.SS1.p5.6.m4.1.1.2.cmml" xref="A2.SS1.p5.6.m4.1.1.2">conv</ci><ci id="A2.SS1.p5.6.m4.1.1.3.cmml" xref="A2.SS1.p5.6.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.6.m4.1c">\mathrm{conv}_{X}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.6.m4.1d">roman_conv start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathrm{conv}_{B}" class="ltx_Math" display="inline" id="A2.SS1.p5.7.m5.1"><semantics id="A2.SS1.p5.7.m5.1a"><msub id="A2.SS1.p5.7.m5.1.1" xref="A2.SS1.p5.7.m5.1.1.cmml"><mi id="A2.SS1.p5.7.m5.1.1.2" xref="A2.SS1.p5.7.m5.1.1.2.cmml">conv</mi><mi id="A2.SS1.p5.7.m5.1.1.3" xref="A2.SS1.p5.7.m5.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.7.m5.1b"><apply id="A2.SS1.p5.7.m5.1.1.cmml" xref="A2.SS1.p5.7.m5.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.7.m5.1.1.1.cmml" xref="A2.SS1.p5.7.m5.1.1">subscript</csymbol><ci id="A2.SS1.p5.7.m5.1.1.2.cmml" xref="A2.SS1.p5.7.m5.1.1.2">conv</ci><ci id="A2.SS1.p5.7.m5.1.1.3.cmml" xref="A2.SS1.p5.7.m5.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.7.m5.1c">\mathrm{conv}_{B}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.7.m5.1d">roman_conv start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>) without affecting the model’s capability of learning the task (or, equivalently, the left-most kernel coefficients of <math alttext="\mathrm{conv}_{C}" class="ltx_Math" display="inline" id="A2.SS1.p5.8.m6.1"><semantics id="A2.SS1.p5.8.m6.1a"><msub id="A2.SS1.p5.8.m6.1.1" xref="A2.SS1.p5.8.m6.1.1.cmml"><mi id="A2.SS1.p5.8.m6.1.1.2" xref="A2.SS1.p5.8.m6.1.1.2.cmml">conv</mi><mi id="A2.SS1.p5.8.m6.1.1.3" xref="A2.SS1.p5.8.m6.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.8.m6.1b"><apply id="A2.SS1.p5.8.m6.1.1.cmml" xref="A2.SS1.p5.8.m6.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.8.m6.1.1.1.cmml" xref="A2.SS1.p5.8.m6.1.1">subscript</csymbol><ci id="A2.SS1.p5.8.m6.1.1.2.cmml" xref="A2.SS1.p5.8.m6.1.1.2">conv</ci><ci id="A2.SS1.p5.8.m6.1.1.3.cmml" xref="A2.SS1.p5.8.m6.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.8.m6.1c">\mathrm{conv}_{C}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.8.m6.1d">roman_conv start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> can be taken to be zero). Intuitively, this is because the role of <math alttext="\bm{c}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p5.9.m7.1"><semantics id="A2.SS1.p5.9.m7.1a"><msub id="A2.SS1.p5.9.m7.1.1" xref="A2.SS1.p5.9.m7.1.1.cmml"><mi id="A2.SS1.p5.9.m7.1.1.2" xref="A2.SS1.p5.9.m7.1.1.2.cmml">𝒄</mi><mi id="A2.SS1.p5.9.m7.1.1.3" xref="A2.SS1.p5.9.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.9.m7.1b"><apply id="A2.SS1.p5.9.m7.1.1.cmml" xref="A2.SS1.p5.9.m7.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.9.m7.1.1.1.cmml" xref="A2.SS1.p5.9.m7.1.1">subscript</csymbol><ci id="A2.SS1.p5.9.m7.1.1.2.cmml" xref="A2.SS1.p5.9.m7.1.1.2">𝒄</ci><ci id="A2.SS1.p5.9.m7.1.1.3.cmml" xref="A2.SS1.p5.9.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.9.m7.1c">\bm{c}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.9.m7.1d">bold_italic_c start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in the state projection is to select the correct transition counts for the computation of the estimator, distilled into <math alttext="\bm{y}_{t}" class="ltx_Math" display="inline" id="A2.SS1.p5.10.m8.1"><semantics id="A2.SS1.p5.10.m8.1a"><msub id="A2.SS1.p5.10.m8.1.1" xref="A2.SS1.p5.10.m8.1.1.cmml"><mi id="A2.SS1.p5.10.m8.1.1.2" xref="A2.SS1.p5.10.m8.1.1.2.cmml">𝒚</mi><mi id="A2.SS1.p5.10.m8.1.1.3" xref="A2.SS1.p5.10.m8.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.10.m8.1b"><apply id="A2.SS1.p5.10.m8.1.1.cmml" xref="A2.SS1.p5.10.m8.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.10.m8.1.1.1.cmml" xref="A2.SS1.p5.10.m8.1.1">subscript</csymbol><ci id="A2.SS1.p5.10.m8.1.1.2.cmml" xref="A2.SS1.p5.10.m8.1.1.2">𝒚</ci><ci id="A2.SS1.p5.10.m8.1.1.3.cmml" xref="A2.SS1.p5.10.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.10.m8.1c">\bm{y}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.10.m8.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. In order to do so, it is sufficient to know the length-<math alttext="k" class="ltx_Math" display="inline" id="A2.SS1.p5.11.m9.1"><semantics id="A2.SS1.p5.11.m9.1a"><mi id="A2.SS1.p5.11.m9.1.1" xref="A2.SS1.p5.11.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.11.m9.1b"><ci id="A2.SS1.p5.11.m9.1.1.cmml" xref="A2.SS1.p5.11.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.11.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.11.m9.1d">italic_k</annotation></semantics></math> context of the current symbol <math alttext="x_{t}" class="ltx_Math" display="inline" id="A2.SS1.p5.12.m10.1"><semantics id="A2.SS1.p5.12.m10.1a"><msub id="A2.SS1.p5.12.m10.1.1" xref="A2.SS1.p5.12.m10.1.1.cmml"><mi id="A2.SS1.p5.12.m10.1.1.2" xref="A2.SS1.p5.12.m10.1.1.2.cmml">x</mi><mi id="A2.SS1.p5.12.m10.1.1.3" xref="A2.SS1.p5.12.m10.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.12.m10.1b"><apply id="A2.SS1.p5.12.m10.1.1.cmml" xref="A2.SS1.p5.12.m10.1.1"><csymbol cd="ambiguous" id="A2.SS1.p5.12.m10.1.1.1.cmml" xref="A2.SS1.p5.12.m10.1.1">subscript</csymbol><ci id="A2.SS1.p5.12.m10.1.1.2.cmml" xref="A2.SS1.p5.12.m10.1.1.2">𝑥</ci><ci id="A2.SS1.p5.12.m10.1.1.3.cmml" xref="A2.SS1.p5.12.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.12.m10.1c">x_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.12.m10.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, which can be encoded by a convolution with window size <math alttext="k" class="ltx_Math" display="inline" id="A2.SS1.p5.13.m11.1"><semantics id="A2.SS1.p5.13.m11.1a"><mi id="A2.SS1.p5.13.m11.1.1" xref="A2.SS1.p5.13.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p5.13.m11.1b"><ci id="A2.SS1.p5.13.m11.1.1.cmml" xref="A2.SS1.p5.13.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p5.13.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.p5.13.m11.1d">italic_k</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="A2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.2 </span>Proof of Thm. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#Thmtheorem1" title="Theorem 1 (𝖬𝖺𝗆𝖻𝖺𝖹𝖾𝗋𝗈 represents order-1 Laplacian smoothing). ‣ 4.2 Main theorem for first-order Markov ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">1</span></a> </h3> <div class="ltx_para" id="A2.SS2.p1"> <p class="ltx_p" id="A2.SS2.p1.9">Fix <math alttext="\epsilon>0" class="ltx_Math" display="inline" id="A2.SS2.p1.1.m1.1"><semantics id="A2.SS2.p1.1.m1.1a"><mrow id="A2.SS2.p1.1.m1.1.1" xref="A2.SS2.p1.1.m1.1.1.cmml"><mi id="A2.SS2.p1.1.m1.1.1.2" xref="A2.SS2.p1.1.m1.1.1.2.cmml">ϵ</mi><mo id="A2.SS2.p1.1.m1.1.1.1" xref="A2.SS2.p1.1.m1.1.1.1.cmml">></mo><mn id="A2.SS2.p1.1.m1.1.1.3" xref="A2.SS2.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.1.m1.1b"><apply id="A2.SS2.p1.1.m1.1.1.cmml" xref="A2.SS2.p1.1.m1.1.1"><gt id="A2.SS2.p1.1.m1.1.1.1.cmml" xref="A2.SS2.p1.1.m1.1.1.1"></gt><ci id="A2.SS2.p1.1.m1.1.1.2.cmml" xref="A2.SS2.p1.1.m1.1.1.2">italic-ϵ</ci><cn id="A2.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="A2.SS2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.1.m1.1c">\epsilon>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.1.m1.1d">italic_ϵ > 0</annotation></semantics></math> and let <math alttext="\beta>0" class="ltx_Math" display="inline" id="A2.SS2.p1.2.m2.1"><semantics id="A2.SS2.p1.2.m2.1a"><mrow id="A2.SS2.p1.2.m2.1.1" xref="A2.SS2.p1.2.m2.1.1.cmml"><mi id="A2.SS2.p1.2.m2.1.1.2" xref="A2.SS2.p1.2.m2.1.1.2.cmml">β</mi><mo id="A2.SS2.p1.2.m2.1.1.1" xref="A2.SS2.p1.2.m2.1.1.1.cmml">></mo><mn id="A2.SS2.p1.2.m2.1.1.3" xref="A2.SS2.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.2.m2.1b"><apply id="A2.SS2.p1.2.m2.1.1.cmml" xref="A2.SS2.p1.2.m2.1.1"><gt id="A2.SS2.p1.2.m2.1.1.1.cmml" xref="A2.SS2.p1.2.m2.1.1.1"></gt><ci id="A2.SS2.p1.2.m2.1.1.2.cmml" xref="A2.SS2.p1.2.m2.1.1.2">𝛽</ci><cn id="A2.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="A2.SS2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.2.m2.1c">\beta>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.2.m2.1d">italic_β > 0</annotation></semantics></math> be the constant of the considered add-constant estimator. Let us fix <math alttext="a=0" class="ltx_Math" display="inline" id="A2.SS2.p1.3.m3.1"><semantics id="A2.SS2.p1.3.m3.1a"><mrow id="A2.SS2.p1.3.m3.1.1" xref="A2.SS2.p1.3.m3.1.1.cmml"><mi id="A2.SS2.p1.3.m3.1.1.2" xref="A2.SS2.p1.3.m3.1.1.2.cmml">a</mi><mo id="A2.SS2.p1.3.m3.1.1.1" xref="A2.SS2.p1.3.m3.1.1.1.cmml">=</mo><mn id="A2.SS2.p1.3.m3.1.1.3" xref="A2.SS2.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.3.m3.1b"><apply id="A2.SS2.p1.3.m3.1.1.cmml" xref="A2.SS2.p1.3.m3.1.1"><eq id="A2.SS2.p1.3.m3.1.1.1.cmml" xref="A2.SS2.p1.3.m3.1.1.1"></eq><ci id="A2.SS2.p1.3.m3.1.1.2.cmml" xref="A2.SS2.p1.3.m3.1.1.2">𝑎</ci><cn id="A2.SS2.p1.3.m3.1.1.3.cmml" type="integer" xref="A2.SS2.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.3.m3.1c">a=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.3.m3.1d">italic_a = 0</annotation></semantics></math> and <math alttext="\Delta_{t}=1" class="ltx_Math" display="inline" id="A2.SS2.p1.4.m4.1"><semantics id="A2.SS2.p1.4.m4.1a"><mrow id="A2.SS2.p1.4.m4.1.1" xref="A2.SS2.p1.4.m4.1.1.cmml"><msub id="A2.SS2.p1.4.m4.1.1.2" xref="A2.SS2.p1.4.m4.1.1.2.cmml"><mi id="A2.SS2.p1.4.m4.1.1.2.2" mathvariant="normal" xref="A2.SS2.p1.4.m4.1.1.2.2.cmml">Δ</mi><mi id="A2.SS2.p1.4.m4.1.1.2.3" xref="A2.SS2.p1.4.m4.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS2.p1.4.m4.1.1.1" xref="A2.SS2.p1.4.m4.1.1.1.cmml">=</mo><mn id="A2.SS2.p1.4.m4.1.1.3" xref="A2.SS2.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.4.m4.1b"><apply id="A2.SS2.p1.4.m4.1.1.cmml" xref="A2.SS2.p1.4.m4.1.1"><eq id="A2.SS2.p1.4.m4.1.1.1.cmml" xref="A2.SS2.p1.4.m4.1.1.1"></eq><apply id="A2.SS2.p1.4.m4.1.1.2.cmml" xref="A2.SS2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p1.4.m4.1.1.2.1.cmml" xref="A2.SS2.p1.4.m4.1.1.2">subscript</csymbol><ci id="A2.SS2.p1.4.m4.1.1.2.2.cmml" xref="A2.SS2.p1.4.m4.1.1.2.2">Δ</ci><ci id="A2.SS2.p1.4.m4.1.1.2.3.cmml" xref="A2.SS2.p1.4.m4.1.1.2.3">𝑡</ci></apply><cn id="A2.SS2.p1.4.m4.1.1.3.cmml" type="integer" xref="A2.SS2.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.4.m4.1c">\Delta_{t}=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.4.m4.1d">roman_Δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math>, so that <math alttext="a_{t}=1" class="ltx_Math" display="inline" id="A2.SS2.p1.5.m5.1"><semantics id="A2.SS2.p1.5.m5.1a"><mrow id="A2.SS2.p1.5.m5.1.1" xref="A2.SS2.p1.5.m5.1.1.cmml"><msub id="A2.SS2.p1.5.m5.1.1.2" xref="A2.SS2.p1.5.m5.1.1.2.cmml"><mi id="A2.SS2.p1.5.m5.1.1.2.2" xref="A2.SS2.p1.5.m5.1.1.2.2.cmml">a</mi><mi id="A2.SS2.p1.5.m5.1.1.2.3" xref="A2.SS2.p1.5.m5.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS2.p1.5.m5.1.1.1" xref="A2.SS2.p1.5.m5.1.1.1.cmml">=</mo><mn id="A2.SS2.p1.5.m5.1.1.3" xref="A2.SS2.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.5.m5.1b"><apply id="A2.SS2.p1.5.m5.1.1.cmml" xref="A2.SS2.p1.5.m5.1.1"><eq id="A2.SS2.p1.5.m5.1.1.1.cmml" xref="A2.SS2.p1.5.m5.1.1.1"></eq><apply id="A2.SS2.p1.5.m5.1.1.2.cmml" xref="A2.SS2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p1.5.m5.1.1.2.1.cmml" xref="A2.SS2.p1.5.m5.1.1.2">subscript</csymbol><ci id="A2.SS2.p1.5.m5.1.1.2.2.cmml" xref="A2.SS2.p1.5.m5.1.1.2.2">𝑎</ci><ci id="A2.SS2.p1.5.m5.1.1.2.3.cmml" xref="A2.SS2.p1.5.m5.1.1.2.3">𝑡</ci></apply><cn id="A2.SS2.p1.5.m5.1.1.3.cmml" type="integer" xref="A2.SS2.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.5.m5.1c">a_{t}=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math>, for all <math alttext="t\geq 1" class="ltx_Math" display="inline" id="A2.SS2.p1.6.m6.1"><semantics id="A2.SS2.p1.6.m6.1a"><mrow id="A2.SS2.p1.6.m6.1.1" xref="A2.SS2.p1.6.m6.1.1.cmml"><mi id="A2.SS2.p1.6.m6.1.1.2" xref="A2.SS2.p1.6.m6.1.1.2.cmml">t</mi><mo id="A2.SS2.p1.6.m6.1.1.1" xref="A2.SS2.p1.6.m6.1.1.1.cmml">≥</mo><mn id="A2.SS2.p1.6.m6.1.1.3" xref="A2.SS2.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.6.m6.1b"><apply id="A2.SS2.p1.6.m6.1.1.cmml" xref="A2.SS2.p1.6.m6.1.1"><geq id="A2.SS2.p1.6.m6.1.1.1.cmml" xref="A2.SS2.p1.6.m6.1.1.1"></geq><ci id="A2.SS2.p1.6.m6.1.1.2.cmml" xref="A2.SS2.p1.6.m6.1.1.2">𝑡</ci><cn id="A2.SS2.p1.6.m6.1.1.3.cmml" type="integer" xref="A2.SS2.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.6.m6.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.6.m6.1d">italic_t ≥ 1</annotation></semantics></math>. This can be done by picking, e.g., <math alttext="\bm{w}_{\Delta}=\bm{0}" class="ltx_Math" display="inline" id="A2.SS2.p1.7.m7.1"><semantics id="A2.SS2.p1.7.m7.1a"><mrow id="A2.SS2.p1.7.m7.1.1" xref="A2.SS2.p1.7.m7.1.1.cmml"><msub id="A2.SS2.p1.7.m7.1.1.2" xref="A2.SS2.p1.7.m7.1.1.2.cmml"><mi id="A2.SS2.p1.7.m7.1.1.2.2" xref="A2.SS2.p1.7.m7.1.1.2.2.cmml">𝒘</mi><mi id="A2.SS2.p1.7.m7.1.1.2.3" mathvariant="normal" xref="A2.SS2.p1.7.m7.1.1.2.3.cmml">Δ</mi></msub><mo id="A2.SS2.p1.7.m7.1.1.1" xref="A2.SS2.p1.7.m7.1.1.1.cmml">=</mo><mn id="A2.SS2.p1.7.m7.1.1.3" xref="A2.SS2.p1.7.m7.1.1.3.cmml">𝟎</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.7.m7.1b"><apply id="A2.SS2.p1.7.m7.1.1.cmml" xref="A2.SS2.p1.7.m7.1.1"><eq id="A2.SS2.p1.7.m7.1.1.1.cmml" xref="A2.SS2.p1.7.m7.1.1.1"></eq><apply id="A2.SS2.p1.7.m7.1.1.2.cmml" xref="A2.SS2.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p1.7.m7.1.1.2.1.cmml" xref="A2.SS2.p1.7.m7.1.1.2">subscript</csymbol><ci id="A2.SS2.p1.7.m7.1.1.2.2.cmml" xref="A2.SS2.p1.7.m7.1.1.2.2">𝒘</ci><ci id="A2.SS2.p1.7.m7.1.1.2.3.cmml" xref="A2.SS2.p1.7.m7.1.1.2.3">Δ</ci></apply><cn id="A2.SS2.p1.7.m7.1.1.3.cmml" type="integer" xref="A2.SS2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.7.m7.1c">\bm{w}_{\Delta}=\bm{0}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.7.m7.1d">bold_italic_w start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT = bold_0</annotation></semantics></math> and <math alttext="\delta" class="ltx_Math" display="inline" id="A2.SS2.p1.8.m8.1"><semantics id="A2.SS2.p1.8.m8.1a"><mi id="A2.SS2.p1.8.m8.1.1" xref="A2.SS2.p1.8.m8.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.8.m8.1b"><ci id="A2.SS2.p1.8.m8.1.1.cmml" xref="A2.SS2.p1.8.m8.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.8.m8.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.8.m8.1d">italic_δ</annotation></semantics></math> such that <math alttext="\mathrm{softplus}(\delta)=1" class="ltx_Math" display="inline" id="A2.SS2.p1.9.m9.1"><semantics id="A2.SS2.p1.9.m9.1a"><mrow id="A2.SS2.p1.9.m9.1.2" xref="A2.SS2.p1.9.m9.1.2.cmml"><mrow id="A2.SS2.p1.9.m9.1.2.2" xref="A2.SS2.p1.9.m9.1.2.2.cmml"><mi id="A2.SS2.p1.9.m9.1.2.2.2" xref="A2.SS2.p1.9.m9.1.2.2.2.cmml">softplus</mi><mo id="A2.SS2.p1.9.m9.1.2.2.1" xref="A2.SS2.p1.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS2.p1.9.m9.1.2.2.3.2" xref="A2.SS2.p1.9.m9.1.2.2.cmml"><mo id="A2.SS2.p1.9.m9.1.2.2.3.2.1" stretchy="false" xref="A2.SS2.p1.9.m9.1.2.2.cmml">(</mo><mi id="A2.SS2.p1.9.m9.1.1" xref="A2.SS2.p1.9.m9.1.1.cmml">δ</mi><mo id="A2.SS2.p1.9.m9.1.2.2.3.2.2" stretchy="false" xref="A2.SS2.p1.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.p1.9.m9.1.2.1" xref="A2.SS2.p1.9.m9.1.2.1.cmml">=</mo><mn id="A2.SS2.p1.9.m9.1.2.3" xref="A2.SS2.p1.9.m9.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.9.m9.1b"><apply id="A2.SS2.p1.9.m9.1.2.cmml" xref="A2.SS2.p1.9.m9.1.2"><eq id="A2.SS2.p1.9.m9.1.2.1.cmml" xref="A2.SS2.p1.9.m9.1.2.1"></eq><apply id="A2.SS2.p1.9.m9.1.2.2.cmml" xref="A2.SS2.p1.9.m9.1.2.2"><times id="A2.SS2.p1.9.m9.1.2.2.1.cmml" xref="A2.SS2.p1.9.m9.1.2.2.1"></times><ci id="A2.SS2.p1.9.m9.1.2.2.2.cmml" xref="A2.SS2.p1.9.m9.1.2.2.2">softplus</ci><ci id="A2.SS2.p1.9.m9.1.1.cmml" xref="A2.SS2.p1.9.m9.1.1">𝛿</ci></apply><cn id="A2.SS2.p1.9.m9.1.2.3.cmml" type="integer" xref="A2.SS2.p1.9.m9.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.9.m9.1c">\mathrm{softplus}(\delta)=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.9.m9.1d">roman_softplus ( italic_δ ) = 1</annotation></semantics></math>. Let us compactly denote the convolution kernels as</p> <table class="ltx_equation ltx_eqn_table" id="A2.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathrm{conv}_{X}=\begin{pmatrix}\alpha_{00}&\alpha_{01}\\ \alpha_{10}&\alpha_{11}\end{pmatrix},\qquad\mathrm{conv}_{B}=\begin{pmatrix}% \gamma_{00}&\gamma_{01}\\ \gamma_{10}&\gamma_{11}\end{pmatrix}" class="ltx_Math" display="block" id="A2.E7.m1.4"><semantics id="A2.E7.m1.4a"><mrow id="A2.E7.m1.4.4.2" xref="A2.E7.m1.4.4.3.cmml"><mrow id="A2.E7.m1.3.3.1.1" xref="A2.E7.m1.3.3.1.1.cmml"><msub id="A2.E7.m1.3.3.1.1.2" xref="A2.E7.m1.3.3.1.1.2.cmml"><mi id="A2.E7.m1.3.3.1.1.2.2" xref="A2.E7.m1.3.3.1.1.2.2.cmml">conv</mi><mi id="A2.E7.m1.3.3.1.1.2.3" xref="A2.E7.m1.3.3.1.1.2.3.cmml">X</mi></msub><mo id="A2.E7.m1.3.3.1.1.1" xref="A2.E7.m1.3.3.1.1.1.cmml">=</mo><mrow id="A2.E7.m1.1.1.3" xref="A2.E7.m1.1.1.2.cmml"><mo id="A2.E7.m1.1.1.3.1" xref="A2.E7.m1.1.1.2.1.cmml">(</mo><mtable columnspacing="5pt" displaystyle="true" id="A2.E7.m1.1.1.1.1" rowspacing="0pt" xref="A2.E7.m1.1.1.1.1.cmml"><mtr id="A2.E7.m1.1.1.1.1a" xref="A2.E7.m1.1.1.1.1.cmml"><mtd id="A2.E7.m1.1.1.1.1b" xref="A2.E7.m1.1.1.1.1.cmml"><msub id="A2.E7.m1.1.1.1.1.1.1.1" xref="A2.E7.m1.1.1.1.1.1.1.1.cmml"><mi id="A2.E7.m1.1.1.1.1.1.1.1.2" xref="A2.E7.m1.1.1.1.1.1.1.1.2.cmml">α</mi><mn id="A2.E7.m1.1.1.1.1.1.1.1.3" xref="A2.E7.m1.1.1.1.1.1.1.1.3.cmml">00</mn></msub></mtd><mtd id="A2.E7.m1.1.1.1.1c" xref="A2.E7.m1.1.1.1.1.cmml"><msub id="A2.E7.m1.1.1.1.1.1.2.1" xref="A2.E7.m1.1.1.1.1.1.2.1.cmml"><mi id="A2.E7.m1.1.1.1.1.1.2.1.2" xref="A2.E7.m1.1.1.1.1.1.2.1.2.cmml">α</mi><mn id="A2.E7.m1.1.1.1.1.1.2.1.3" xref="A2.E7.m1.1.1.1.1.1.2.1.3.cmml">01</mn></msub></mtd></mtr><mtr id="A2.E7.m1.1.1.1.1d" xref="A2.E7.m1.1.1.1.1.cmml"><mtd id="A2.E7.m1.1.1.1.1e" xref="A2.E7.m1.1.1.1.1.cmml"><msub id="A2.E7.m1.1.1.1.1.2.1.1" xref="A2.E7.m1.1.1.1.1.2.1.1.cmml"><mi id="A2.E7.m1.1.1.1.1.2.1.1.2" xref="A2.E7.m1.1.1.1.1.2.1.1.2.cmml">α</mi><mn id="A2.E7.m1.1.1.1.1.2.1.1.3" xref="A2.E7.m1.1.1.1.1.2.1.1.3.cmml">10</mn></msub></mtd><mtd id="A2.E7.m1.1.1.1.1f" xref="A2.E7.m1.1.1.1.1.cmml"><msub id="A2.E7.m1.1.1.1.1.2.2.1" xref="A2.E7.m1.1.1.1.1.2.2.1.cmml"><mi id="A2.E7.m1.1.1.1.1.2.2.1.2" xref="A2.E7.m1.1.1.1.1.2.2.1.2.cmml">α</mi><mn id="A2.E7.m1.1.1.1.1.2.2.1.3" xref="A2.E7.m1.1.1.1.1.2.2.1.3.cmml">11</mn></msub></mtd></mtr></mtable><mo id="A2.E7.m1.1.1.3.2" xref="A2.E7.m1.1.1.2.1.cmml">)</mo></mrow></mrow><mo id="A2.E7.m1.4.4.2.3" rspace="2.167em" xref="A2.E7.m1.4.4.3a.cmml">,</mo><mrow id="A2.E7.m1.4.4.2.2" xref="A2.E7.m1.4.4.2.2.cmml"><msub id="A2.E7.m1.4.4.2.2.2" xref="A2.E7.m1.4.4.2.2.2.cmml"><mi id="A2.E7.m1.4.4.2.2.2.2" xref="A2.E7.m1.4.4.2.2.2.2.cmml">conv</mi><mi id="A2.E7.m1.4.4.2.2.2.3" xref="A2.E7.m1.4.4.2.2.2.3.cmml">B</mi></msub><mo id="A2.E7.m1.4.4.2.2.1" xref="A2.E7.m1.4.4.2.2.1.cmml">=</mo><mrow id="A2.E7.m1.2.2.3" xref="A2.E7.m1.2.2.2.cmml"><mo id="A2.E7.m1.2.2.3.1" xref="A2.E7.m1.2.2.2.1.cmml">(</mo><mtable columnspacing="5pt" displaystyle="true" id="A2.E7.m1.2.2.1.1" rowspacing="0pt" xref="A2.E7.m1.2.2.1.1.cmml"><mtr id="A2.E7.m1.2.2.1.1a" xref="A2.E7.m1.2.2.1.1.cmml"><mtd id="A2.E7.m1.2.2.1.1b" xref="A2.E7.m1.2.2.1.1.cmml"><msub id="A2.E7.m1.2.2.1.1.1.1.1" xref="A2.E7.m1.2.2.1.1.1.1.1.cmml"><mi id="A2.E7.m1.2.2.1.1.1.1.1.2" xref="A2.E7.m1.2.2.1.1.1.1.1.2.cmml">γ</mi><mn id="A2.E7.m1.2.2.1.1.1.1.1.3" xref="A2.E7.m1.2.2.1.1.1.1.1.3.cmml">00</mn></msub></mtd><mtd id="A2.E7.m1.2.2.1.1c" xref="A2.E7.m1.2.2.1.1.cmml"><msub id="A2.E7.m1.2.2.1.1.1.2.1" xref="A2.E7.m1.2.2.1.1.1.2.1.cmml"><mi id="A2.E7.m1.2.2.1.1.1.2.1.2" xref="A2.E7.m1.2.2.1.1.1.2.1.2.cmml">γ</mi><mn id="A2.E7.m1.2.2.1.1.1.2.1.3" xref="A2.E7.m1.2.2.1.1.1.2.1.3.cmml">01</mn></msub></mtd></mtr><mtr id="A2.E7.m1.2.2.1.1d" xref="A2.E7.m1.2.2.1.1.cmml"><mtd 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type="integer" xref="A2.E7.m1.2.2.1.1.2.1.1.3">10</cn></apply><apply id="A2.E7.m1.2.2.1.1.2.2.1.cmml" xref="A2.E7.m1.2.2.1.1.2.2.1"><csymbol cd="ambiguous" id="A2.E7.m1.2.2.1.1.2.2.1.1.cmml" xref="A2.E7.m1.2.2.1.1.2.2.1">subscript</csymbol><ci id="A2.E7.m1.2.2.1.1.2.2.1.2.cmml" xref="A2.E7.m1.2.2.1.1.2.2.1.2">𝛾</ci><cn id="A2.E7.m1.2.2.1.1.2.2.1.3.cmml" type="integer" xref="A2.E7.m1.2.2.1.1.2.2.1.3">11</cn></apply></matrixrow></matrix></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E7.m1.4c">\mathrm{conv}_{X}=\begin{pmatrix}\alpha_{00}&\alpha_{01}\\ \alpha_{10}&\alpha_{11}\end{pmatrix},\qquad\mathrm{conv}_{B}=\begin{pmatrix}% \gamma_{00}&\gamma_{01}\\ \gamma_{10}&\gamma_{11}\end{pmatrix}</annotation><annotation encoding="application/x-llamapun" id="A2.E7.m1.4d">roman_conv start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_α start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_CELL start_CELL italic_α start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL start_CELL italic_α start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , roman_conv start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_CELL start_CELL italic_γ start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL start_CELL italic_γ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p1.13">where each row corresponds to the kernel weights applied time-wise to each coordinate of the input sequence <math alttext="(\bm{x}_{t})_{t\geq 1}" class="ltx_Math" display="inline" id="A2.SS2.p1.10.m1.1"><semantics id="A2.SS2.p1.10.m1.1a"><msub id="A2.SS2.p1.10.m1.1.1" xref="A2.SS2.p1.10.m1.1.1.cmml"><mrow id="A2.SS2.p1.10.m1.1.1.1.1" xref="A2.SS2.p1.10.m1.1.1.1.1.1.cmml"><mo id="A2.SS2.p1.10.m1.1.1.1.1.2" stretchy="false" xref="A2.SS2.p1.10.m1.1.1.1.1.1.cmml">(</mo><msub id="A2.SS2.p1.10.m1.1.1.1.1.1" xref="A2.SS2.p1.10.m1.1.1.1.1.1.cmml"><mi id="A2.SS2.p1.10.m1.1.1.1.1.1.2" xref="A2.SS2.p1.10.m1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A2.SS2.p1.10.m1.1.1.1.1.1.3" xref="A2.SS2.p1.10.m1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A2.SS2.p1.10.m1.1.1.1.1.3" stretchy="false" xref="A2.SS2.p1.10.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A2.SS2.p1.10.m1.1.1.3" xref="A2.SS2.p1.10.m1.1.1.3.cmml"><mi id="A2.SS2.p1.10.m1.1.1.3.2" xref="A2.SS2.p1.10.m1.1.1.3.2.cmml">t</mi><mo id="A2.SS2.p1.10.m1.1.1.3.1" xref="A2.SS2.p1.10.m1.1.1.3.1.cmml">≥</mo><mn id="A2.SS2.p1.10.m1.1.1.3.3" xref="A2.SS2.p1.10.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.10.m1.1b"><apply id="A2.SS2.p1.10.m1.1.1.cmml" xref="A2.SS2.p1.10.m1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.10.m1.1.1.2.cmml" xref="A2.SS2.p1.10.m1.1.1">subscript</csymbol><apply id="A2.SS2.p1.10.m1.1.1.1.1.1.cmml" xref="A2.SS2.p1.10.m1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.10.m1.1.1.1.1.1.1.cmml" xref="A2.SS2.p1.10.m1.1.1.1.1">subscript</csymbol><ci id="A2.SS2.p1.10.m1.1.1.1.1.1.2.cmml" xref="A2.SS2.p1.10.m1.1.1.1.1.1.2">𝒙</ci><ci id="A2.SS2.p1.10.m1.1.1.1.1.1.3.cmml" xref="A2.SS2.p1.10.m1.1.1.1.1.1.3">𝑡</ci></apply><apply id="A2.SS2.p1.10.m1.1.1.3.cmml" xref="A2.SS2.p1.10.m1.1.1.3"><geq id="A2.SS2.p1.10.m1.1.1.3.1.cmml" xref="A2.SS2.p1.10.m1.1.1.3.1"></geq><ci id="A2.SS2.p1.10.m1.1.1.3.2.cmml" xref="A2.SS2.p1.10.m1.1.1.3.2">𝑡</ci><cn id="A2.SS2.p1.10.m1.1.1.3.3.cmml" type="integer" xref="A2.SS2.p1.10.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.10.m1.1c">(\bm{x}_{t})_{t\geq 1}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.10.m1.1d">( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t ≥ 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Since the window for <math alttext="\mathrm{conv}_{C}" class="ltx_Math" display="inline" id="A2.SS2.p1.11.m2.1"><semantics id="A2.SS2.p1.11.m2.1a"><msub id="A2.SS2.p1.11.m2.1.1" xref="A2.SS2.p1.11.m2.1.1.cmml"><mi id="A2.SS2.p1.11.m2.1.1.2" xref="A2.SS2.p1.11.m2.1.1.2.cmml">conv</mi><mi id="A2.SS2.p1.11.m2.1.1.3" xref="A2.SS2.p1.11.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.11.m2.1b"><apply id="A2.SS2.p1.11.m2.1.1.cmml" xref="A2.SS2.p1.11.m2.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.11.m2.1.1.1.cmml" xref="A2.SS2.p1.11.m2.1.1">subscript</csymbol><ci id="A2.SS2.p1.11.m2.1.1.2.cmml" xref="A2.SS2.p1.11.m2.1.1.2">conv</ci><ci id="A2.SS2.p1.11.m2.1.1.3.cmml" xref="A2.SS2.p1.11.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.11.m2.1c">\mathrm{conv}_{C}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.11.m2.1d">roman_conv start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="w_{C}=1" class="ltx_Math" display="inline" id="A2.SS2.p1.12.m3.1"><semantics id="A2.SS2.p1.12.m3.1a"><mrow id="A2.SS2.p1.12.m3.1.1" xref="A2.SS2.p1.12.m3.1.1.cmml"><msub id="A2.SS2.p1.12.m3.1.1.2" xref="A2.SS2.p1.12.m3.1.1.2.cmml"><mi id="A2.SS2.p1.12.m3.1.1.2.2" xref="A2.SS2.p1.12.m3.1.1.2.2.cmml">w</mi><mi id="A2.SS2.p1.12.m3.1.1.2.3" xref="A2.SS2.p1.12.m3.1.1.2.3.cmml">C</mi></msub><mo id="A2.SS2.p1.12.m3.1.1.1" xref="A2.SS2.p1.12.m3.1.1.1.cmml">=</mo><mn id="A2.SS2.p1.12.m3.1.1.3" xref="A2.SS2.p1.12.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.12.m3.1b"><apply id="A2.SS2.p1.12.m3.1.1.cmml" xref="A2.SS2.p1.12.m3.1.1"><eq id="A2.SS2.p1.12.m3.1.1.1.cmml" xref="A2.SS2.p1.12.m3.1.1.1"></eq><apply id="A2.SS2.p1.12.m3.1.1.2.cmml" xref="A2.SS2.p1.12.m3.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p1.12.m3.1.1.2.1.cmml" xref="A2.SS2.p1.12.m3.1.1.2">subscript</csymbol><ci id="A2.SS2.p1.12.m3.1.1.2.2.cmml" xref="A2.SS2.p1.12.m3.1.1.2.2">𝑤</ci><ci id="A2.SS2.p1.12.m3.1.1.2.3.cmml" xref="A2.SS2.p1.12.m3.1.1.2.3">𝐶</ci></apply><cn id="A2.SS2.p1.12.m3.1.1.3.cmml" type="integer" xref="A2.SS2.p1.12.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.12.m3.1c">w_{C}=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.12.m3.1d">italic_w start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = 1</annotation></semantics></math>, we can simply assume w.l.o.g. that <math alttext="C_{t}=W_{C}\bm{x}_{t}" class="ltx_Math" display="inline" id="A2.SS2.p1.13.m4.1"><semantics id="A2.SS2.p1.13.m4.1a"><mrow id="A2.SS2.p1.13.m4.1.1" xref="A2.SS2.p1.13.m4.1.1.cmml"><msub id="A2.SS2.p1.13.m4.1.1.2" xref="A2.SS2.p1.13.m4.1.1.2.cmml"><mi id="A2.SS2.p1.13.m4.1.1.2.2" xref="A2.SS2.p1.13.m4.1.1.2.2.cmml">C</mi><mi id="A2.SS2.p1.13.m4.1.1.2.3" xref="A2.SS2.p1.13.m4.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS2.p1.13.m4.1.1.1" xref="A2.SS2.p1.13.m4.1.1.1.cmml">=</mo><mrow id="A2.SS2.p1.13.m4.1.1.3" xref="A2.SS2.p1.13.m4.1.1.3.cmml"><msub id="A2.SS2.p1.13.m4.1.1.3.2" xref="A2.SS2.p1.13.m4.1.1.3.2.cmml"><mi id="A2.SS2.p1.13.m4.1.1.3.2.2" xref="A2.SS2.p1.13.m4.1.1.3.2.2.cmml">W</mi><mi id="A2.SS2.p1.13.m4.1.1.3.2.3" xref="A2.SS2.p1.13.m4.1.1.3.2.3.cmml">C</mi></msub><mo id="A2.SS2.p1.13.m4.1.1.3.1" xref="A2.SS2.p1.13.m4.1.1.3.1.cmml">⁢</mo><msub id="A2.SS2.p1.13.m4.1.1.3.3" xref="A2.SS2.p1.13.m4.1.1.3.3.cmml"><mi id="A2.SS2.p1.13.m4.1.1.3.3.2" xref="A2.SS2.p1.13.m4.1.1.3.3.2.cmml">𝒙</mi><mi id="A2.SS2.p1.13.m4.1.1.3.3.3" xref="A2.SS2.p1.13.m4.1.1.3.3.3.cmml">t</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.13.m4.1b"><apply id="A2.SS2.p1.13.m4.1.1.cmml" xref="A2.SS2.p1.13.m4.1.1"><eq id="A2.SS2.p1.13.m4.1.1.1.cmml" xref="A2.SS2.p1.13.m4.1.1.1"></eq><apply id="A2.SS2.p1.13.m4.1.1.2.cmml" xref="A2.SS2.p1.13.m4.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p1.13.m4.1.1.2.1.cmml" xref="A2.SS2.p1.13.m4.1.1.2">subscript</csymbol><ci id="A2.SS2.p1.13.m4.1.1.2.2.cmml" xref="A2.SS2.p1.13.m4.1.1.2.2">𝐶</ci><ci id="A2.SS2.p1.13.m4.1.1.2.3.cmml" xref="A2.SS2.p1.13.m4.1.1.2.3">𝑡</ci></apply><apply id="A2.SS2.p1.13.m4.1.1.3.cmml" xref="A2.SS2.p1.13.m4.1.1.3"><times id="A2.SS2.p1.13.m4.1.1.3.1.cmml" xref="A2.SS2.p1.13.m4.1.1.3.1"></times><apply id="A2.SS2.p1.13.m4.1.1.3.2.cmml" xref="A2.SS2.p1.13.m4.1.1.3.2"><csymbol cd="ambiguous" id="A2.SS2.p1.13.m4.1.1.3.2.1.cmml" xref="A2.SS2.p1.13.m4.1.1.3.2">subscript</csymbol><ci id="A2.SS2.p1.13.m4.1.1.3.2.2.cmml" xref="A2.SS2.p1.13.m4.1.1.3.2.2">𝑊</ci><ci id="A2.SS2.p1.13.m4.1.1.3.2.3.cmml" xref="A2.SS2.p1.13.m4.1.1.3.2.3">𝐶</ci></apply><apply id="A2.SS2.p1.13.m4.1.1.3.3.cmml" xref="A2.SS2.p1.13.m4.1.1.3.3"><csymbol cd="ambiguous" id="A2.SS2.p1.13.m4.1.1.3.3.1.cmml" xref="A2.SS2.p1.13.m4.1.1.3.3">subscript</csymbol><ci id="A2.SS2.p1.13.m4.1.1.3.3.2.cmml" xref="A2.SS2.p1.13.m4.1.1.3.3.2">𝒙</ci><ci id="A2.SS2.p1.13.m4.1.1.3.3.3.cmml" xref="A2.SS2.p1.13.m4.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.13.m4.1c">C_{t}=W_{C}\bm{x}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.13.m4.1d">italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.SS2.p2"> <p class="ltx_p" id="A2.SS2.p2.3">Let us denote the embedding vectors to be <math alttext="\bm{e}_{0}=(e_{00},e_{01})^{\top}" class="ltx_Math" display="inline" id="A2.SS2.p2.1.m1.2"><semantics id="A2.SS2.p2.1.m1.2a"><mrow id="A2.SS2.p2.1.m1.2.2" xref="A2.SS2.p2.1.m1.2.2.cmml"><msub id="A2.SS2.p2.1.m1.2.2.4" xref="A2.SS2.p2.1.m1.2.2.4.cmml"><mi id="A2.SS2.p2.1.m1.2.2.4.2" xref="A2.SS2.p2.1.m1.2.2.4.2.cmml">𝒆</mi><mn id="A2.SS2.p2.1.m1.2.2.4.3" xref="A2.SS2.p2.1.m1.2.2.4.3.cmml">0</mn></msub><mo id="A2.SS2.p2.1.m1.2.2.3" xref="A2.SS2.p2.1.m1.2.2.3.cmml">=</mo><msup id="A2.SS2.p2.1.m1.2.2.2" xref="A2.SS2.p2.1.m1.2.2.2.cmml"><mrow id="A2.SS2.p2.1.m1.2.2.2.2.2" xref="A2.SS2.p2.1.m1.2.2.2.2.3.cmml"><mo id="A2.SS2.p2.1.m1.2.2.2.2.2.3" stretchy="false" xref="A2.SS2.p2.1.m1.2.2.2.2.3.cmml">(</mo><msub id="A2.SS2.p2.1.m1.1.1.1.1.1.1" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="A2.SS2.p2.1.m1.1.1.1.1.1.1.2" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1.2.cmml">e</mi><mn id="A2.SS2.p2.1.m1.1.1.1.1.1.1.3" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1.3.cmml">00</mn></msub><mo id="A2.SS2.p2.1.m1.2.2.2.2.2.4" xref="A2.SS2.p2.1.m1.2.2.2.2.3.cmml">,</mo><msub id="A2.SS2.p2.1.m1.2.2.2.2.2.2" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2.cmml"><mi id="A2.SS2.p2.1.m1.2.2.2.2.2.2.2" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2.2.cmml">e</mi><mn id="A2.SS2.p2.1.m1.2.2.2.2.2.2.3" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2.3.cmml">01</mn></msub><mo id="A2.SS2.p2.1.m1.2.2.2.2.2.5" stretchy="false" xref="A2.SS2.p2.1.m1.2.2.2.2.3.cmml">)</mo></mrow><mo id="A2.SS2.p2.1.m1.2.2.2.4" xref="A2.SS2.p2.1.m1.2.2.2.4.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.1.m1.2b"><apply id="A2.SS2.p2.1.m1.2.2.cmml" xref="A2.SS2.p2.1.m1.2.2"><eq id="A2.SS2.p2.1.m1.2.2.3.cmml" xref="A2.SS2.p2.1.m1.2.2.3"></eq><apply id="A2.SS2.p2.1.m1.2.2.4.cmml" xref="A2.SS2.p2.1.m1.2.2.4"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.2.2.4.1.cmml" xref="A2.SS2.p2.1.m1.2.2.4">subscript</csymbol><ci id="A2.SS2.p2.1.m1.2.2.4.2.cmml" xref="A2.SS2.p2.1.m1.2.2.4.2">𝒆</ci><cn id="A2.SS2.p2.1.m1.2.2.4.3.cmml" type="integer" xref="A2.SS2.p2.1.m1.2.2.4.3">0</cn></apply><apply id="A2.SS2.p2.1.m1.2.2.2.cmml" xref="A2.SS2.p2.1.m1.2.2.2"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.2.2.2.3.cmml" xref="A2.SS2.p2.1.m1.2.2.2">superscript</csymbol><interval closure="open" id="A2.SS2.p2.1.m1.2.2.2.2.3.cmml" xref="A2.SS2.p2.1.m1.2.2.2.2.2"><apply id="A2.SS2.p2.1.m1.1.1.1.1.1.1.cmml" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.SS2.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1.2">𝑒</ci><cn id="A2.SS2.p2.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.SS2.p2.1.m1.1.1.1.1.1.1.3">00</cn></apply><apply id="A2.SS2.p2.1.m1.2.2.2.2.2.2.cmml" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.2.2.2.2.2.2.1.cmml" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="A2.SS2.p2.1.m1.2.2.2.2.2.2.2.cmml" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2.2">𝑒</ci><cn id="A2.SS2.p2.1.m1.2.2.2.2.2.2.3.cmml" type="integer" xref="A2.SS2.p2.1.m1.2.2.2.2.2.2.3">01</cn></apply></interval><csymbol cd="latexml" id="A2.SS2.p2.1.m1.2.2.2.4.cmml" xref="A2.SS2.p2.1.m1.2.2.2.4">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.1.m1.2c">\bm{e}_{0}=(e_{00},e_{01})^{\top}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.1.m1.2d">bold_italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( italic_e start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\bm{e}_{1}=(e_{10},e_{11})^{\top}" class="ltx_Math" display="inline" id="A2.SS2.p2.2.m2.2"><semantics id="A2.SS2.p2.2.m2.2a"><mrow id="A2.SS2.p2.2.m2.2.2" xref="A2.SS2.p2.2.m2.2.2.cmml"><msub id="A2.SS2.p2.2.m2.2.2.4" xref="A2.SS2.p2.2.m2.2.2.4.cmml"><mi id="A2.SS2.p2.2.m2.2.2.4.2" xref="A2.SS2.p2.2.m2.2.2.4.2.cmml">𝒆</mi><mn id="A2.SS2.p2.2.m2.2.2.4.3" xref="A2.SS2.p2.2.m2.2.2.4.3.cmml">1</mn></msub><mo id="A2.SS2.p2.2.m2.2.2.3" xref="A2.SS2.p2.2.m2.2.2.3.cmml">=</mo><msup id="A2.SS2.p2.2.m2.2.2.2" xref="A2.SS2.p2.2.m2.2.2.2.cmml"><mrow id="A2.SS2.p2.2.m2.2.2.2.2.2" xref="A2.SS2.p2.2.m2.2.2.2.2.3.cmml"><mo id="A2.SS2.p2.2.m2.2.2.2.2.2.3" stretchy="false" xref="A2.SS2.p2.2.m2.2.2.2.2.3.cmml">(</mo><msub id="A2.SS2.p2.2.m2.1.1.1.1.1.1" xref="A2.SS2.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="A2.SS2.p2.2.m2.1.1.1.1.1.1.2" xref="A2.SS2.p2.2.m2.1.1.1.1.1.1.2.cmml">e</mi><mn id="A2.SS2.p2.2.m2.1.1.1.1.1.1.3" xref="A2.SS2.p2.2.m2.1.1.1.1.1.1.3.cmml">10</mn></msub><mo id="A2.SS2.p2.2.m2.2.2.2.2.2.4" xref="A2.SS2.p2.2.m2.2.2.2.2.3.cmml">,</mo><msub id="A2.SS2.p2.2.m2.2.2.2.2.2.2" xref="A2.SS2.p2.2.m2.2.2.2.2.2.2.cmml"><mi id="A2.SS2.p2.2.m2.2.2.2.2.2.2.2" xref="A2.SS2.p2.2.m2.2.2.2.2.2.2.2.cmml">e</mi><mn id="A2.SS2.p2.2.m2.2.2.2.2.2.2.3" xref="A2.SS2.p2.2.m2.2.2.2.2.2.2.3.cmml">11</mn></msub><mo id="A2.SS2.p2.2.m2.2.2.2.2.2.5" stretchy="false" xref="A2.SS2.p2.2.m2.2.2.2.2.3.cmml">)</mo></mrow><mo id="A2.SS2.p2.2.m2.2.2.2.4" xref="A2.SS2.p2.2.m2.2.2.2.4.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.2.m2.2b"><apply id="A2.SS2.p2.2.m2.2.2.cmml" xref="A2.SS2.p2.2.m2.2.2"><eq id="A2.SS2.p2.2.m2.2.2.3.cmml" xref="A2.SS2.p2.2.m2.2.2.3"></eq><apply id="A2.SS2.p2.2.m2.2.2.4.cmml" xref="A2.SS2.p2.2.m2.2.2.4"><csymbol cd="ambiguous" id="A2.SS2.p2.2.m2.2.2.4.1.cmml" xref="A2.SS2.p2.2.m2.2.2.4">subscript</csymbol><ci id="A2.SS2.p2.2.m2.2.2.4.2.cmml" xref="A2.SS2.p2.2.m2.2.2.4.2">𝒆</ci><cn id="A2.SS2.p2.2.m2.2.2.4.3.cmml" type="integer" xref="A2.SS2.p2.2.m2.2.2.4.3">1</cn></apply><apply id="A2.SS2.p2.2.m2.2.2.2.cmml" xref="A2.SS2.p2.2.m2.2.2.2"><csymbol cd="ambiguous" id="A2.SS2.p2.2.m2.2.2.2.3.cmml" xref="A2.SS2.p2.2.m2.2.2.2">superscript</csymbol><interval closure="open" id="A2.SS2.p2.2.m2.2.2.2.2.3.cmml" xref="A2.SS2.p2.2.m2.2.2.2.2.2"><apply id="A2.SS2.p2.2.m2.1.1.1.1.1.1.cmml" xref="A2.SS2.p2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="A2.SS2.p2.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="A2.SS2.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="A2.SS2.p2.2.m2.1.1.1.1.1.1.2">𝑒</ci><cn id="A2.SS2.p2.2.m2.1.1.1.1.1.1.3.cmml" type="integer" 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and assume that the vectors are not collinear. Take also <math alttext="W_{X}=W_{B}" class="ltx_Math" display="inline" id="A2.SS2.p2.3.m3.1"><semantics id="A2.SS2.p2.3.m3.1a"><mrow id="A2.SS2.p2.3.m3.1.1" xref="A2.SS2.p2.3.m3.1.1.cmml"><msub id="A2.SS2.p2.3.m3.1.1.2" xref="A2.SS2.p2.3.m3.1.1.2.cmml"><mi id="A2.SS2.p2.3.m3.1.1.2.2" xref="A2.SS2.p2.3.m3.1.1.2.2.cmml">W</mi><mi id="A2.SS2.p2.3.m3.1.1.2.3" xref="A2.SS2.p2.3.m3.1.1.2.3.cmml">X</mi></msub><mo id="A2.SS2.p2.3.m3.1.1.1" xref="A2.SS2.p2.3.m3.1.1.1.cmml">=</mo><msub id="A2.SS2.p2.3.m3.1.1.3" xref="A2.SS2.p2.3.m3.1.1.3.cmml"><mi id="A2.SS2.p2.3.m3.1.1.3.2" xref="A2.SS2.p2.3.m3.1.1.3.2.cmml">W</mi><mi id="A2.SS2.p2.3.m3.1.1.3.3" xref="A2.SS2.p2.3.m3.1.1.3.3.cmml">B</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.3.m3.1b"><apply id="A2.SS2.p2.3.m3.1.1.cmml" xref="A2.SS2.p2.3.m3.1.1"><eq id="A2.SS2.p2.3.m3.1.1.1.cmml" xref="A2.SS2.p2.3.m3.1.1.1"></eq><apply id="A2.SS2.p2.3.m3.1.1.2.cmml" xref="A2.SS2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.3.m3.1.1.2.1.cmml" xref="A2.SS2.p2.3.m3.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.3.m3.1.1.2.2.cmml" xref="A2.SS2.p2.3.m3.1.1.2.2">𝑊</ci><ci id="A2.SS2.p2.3.m3.1.1.2.3.cmml" xref="A2.SS2.p2.3.m3.1.1.2.3">𝑋</ci></apply><apply id="A2.SS2.p2.3.m3.1.1.3.cmml" xref="A2.SS2.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p2.3.m3.1.1.3.1.cmml" xref="A2.SS2.p2.3.m3.1.1.3">subscript</csymbol><ci id="A2.SS2.p2.3.m3.1.1.3.2.cmml" xref="A2.SS2.p2.3.m3.1.1.3.2">𝑊</ci><ci id="A2.SS2.p2.3.m3.1.1.3.3.cmml" xref="A2.SS2.p2.3.m3.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.3.m3.1c">W_{X}=W_{B}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.3.m3.1d">italic_W start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="A2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row 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start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) , italic_W start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT bold_italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL 1 end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.4">and take <math alttext="W_{C}" class="ltx_Math" display="inline" id="A2.SS2.p2.4.m1.1"><semantics id="A2.SS2.p2.4.m1.1a"><msub id="A2.SS2.p2.4.m1.1.1" xref="A2.SS2.p2.4.m1.1.1.cmml"><mi id="A2.SS2.p2.4.m1.1.1.2" xref="A2.SS2.p2.4.m1.1.1.2.cmml">W</mi><mi id="A2.SS2.p2.4.m1.1.1.3" xref="A2.SS2.p2.4.m1.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" 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start_ARG start_ROW start_CELL italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) , italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT bold_italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.6">Let us also take the kernels of <math alttext="\mathrm{conv}_{X}" class="ltx_Math" display="inline" id="A2.SS2.p2.5.m1.1"><semantics id="A2.SS2.p2.5.m1.1a"><msub id="A2.SS2.p2.5.m1.1.1" xref="A2.SS2.p2.5.m1.1.1.cmml"><mi id="A2.SS2.p2.5.m1.1.1.2" xref="A2.SS2.p2.5.m1.1.1.2.cmml">conv</mi><mi id="A2.SS2.p2.5.m1.1.1.3" xref="A2.SS2.p2.5.m1.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.5.m1.1b"><apply id="A2.SS2.p2.5.m1.1.1.cmml" xref="A2.SS2.p2.5.m1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p2.5.m1.1.1.1.cmml" xref="A2.SS2.p2.5.m1.1.1">subscript</csymbol><ci id="A2.SS2.p2.5.m1.1.1.2.cmml" xref="A2.SS2.p2.5.m1.1.1.2">conv</ci><ci id="A2.SS2.p2.5.m1.1.1.3.cmml" xref="A2.SS2.p2.5.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.5.m1.1c">\mathrm{conv}_{X}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.5.m1.1d">roman_conv start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathrm{conv}_{B}" class="ltx_Math" display="inline" id="A2.SS2.p2.6.m2.1"><semantics id="A2.SS2.p2.6.m2.1a"><msub id="A2.SS2.p2.6.m2.1.1" xref="A2.SS2.p2.6.m2.1.1.cmml"><mi id="A2.SS2.p2.6.m2.1.1.2" xref="A2.SS2.p2.6.m2.1.1.2.cmml">conv</mi><mi id="A2.SS2.p2.6.m2.1.1.3" xref="A2.SS2.p2.6.m2.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.6.m2.1b"><apply id="A2.SS2.p2.6.m2.1.1.cmml" xref="A2.SS2.p2.6.m2.1.1"><csymbol cd="ambiguous" id="A2.SS2.p2.6.m2.1.1.1.cmml" xref="A2.SS2.p2.6.m2.1.1">subscript</csymbol><ci id="A2.SS2.p2.6.m2.1.1.2.cmml" xref="A2.SS2.p2.6.m2.1.1.2">conv</ci><ci id="A2.SS2.p2.6.m2.1.1.3.cmml" xref="A2.SS2.p2.6.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.6.m2.1c">\mathrm{conv}_{B}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.6.m2.1d">roman_conv start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> to be the same across coordinates, i.e.,</p> <table class="ltx_equation ltx_eqn_table" id="A2.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , roman_conv start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL start_CELL italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL start_CELL italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.20">such that the following conditions are satisfied:</p> <table class="ltx_equation ltx_eqn_table" id="A2.E11"> <tbody><tr class="ltx_equation ltx_eqn_row 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encoding="application/x-llamapun" id="A2.E11.m1.4d">{ start_ROW start_CELL italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = - italic_β italic_ϵ end_CELL start_CELL end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.21">Note that, with such a choice of parameters, we have</p> <table class="ltx_equation ltx_eqn_table" id="A2.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="X^{(0)}=\begin{pmatrix}\alpha_{1}\\ 0\end{pmatrix},\qquad X^{(1)}=\begin{pmatrix}0\\ \alpha_{1}\end{pmatrix},\qquad B^{(0)}=\begin{pmatrix}\gamma_{1}\\ 0\end{pmatrix},\qquad B^{(1)}=\begin{pmatrix}0\\ \gamma_{1}\end{pmatrix}" class="ltx_Math" 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id="A2.E12.m1.10c">X^{(0)}=\begin{pmatrix}\alpha_{1}\\ 0\end{pmatrix},\qquad X^{(1)}=\begin{pmatrix}0\\ \alpha_{1}\end{pmatrix},\qquad B^{(0)}=\begin{pmatrix}\gamma_{1}\\ 0\end{pmatrix},\qquad B^{(1)}=\begin{pmatrix}0\\ \gamma_{1}\end{pmatrix}</annotation><annotation encoding="application/x-llamapun" id="A2.E12.m1.10d">italic_X start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) , italic_X start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , italic_B start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) , italic_B start_POSTSUPERSCRIPT ( 1 ) 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start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , italic_X start_POSTSUPERSCRIPT ( 10 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , italic_X start_POSTSUPERSCRIPT ( 11 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="A2.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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\gamma_{0}+\gamma_{1}\end{pmatrix}.</annotation><annotation encoding="application/x-llamapun" id="A2.E15.m1.9d">italic_B start_POSTSUPERSCRIPT ( 00 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) , italic_B start_POSTSUPERSCRIPT ( 01 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , italic_B start_POSTSUPERSCRIPT ( 10 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , italic_B start_POSTSUPERSCRIPT ( 11 ) end_POSTSUPERSCRIPT = ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.9">(We replaced the vector notation of Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4" title="4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">4</span></a> with matrix notation, so that <math alttext="X^{(0)}" class="ltx_Math" display="inline" id="A2.SS2.p2.7.m1.1"><semantics id="A2.SS2.p2.7.m1.1a"><msup id="A2.SS2.p2.7.m1.1.2" xref="A2.SS2.p2.7.m1.1.2.cmml"><mi id="A2.SS2.p2.7.m1.1.2.2" xref="A2.SS2.p2.7.m1.1.2.2.cmml">X</mi><mrow id="A2.SS2.p2.7.m1.1.1.1.3" xref="A2.SS2.p2.7.m1.1.2.cmml"><mo id="A2.SS2.p2.7.m1.1.1.1.3.1" stretchy="false" xref="A2.SS2.p2.7.m1.1.2.cmml">(</mo><mn id="A2.SS2.p2.7.m1.1.1.1.1" xref="A2.SS2.p2.7.m1.1.1.1.1.cmml">0</mn><mo id="A2.SS2.p2.7.m1.1.1.1.3.2" stretchy="false" xref="A2.SS2.p2.7.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.7.m1.1b"><apply id="A2.SS2.p2.7.m1.1.2.cmml" xref="A2.SS2.p2.7.m1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.7.m1.1.2.1.cmml" xref="A2.SS2.p2.7.m1.1.2">superscript</csymbol><ci id="A2.SS2.p2.7.m1.1.2.2.cmml" xref="A2.SS2.p2.7.m1.1.2.2">𝑋</ci><cn id="A2.SS2.p2.7.m1.1.1.1.1.cmml" type="integer" xref="A2.SS2.p2.7.m1.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.7.m1.1c">X^{(0)}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.7.m1.1d">italic_X start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math> has to be intended as <math alttext="\widetilde{\bm{x}}^{(0)}" class="ltx_Math" display="inline" id="A2.SS2.p2.8.m2.1"><semantics id="A2.SS2.p2.8.m2.1a"><msup id="A2.SS2.p2.8.m2.1.2" xref="A2.SS2.p2.8.m2.1.2.cmml"><mover accent="true" id="A2.SS2.p2.8.m2.1.2.2" xref="A2.SS2.p2.8.m2.1.2.2.cmml"><mi id="A2.SS2.p2.8.m2.1.2.2.2" xref="A2.SS2.p2.8.m2.1.2.2.2.cmml">𝒙</mi><mo id="A2.SS2.p2.8.m2.1.2.2.1" xref="A2.SS2.p2.8.m2.1.2.2.1.cmml">~</mo></mover><mrow id="A2.SS2.p2.8.m2.1.1.1.3" xref="A2.SS2.p2.8.m2.1.2.cmml"><mo id="A2.SS2.p2.8.m2.1.1.1.3.1" stretchy="false" xref="A2.SS2.p2.8.m2.1.2.cmml">(</mo><mn id="A2.SS2.p2.8.m2.1.1.1.1" xref="A2.SS2.p2.8.m2.1.1.1.1.cmml">0</mn><mo id="A2.SS2.p2.8.m2.1.1.1.3.2" stretchy="false" xref="A2.SS2.p2.8.m2.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.8.m2.1b"><apply id="A2.SS2.p2.8.m2.1.2.cmml" xref="A2.SS2.p2.8.m2.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.8.m2.1.2.1.cmml" xref="A2.SS2.p2.8.m2.1.2">superscript</csymbol><apply id="A2.SS2.p2.8.m2.1.2.2.cmml" xref="A2.SS2.p2.8.m2.1.2.2"><ci id="A2.SS2.p2.8.m2.1.2.2.1.cmml" xref="A2.SS2.p2.8.m2.1.2.2.1">~</ci><ci id="A2.SS2.p2.8.m2.1.2.2.2.cmml" xref="A2.SS2.p2.8.m2.1.2.2.2">𝒙</ci></apply><cn id="A2.SS2.p2.8.m2.1.1.1.1.cmml" type="integer" xref="A2.SS2.p2.8.m2.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.8.m2.1c">\widetilde{\bm{x}}^{(0)}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.8.m2.1d">over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and so on.) Take also <math alttext="W_{o}=W_{\ell}=I" class="ltx_Math" display="inline" id="A2.SS2.p2.9.m3.1"><semantics id="A2.SS2.p2.9.m3.1a"><mrow id="A2.SS2.p2.9.m3.1.1" xref="A2.SS2.p2.9.m3.1.1.cmml"><msub id="A2.SS2.p2.9.m3.1.1.2" xref="A2.SS2.p2.9.m3.1.1.2.cmml"><mi id="A2.SS2.p2.9.m3.1.1.2.2" xref="A2.SS2.p2.9.m3.1.1.2.2.cmml">W</mi><mi id="A2.SS2.p2.9.m3.1.1.2.3" xref="A2.SS2.p2.9.m3.1.1.2.3.cmml">o</mi></msub><mo id="A2.SS2.p2.9.m3.1.1.3" xref="A2.SS2.p2.9.m3.1.1.3.cmml">=</mo><msub id="A2.SS2.p2.9.m3.1.1.4" xref="A2.SS2.p2.9.m3.1.1.4.cmml"><mi id="A2.SS2.p2.9.m3.1.1.4.2" xref="A2.SS2.p2.9.m3.1.1.4.2.cmml">W</mi><mi id="A2.SS2.p2.9.m3.1.1.4.3" mathvariant="normal" xref="A2.SS2.p2.9.m3.1.1.4.3.cmml">ℓ</mi></msub><mo id="A2.SS2.p2.9.m3.1.1.5" xref="A2.SS2.p2.9.m3.1.1.5.cmml">=</mo><mi id="A2.SS2.p2.9.m3.1.1.6" xref="A2.SS2.p2.9.m3.1.1.6.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.9.m3.1b"><apply id="A2.SS2.p2.9.m3.1.1.cmml" xref="A2.SS2.p2.9.m3.1.1"><and id="A2.SS2.p2.9.m3.1.1a.cmml" xref="A2.SS2.p2.9.m3.1.1"></and><apply id="A2.SS2.p2.9.m3.1.1b.cmml" xref="A2.SS2.p2.9.m3.1.1"><eq id="A2.SS2.p2.9.m3.1.1.3.cmml" xref="A2.SS2.p2.9.m3.1.1.3"></eq><apply id="A2.SS2.p2.9.m3.1.1.2.cmml" xref="A2.SS2.p2.9.m3.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.9.m3.1.1.2.1.cmml" xref="A2.SS2.p2.9.m3.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.9.m3.1.1.2.2.cmml" xref="A2.SS2.p2.9.m3.1.1.2.2">𝑊</ci><ci id="A2.SS2.p2.9.m3.1.1.2.3.cmml" xref="A2.SS2.p2.9.m3.1.1.2.3">𝑜</ci></apply><apply id="A2.SS2.p2.9.m3.1.1.4.cmml" xref="A2.SS2.p2.9.m3.1.1.4"><csymbol cd="ambiguous" id="A2.SS2.p2.9.m3.1.1.4.1.cmml" xref="A2.SS2.p2.9.m3.1.1.4">subscript</csymbol><ci id="A2.SS2.p2.9.m3.1.1.4.2.cmml" xref="A2.SS2.p2.9.m3.1.1.4.2">𝑊</ci><ci id="A2.SS2.p2.9.m3.1.1.4.3.cmml" xref="A2.SS2.p2.9.m3.1.1.4.3">ℓ</ci></apply></apply><apply id="A2.SS2.p2.9.m3.1.1c.cmml" xref="A2.SS2.p2.9.m3.1.1"><eq id="A2.SS2.p2.9.m3.1.1.5.cmml" xref="A2.SS2.p2.9.m3.1.1.5"></eq><share href="https://arxiv.org/html/2502.10178v1#A2.SS2.p2.9.m3.1.1.4.cmml" id="A2.SS2.p2.9.m3.1.1d.cmml" xref="A2.SS2.p2.9.m3.1.1"></share><ci id="A2.SS2.p2.9.m3.1.1.6.cmml" xref="A2.SS2.p2.9.m3.1.1.6">𝐼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.9.m3.1c">W_{o}=W_{\ell}=I</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.9.m3.1d">italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = italic_I</annotation></semantics></math>. With this choice of parameters, the final logit vector becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx12"> <tbody id="A2.Ex37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathrm{logit}_{t}" class="ltx_Math" display="inline" id="A2.Ex37.m1.1"><semantics id="A2.Ex37.m1.1a"><msub id="A2.Ex37.m1.1.1" xref="A2.Ex37.m1.1.1.cmml"><mi id="A2.Ex37.m1.1.1.2" xref="A2.Ex37.m1.1.1.2.cmml">logit</mi><mi id="A2.Ex37.m1.1.1.3" xref="A2.Ex37.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.Ex37.m1.1b"><apply id="A2.Ex37.m1.1.1.cmml" xref="A2.Ex37.m1.1.1"><csymbol cd="ambiguous" id="A2.Ex37.m1.1.1.1.cmml" xref="A2.Ex37.m1.1.1">subscript</csymbol><ci id="A2.Ex37.m1.1.1.2.cmml" xref="A2.Ex37.m1.1.1.2">logit</ci><ci id="A2.Ex37.m1.1.1.3.cmml" xref="A2.Ex37.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex37.m1.1c">\displaystyle\mathrm{logit}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex37.m1.1d">roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=W_{\ell}\bm{x}_{t}+W_{\ell}W_{o}X^{(x_{1})}B^{(x_{1})^{\top}}C^{% (x_{t})}+\mathbbm{1}_{\{x_{1}\neq x_{t}\}}W_{\ell}W_{o}X^{(x_{1}x_{t})}B^{(x_{% 1}x_{t})\top}C^{(x_{t})}" class="ltx_Math" display="inline" id="A2.Ex37.m2.7"><semantics id="A2.Ex37.m2.7a"><mrow id="A2.Ex37.m2.7.8" xref="A2.Ex37.m2.7.8.cmml"><mi id="A2.Ex37.m2.7.8.2" xref="A2.Ex37.m2.7.8.2.cmml"></mi><mo id="A2.Ex37.m2.7.8.1" xref="A2.Ex37.m2.7.8.1.cmml">=</mo><mrow id="A2.Ex37.m2.7.8.3" xref="A2.Ex37.m2.7.8.3.cmml"><mrow id="A2.Ex37.m2.7.8.3.2" xref="A2.Ex37.m2.7.8.3.2.cmml"><msub id="A2.Ex37.m2.7.8.3.2.2" 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id="A2.Ex37.m2.7.8.3.3.5.2" xref="A2.Ex37.m2.7.8.3.3.5.2.cmml">B</mi><msup id="A2.Ex37.m2.2.2.1" xref="A2.Ex37.m2.2.2.1.cmml"><mrow id="A2.Ex37.m2.2.2.1.1.1" xref="A2.Ex37.m2.2.2.1.1.1.1.cmml"><mo id="A2.Ex37.m2.2.2.1.1.1.2" stretchy="false" xref="A2.Ex37.m2.2.2.1.1.1.1.cmml">(</mo><msub id="A2.Ex37.m2.2.2.1.1.1.1" xref="A2.Ex37.m2.2.2.1.1.1.1.cmml"><mi id="A2.Ex37.m2.2.2.1.1.1.1.2" xref="A2.Ex37.m2.2.2.1.1.1.1.2.cmml">x</mi><mn id="A2.Ex37.m2.2.2.1.1.1.1.3" xref="A2.Ex37.m2.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="A2.Ex37.m2.2.2.1.1.1.3" stretchy="false" xref="A2.Ex37.m2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="A2.Ex37.m2.2.2.1.3" xref="A2.Ex37.m2.2.2.1.3.cmml">⊤</mo></msup></msup><mo id="A2.Ex37.m2.7.8.3.3.1c" xref="A2.Ex37.m2.7.8.3.3.1.cmml">⁢</mo><msup id="A2.Ex37.m2.7.8.3.3.6" xref="A2.Ex37.m2.7.8.3.3.6.cmml"><mi id="A2.Ex37.m2.7.8.3.3.6.2" xref="A2.Ex37.m2.7.8.3.3.6.2.cmml">C</mi><mrow id="A2.Ex37.m2.3.3.1.1" xref="A2.Ex37.m2.3.3.1.1.1.cmml"><mo id="A2.Ex37.m2.3.3.1.1.2" 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end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_C start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ⊤ end_POSTSUPERSCRIPT italic_C start_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) 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id="A2.Ex38.m1.4c">\displaystyle=\begin{pmatrix}e_{00}+c_{0}\alpha_{1}\gamma_{1}\\ e_{01}\end{pmatrix}+\mathbbm{1}_{\{x_{1}=1\}}\cdot\begin{pmatrix}0\\ c_{0}\alpha_{0}\gamma_{1}\end{pmatrix}+n_{00}\cdot\begin{pmatrix}(\alpha_{0}+% \alpha_{1})(\gamma_{0}+\gamma_{1})c_{0}\\ 0\end{pmatrix}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex38.m1.4d">= ( start_ARG start_ROW start_CELL italic_e start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_e start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 } end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) + italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hskip 260.0004pt+n_{01}\cdot\begin{pmatrix}(\alpha_{0}\gamma_{0}% +\alpha_{1}\gamma_{1})c_{0}\\ (\alpha_{0}\gamma_{1}+\alpha_{1}\gamma_{0})c_{0}\end{pmatrix}" 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end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> <tbody id="A2.Ex39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\begin{pmatrix}e_{00}+c_{0}\alpha_{1}\gamma_{1}\\ 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encoding="application/x-tex" id="A2.Ex40.m1.1c">\displaystyle\mathrm{logit}_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex40.m1.1d">roman_logit start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\begin{pmatrix}e_{10}\\ e_{11}+c_{1}\alpha_{1}\gamma_{1}\end{pmatrix}+\mathbbm{1}_{\{x_{1}=0\}}\cdot% \begin{pmatrix}c_{1}\alpha_{0}\gamma_{1}\\ 0\end{pmatrix}+n_{10}\cdot\begin{pmatrix}(\alpha_{0}\gamma_{1}+\alpha_{1}% \gamma_{0})c_{1}\\ (\alpha_{0}\gamma_{0}+\alpha_{1}\gamma_{1})c_{1}\end{pmatrix}" class="ltx_Math" display="inline" id="A2.Ex40.m2.4"><semantics id="A2.Ex40.m2.4a"><mrow id="A2.Ex40.m2.4.5" xref="A2.Ex40.m2.4.5.cmml"><mi id="A2.Ex40.m2.4.5.2" xref="A2.Ex40.m2.4.5.2.cmml"></mi><mo id="A2.Ex40.m2.4.5.1" xref="A2.Ex40.m2.4.5.1.cmml">=</mo><mrow id="A2.Ex40.m2.4.5.3" xref="A2.Ex40.m2.4.5.3.cmml"><mrow id="A2.Ex40.m2.1.1a.3" 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xref="A2.Ex40.m2.2.2a.2.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex40.m2.4.5.3.1a" xref="A2.Ex40.m2.4.5.3.1.cmml">+</mo><mrow id="A2.Ex40.m2.4.5.3.3" xref="A2.Ex40.m2.4.5.3.3.cmml"><msub id="A2.Ex40.m2.4.5.3.3.2" xref="A2.Ex40.m2.4.5.3.3.2.cmml"><mi id="A2.Ex40.m2.4.5.3.3.2.2" xref="A2.Ex40.m2.4.5.3.3.2.2.cmml">n</mi><mn id="A2.Ex40.m2.4.5.3.3.2.3" xref="A2.Ex40.m2.4.5.3.3.2.3.cmml">10</mn></msub><mo id="A2.Ex40.m2.4.5.3.3.1" lspace="0.222em" rspace="0.222em" xref="A2.Ex40.m2.4.5.3.3.1.cmml">⋅</mo><mrow id="A2.Ex40.m2.3.3a.3" xref="A2.Ex40.m2.3.3a.2.cmml"><mo id="A2.Ex40.m2.3.3a.3.1" xref="A2.Ex40.m2.3.3a.2.1.cmml">(</mo><mtable id="A2.Ex40.m2.3.3.1.1" rowspacing="0pt" xref="A2.Ex40.m2.3.3.1.1.cmml"><mtr id="A2.Ex40.m2.3.3.1.1a" xref="A2.Ex40.m2.3.3.1.1.cmml"><mtd id="A2.Ex40.m2.3.3.1.1b" xref="A2.Ex40.m2.3.3.1.1.cmml"><mrow id="A2.Ex40.m2.3.3.1.1.1.1.1.1" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.cmml"><mrow id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.cmml"><mo 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xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.1" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.cmml"><msub id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.2" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.2.2" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.2.2.cmml">α</mi><mn id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.2.3" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.2.3.cmml">1</mn></msub><mo id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.1" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.3" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.3.2" xref="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.3.2.cmml">γ</mi><mn id="A2.Ex40.m2.3.3.1.1.1.1.1.1.1.1.1.3.3.3" 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e_{11}+c_{1}\alpha_{1}\gamma_{1}\end{pmatrix}+\mathbbm{1}_{\{x_{1}=0\}}\cdot% \begin{pmatrix}c_{1}\alpha_{0}\gamma_{1}\\ 0\end{pmatrix}+n_{10}\cdot\begin{pmatrix}(\alpha_{0}\gamma_{1}+\alpha_{1}% \gamma_{0})c_{1}\\ (\alpha_{0}\gamma_{0}+\alpha_{1}\gamma_{1})c_{1}\end{pmatrix}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex40.m2.4d">= ( start_ARG start_ROW start_CELL italic_e start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_e start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 } end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) + italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.E19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td 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encoding="application/x-tex" id="A2.Ex41.m1.4c">\displaystyle=\begin{pmatrix}e_{10}\\ e_{11}+c_{1}\alpha_{1}\gamma_{1}\end{pmatrix}+\mathbbm{1}_{\{x_{1}=0\}}\cdot% \begin{pmatrix}c_{1}\alpha_{0}\gamma_{1}\\ 0\end{pmatrix}+n_{10}\cdot\begin{pmatrix}(\alpha_{0}\gamma_{1}+\alpha_{1}% \gamma_{0})c_{1}\\ 0\end{pmatrix}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex41.m1.4d">= ( start_ARG start_ROW start_CELL italic_e start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_e start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 } end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ) + italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.E20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hskip 240.00037pt+n_{11}\cdot\begin{pmatrix}0\\ (\alpha_{0}\gamma_{1}+\alpha_{1}\gamma_{0})c_{1}\end{pmatrix}" class="ltx_Math" 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xref="A2.E20.m1.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply></matrixrow></matrix></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E20.m1.1c">\displaystyle\hskip 240.00037pt+n_{11}\cdot\begin{pmatrix}0\\ (\alpha_{0}\gamma_{1}+\alpha_{1}\gamma_{0})c_{1}\end{pmatrix}</annotation><annotation encoding="application/x-llamapun" id="A2.E20.m1.1d">+ italic_n start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ⋅ ( start_ARG start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.11">if <math alttext="x_{t}=1" class="ltx_Math" display="inline" id="A2.SS2.p2.11.m1.1"><semantics id="A2.SS2.p2.11.m1.1a"><mrow id="A2.SS2.p2.11.m1.1.1" xref="A2.SS2.p2.11.m1.1.1.cmml"><msub id="A2.SS2.p2.11.m1.1.1.2" xref="A2.SS2.p2.11.m1.1.1.2.cmml"><mi id="A2.SS2.p2.11.m1.1.1.2.2" xref="A2.SS2.p2.11.m1.1.1.2.2.cmml">x</mi><mi id="A2.SS2.p2.11.m1.1.1.2.3" xref="A2.SS2.p2.11.m1.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS2.p2.11.m1.1.1.1" xref="A2.SS2.p2.11.m1.1.1.1.cmml">=</mo><mn id="A2.SS2.p2.11.m1.1.1.3" xref="A2.SS2.p2.11.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.11.m1.1b"><apply id="A2.SS2.p2.11.m1.1.1.cmml" xref="A2.SS2.p2.11.m1.1.1"><eq id="A2.SS2.p2.11.m1.1.1.1.cmml" xref="A2.SS2.p2.11.m1.1.1.1"></eq><apply id="A2.SS2.p2.11.m1.1.1.2.cmml" xref="A2.SS2.p2.11.m1.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.11.m1.1.1.2.1.cmml" xref="A2.SS2.p2.11.m1.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.11.m1.1.1.2.2.cmml" xref="A2.SS2.p2.11.m1.1.1.2.2">𝑥</ci><ci id="A2.SS2.p2.11.m1.1.1.2.3.cmml" xref="A2.SS2.p2.11.m1.1.1.2.3">𝑡</ci></apply><cn id="A2.SS2.p2.11.m1.1.1.3.cmml" type="integer" xref="A2.SS2.p2.11.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.11.m1.1c">x_{t}=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.11.m1.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Take now</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx14"> <tbody id="A2.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle e_{00}" class="ltx_Math" display="inline" id="A2.E21.m1.1"><semantics id="A2.E21.m1.1a"><msub id="A2.E21.m1.1.1" xref="A2.E21.m1.1.1.cmml"><mi id="A2.E21.m1.1.1.2" xref="A2.E21.m1.1.1.2.cmml">e</mi><mn id="A2.E21.m1.1.1.3" xref="A2.E21.m1.1.1.3.cmml">00</mn></msub><annotation-xml encoding="MathML-Content" id="A2.E21.m1.1b"><apply id="A2.E21.m1.1.1.cmml" xref="A2.E21.m1.1.1"><csymbol cd="ambiguous" id="A2.E21.m1.1.1.1.cmml" xref="A2.E21.m1.1.1">subscript</csymbol><ci id="A2.E21.m1.1.1.2.cmml" xref="A2.E21.m1.1.1.2">𝑒</ci><cn id="A2.E21.m1.1.1.3.cmml" type="integer" xref="A2.E21.m1.1.1.3">00</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E21.m1.1c">\displaystyle 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xref="A2.E21.m2.1.1.1.3.4.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E21.m2.1c">\displaystyle=e_{11}=(\alpha_{0}\gamma_{1}+\alpha_{1}\gamma_{0})\beta c_{0}-% \alpha_{1}\gamma_{1}c_{0}</annotation><annotation encoding="application/x-llamapun" id="A2.E21.m2.1d">= italic_e start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_β italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> <tbody id="A2.E22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle e_{01}" class="ltx_Math" display="inline" id="A2.E22.m1.1"><semantics id="A2.E22.m1.1a"><msub id="A2.E22.m1.1.1" xref="A2.E22.m1.1.1.cmml"><mi id="A2.E22.m1.1.1.2" xref="A2.E22.m1.1.1.2.cmml">e</mi><mn id="A2.E22.m1.1.1.3" xref="A2.E22.m1.1.1.3.cmml">01</mn></msub><annotation-xml encoding="MathML-Content" id="A2.E22.m1.1b"><apply id="A2.E22.m1.1.1.cmml" xref="A2.E22.m1.1.1"><csymbol cd="ambiguous" id="A2.E22.m1.1.1.1.cmml" xref="A2.E22.m1.1.1">subscript</csymbol><ci id="A2.E22.m1.1.1.2.cmml" xref="A2.E22.m1.1.1.2">𝑒</ci><cn id="A2.E22.m1.1.1.3.cmml" type="integer" xref="A2.E22.m1.1.1.3">01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E22.m1.1c">\displaystyle 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xref="A2.E22.m2.1.1.1.3.4.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E22.m2.1c">\displaystyle=e_{10}=(\alpha_{0}\gamma_{1}+\alpha_{1}\gamma_{0})\beta c_{0}-% \alpha_{0}\gamma_{1}c_{0}</annotation><annotation encoding="application/x-llamapun" id="A2.E22.m2.1d">= italic_e start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT = ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_β italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.22">With this choice of parameters, after the layer normalization, the final output probability vector is</p> <table class="ltx_equation ltx_eqn_table" id="A2.E23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{\bm{\theta}}(x_{1}^{t})=\left(\frac{n_{00}+\beta}{n_{00}+n_{01}+2\beta+% \mathbbm{1}_{\{x_{1}=0\}}\cdot\beta\epsilon}\ ,\ \frac{n_{01}+\beta+\mathbbm{1% }_{\{x_{1}=0\}}\cdot\beta\epsilon}{n_{00}+n_{01}+2\beta+\mathbbm{1}_{\{x_{1}=0% \}}\cdot\beta\epsilon}\right)^{\top}" class="ltx_Math" display="block" id="A2.E23.m1.4"><semantics id="A2.E23.m1.4a"><mrow id="A2.E23.m1.4.4" xref="A2.E23.m1.4.4.cmml"><mrow id="A2.E23.m1.4.4.1" xref="A2.E23.m1.4.4.1.cmml"><msub id="A2.E23.m1.4.4.1.3" xref="A2.E23.m1.4.4.1.3.cmml"><mi id="A2.E23.m1.4.4.1.3.2" 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start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = ( divide start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 } end_POSTSUBSCRIPT ⋅ italic_β italic_ϵ end_ARG , divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 } end_POSTSUBSCRIPT ⋅ italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 } end_POSTSUBSCRIPT ⋅ italic_β italic_ϵ end_ARG ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.12">if <math alttext="x_{t}=0" class="ltx_Math" display="inline" id="A2.SS2.p2.12.m1.1"><semantics id="A2.SS2.p2.12.m1.1a"><mrow id="A2.SS2.p2.12.m1.1.1" xref="A2.SS2.p2.12.m1.1.1.cmml"><msub id="A2.SS2.p2.12.m1.1.1.2" xref="A2.SS2.p2.12.m1.1.1.2.cmml"><mi id="A2.SS2.p2.12.m1.1.1.2.2" xref="A2.SS2.p2.12.m1.1.1.2.2.cmml">x</mi><mi id="A2.SS2.p2.12.m1.1.1.2.3" xref="A2.SS2.p2.12.m1.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS2.p2.12.m1.1.1.1" xref="A2.SS2.p2.12.m1.1.1.1.cmml">=</mo><mn id="A2.SS2.p2.12.m1.1.1.3" xref="A2.SS2.p2.12.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.12.m1.1b"><apply id="A2.SS2.p2.12.m1.1.1.cmml" xref="A2.SS2.p2.12.m1.1.1"><eq id="A2.SS2.p2.12.m1.1.1.1.cmml" xref="A2.SS2.p2.12.m1.1.1.1"></eq><apply id="A2.SS2.p2.12.m1.1.1.2.cmml" xref="A2.SS2.p2.12.m1.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.12.m1.1.1.2.1.cmml" xref="A2.SS2.p2.12.m1.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.12.m1.1.1.2.2.cmml" xref="A2.SS2.p2.12.m1.1.1.2.2">𝑥</ci><ci id="A2.SS2.p2.12.m1.1.1.2.3.cmml" xref="A2.SS2.p2.12.m1.1.1.2.3">𝑡</ci></apply><cn id="A2.SS2.p2.12.m1.1.1.3.cmml" type="integer" xref="A2.SS2.p2.12.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.12.m1.1c">x_{t}=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.12.m1.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>, and</p> <table class="ltx_equation ltx_eqn_table" id="A2.E24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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encoding="application/x-tex" id="A2.E24.m1.4c">f_{\bm{\theta}}(x_{1}^{t})=\left(\frac{n_{10}+\beta+\mathbbm{1}_{\{x_{1}=1\}}% \cdot\beta\epsilon}{n_{10}+n_{11}+2\beta+\mathbbm{1}_{\{x_{1}=1\}}\cdot\beta% \epsilon}\ ,\ \frac{n_{11}+\beta}{n_{10}+n_{11}+2\beta+\mathbbm{1}_{\{x_{1}=1% \}}\cdot\beta\epsilon}\right)^{\top}</annotation><annotation encoding="application/x-llamapun" id="A2.E24.m1.4d">italic_f start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) = ( divide start_ARG italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT + italic_β + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 } end_POSTSUBSCRIPT ⋅ italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + 2 italic_β + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 } end_POSTSUBSCRIPT ⋅ italic_β italic_ϵ end_ARG , divide start_ARG italic_n start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + 2 italic_β + blackboard_1 start_POSTSUBSCRIPT { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 } end_POSTSUBSCRIPT ⋅ italic_β italic_ϵ end_ARG ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.19">if <math alttext="x_{t}=1" class="ltx_Math" display="inline" id="A2.SS2.p2.13.m1.1"><semantics id="A2.SS2.p2.13.m1.1a"><mrow id="A2.SS2.p2.13.m1.1.1" xref="A2.SS2.p2.13.m1.1.1.cmml"><msub id="A2.SS2.p2.13.m1.1.1.2" xref="A2.SS2.p2.13.m1.1.1.2.cmml"><mi id="A2.SS2.p2.13.m1.1.1.2.2" xref="A2.SS2.p2.13.m1.1.1.2.2.cmml">x</mi><mi id="A2.SS2.p2.13.m1.1.1.2.3" xref="A2.SS2.p2.13.m1.1.1.2.3.cmml">t</mi></msub><mo id="A2.SS2.p2.13.m1.1.1.1" xref="A2.SS2.p2.13.m1.1.1.1.cmml">=</mo><mn id="A2.SS2.p2.13.m1.1.1.3" xref="A2.SS2.p2.13.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.13.m1.1b"><apply id="A2.SS2.p2.13.m1.1.1.cmml" xref="A2.SS2.p2.13.m1.1.1"><eq id="A2.SS2.p2.13.m1.1.1.1.cmml" xref="A2.SS2.p2.13.m1.1.1.1"></eq><apply id="A2.SS2.p2.13.m1.1.1.2.cmml" xref="A2.SS2.p2.13.m1.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.13.m1.1.1.2.1.cmml" xref="A2.SS2.p2.13.m1.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.13.m1.1.1.2.2.cmml" xref="A2.SS2.p2.13.m1.1.1.2.2">𝑥</ci><ci id="A2.SS2.p2.13.m1.1.1.2.3.cmml" xref="A2.SS2.p2.13.m1.1.1.2.3">𝑡</ci></apply><cn id="A2.SS2.p2.13.m1.1.1.3.cmml" type="integer" xref="A2.SS2.p2.13.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.13.m1.1c">x_{t}=1</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.13.m1.1d">italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Note that the resulting predicted probabilities exactly match the add-<math alttext="\beta" class="ltx_Math" display="inline" id="A2.SS2.p2.14.m2.1"><semantics id="A2.SS2.p2.14.m2.1a"><mi id="A2.SS2.p2.14.m2.1.1" xref="A2.SS2.p2.14.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.14.m2.1b"><ci id="A2.SS2.p2.14.m2.1.1.cmml" xref="A2.SS2.p2.14.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.14.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.14.m2.1d">italic_β</annotation></semantics></math> estimator when <math alttext="x_{1}\neq x_{t}" class="ltx_Math" display="inline" id="A2.SS2.p2.15.m3.1"><semantics id="A2.SS2.p2.15.m3.1a"><mrow id="A2.SS2.p2.15.m3.1.1" xref="A2.SS2.p2.15.m3.1.1.cmml"><msub id="A2.SS2.p2.15.m3.1.1.2" xref="A2.SS2.p2.15.m3.1.1.2.cmml"><mi id="A2.SS2.p2.15.m3.1.1.2.2" xref="A2.SS2.p2.15.m3.1.1.2.2.cmml">x</mi><mn id="A2.SS2.p2.15.m3.1.1.2.3" xref="A2.SS2.p2.15.m3.1.1.2.3.cmml">1</mn></msub><mo id="A2.SS2.p2.15.m3.1.1.1" xref="A2.SS2.p2.15.m3.1.1.1.cmml">≠</mo><msub id="A2.SS2.p2.15.m3.1.1.3" xref="A2.SS2.p2.15.m3.1.1.3.cmml"><mi id="A2.SS2.p2.15.m3.1.1.3.2" xref="A2.SS2.p2.15.m3.1.1.3.2.cmml">x</mi><mi id="A2.SS2.p2.15.m3.1.1.3.3" xref="A2.SS2.p2.15.m3.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.15.m3.1b"><apply id="A2.SS2.p2.15.m3.1.1.cmml" xref="A2.SS2.p2.15.m3.1.1"><neq id="A2.SS2.p2.15.m3.1.1.1.cmml" xref="A2.SS2.p2.15.m3.1.1.1"></neq><apply id="A2.SS2.p2.15.m3.1.1.2.cmml" xref="A2.SS2.p2.15.m3.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.15.m3.1.1.2.1.cmml" xref="A2.SS2.p2.15.m3.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.15.m3.1.1.2.2.cmml" xref="A2.SS2.p2.15.m3.1.1.2.2">𝑥</ci><cn id="A2.SS2.p2.15.m3.1.1.2.3.cmml" type="integer" xref="A2.SS2.p2.15.m3.1.1.2.3">1</cn></apply><apply id="A2.SS2.p2.15.m3.1.1.3.cmml" xref="A2.SS2.p2.15.m3.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p2.15.m3.1.1.3.1.cmml" xref="A2.SS2.p2.15.m3.1.1.3">subscript</csymbol><ci id="A2.SS2.p2.15.m3.1.1.3.2.cmml" xref="A2.SS2.p2.15.m3.1.1.3.2">𝑥</ci><ci id="A2.SS2.p2.15.m3.1.1.3.3.cmml" xref="A2.SS2.p2.15.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.15.m3.1c">x_{1}\neq x_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.15.m3.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, but they are slightly different when <math alttext="x_{1}=x_{t}" class="ltx_Math" display="inline" id="A2.SS2.p2.16.m4.1"><semantics id="A2.SS2.p2.16.m4.1a"><mrow id="A2.SS2.p2.16.m4.1.1" xref="A2.SS2.p2.16.m4.1.1.cmml"><msub id="A2.SS2.p2.16.m4.1.1.2" xref="A2.SS2.p2.16.m4.1.1.2.cmml"><mi id="A2.SS2.p2.16.m4.1.1.2.2" xref="A2.SS2.p2.16.m4.1.1.2.2.cmml">x</mi><mn id="A2.SS2.p2.16.m4.1.1.2.3" xref="A2.SS2.p2.16.m4.1.1.2.3.cmml">1</mn></msub><mo id="A2.SS2.p2.16.m4.1.1.1" xref="A2.SS2.p2.16.m4.1.1.1.cmml">=</mo><msub id="A2.SS2.p2.16.m4.1.1.3" xref="A2.SS2.p2.16.m4.1.1.3.cmml"><mi id="A2.SS2.p2.16.m4.1.1.3.2" xref="A2.SS2.p2.16.m4.1.1.3.2.cmml">x</mi><mi id="A2.SS2.p2.16.m4.1.1.3.3" xref="A2.SS2.p2.16.m4.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.16.m4.1b"><apply id="A2.SS2.p2.16.m4.1.1.cmml" xref="A2.SS2.p2.16.m4.1.1"><eq id="A2.SS2.p2.16.m4.1.1.1.cmml" xref="A2.SS2.p2.16.m4.1.1.1"></eq><apply id="A2.SS2.p2.16.m4.1.1.2.cmml" xref="A2.SS2.p2.16.m4.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.16.m4.1.1.2.1.cmml" xref="A2.SS2.p2.16.m4.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.16.m4.1.1.2.2.cmml" xref="A2.SS2.p2.16.m4.1.1.2.2">𝑥</ci><cn id="A2.SS2.p2.16.m4.1.1.2.3.cmml" type="integer" xref="A2.SS2.p2.16.m4.1.1.2.3">1</cn></apply><apply id="A2.SS2.p2.16.m4.1.1.3.cmml" xref="A2.SS2.p2.16.m4.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p2.16.m4.1.1.3.1.cmml" xref="A2.SS2.p2.16.m4.1.1.3">subscript</csymbol><ci id="A2.SS2.p2.16.m4.1.1.3.2.cmml" xref="A2.SS2.p2.16.m4.1.1.3.2">𝑥</ci><ci id="A2.SS2.p2.16.m4.1.1.3.3.cmml" xref="A2.SS2.p2.16.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.16.m4.1c">x_{1}=x_{t}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.16.m4.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> due to the additional <math alttext="\beta\epsilon" class="ltx_Math" display="inline" id="A2.SS2.p2.17.m5.1"><semantics id="A2.SS2.p2.17.m5.1a"><mrow id="A2.SS2.p2.17.m5.1.1" xref="A2.SS2.p2.17.m5.1.1.cmml"><mi id="A2.SS2.p2.17.m5.1.1.2" xref="A2.SS2.p2.17.m5.1.1.2.cmml">β</mi><mo id="A2.SS2.p2.17.m5.1.1.1" xref="A2.SS2.p2.17.m5.1.1.1.cmml">⁢</mo><mi id="A2.SS2.p2.17.m5.1.1.3" xref="A2.SS2.p2.17.m5.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.17.m5.1b"><apply id="A2.SS2.p2.17.m5.1.1.cmml" xref="A2.SS2.p2.17.m5.1.1"><times id="A2.SS2.p2.17.m5.1.1.1.cmml" xref="A2.SS2.p2.17.m5.1.1.1"></times><ci id="A2.SS2.p2.17.m5.1.1.2.cmml" xref="A2.SS2.p2.17.m5.1.1.2">𝛽</ci><ci id="A2.SS2.p2.17.m5.1.1.3.cmml" xref="A2.SS2.p2.17.m5.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.17.m5.1c">\beta\epsilon</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.17.m5.1d">italic_β italic_ϵ</annotation></semantics></math> factor. We now show that, when the additional factor is present, the two predictors nevertheless differ by at most <math alttext="\epsilon" class="ltx_Math" display="inline" id="A2.SS2.p2.18.m6.1"><semantics id="A2.SS2.p2.18.m6.1a"><mi id="A2.SS2.p2.18.m6.1.1" xref="A2.SS2.p2.18.m6.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.18.m6.1b"><ci id="A2.SS2.p2.18.m6.1.1.cmml" xref="A2.SS2.p2.18.m6.1.1">italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.18.m6.1c">\epsilon</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.18.m6.1d">italic_ϵ</annotation></semantics></math> in KL distance. We show it for the case <math alttext="x_{1}=x_{t}=0" class="ltx_Math" display="inline" id="A2.SS2.p2.19.m7.1"><semantics id="A2.SS2.p2.19.m7.1a"><mrow id="A2.SS2.p2.19.m7.1.1" xref="A2.SS2.p2.19.m7.1.1.cmml"><msub id="A2.SS2.p2.19.m7.1.1.2" xref="A2.SS2.p2.19.m7.1.1.2.cmml"><mi id="A2.SS2.p2.19.m7.1.1.2.2" xref="A2.SS2.p2.19.m7.1.1.2.2.cmml">x</mi><mn id="A2.SS2.p2.19.m7.1.1.2.3" xref="A2.SS2.p2.19.m7.1.1.2.3.cmml">1</mn></msub><mo id="A2.SS2.p2.19.m7.1.1.3" xref="A2.SS2.p2.19.m7.1.1.3.cmml">=</mo><msub id="A2.SS2.p2.19.m7.1.1.4" xref="A2.SS2.p2.19.m7.1.1.4.cmml"><mi id="A2.SS2.p2.19.m7.1.1.4.2" xref="A2.SS2.p2.19.m7.1.1.4.2.cmml">x</mi><mi id="A2.SS2.p2.19.m7.1.1.4.3" xref="A2.SS2.p2.19.m7.1.1.4.3.cmml">t</mi></msub><mo id="A2.SS2.p2.19.m7.1.1.5" xref="A2.SS2.p2.19.m7.1.1.5.cmml">=</mo><mn id="A2.SS2.p2.19.m7.1.1.6" xref="A2.SS2.p2.19.m7.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.19.m7.1b"><apply id="A2.SS2.p2.19.m7.1.1.cmml" xref="A2.SS2.p2.19.m7.1.1"><and id="A2.SS2.p2.19.m7.1.1a.cmml" xref="A2.SS2.p2.19.m7.1.1"></and><apply id="A2.SS2.p2.19.m7.1.1b.cmml" xref="A2.SS2.p2.19.m7.1.1"><eq id="A2.SS2.p2.19.m7.1.1.3.cmml" xref="A2.SS2.p2.19.m7.1.1.3"></eq><apply id="A2.SS2.p2.19.m7.1.1.2.cmml" xref="A2.SS2.p2.19.m7.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p2.19.m7.1.1.2.1.cmml" xref="A2.SS2.p2.19.m7.1.1.2">subscript</csymbol><ci id="A2.SS2.p2.19.m7.1.1.2.2.cmml" xref="A2.SS2.p2.19.m7.1.1.2.2">𝑥</ci><cn id="A2.SS2.p2.19.m7.1.1.2.3.cmml" type="integer" xref="A2.SS2.p2.19.m7.1.1.2.3">1</cn></apply><apply id="A2.SS2.p2.19.m7.1.1.4.cmml" xref="A2.SS2.p2.19.m7.1.1.4"><csymbol cd="ambiguous" id="A2.SS2.p2.19.m7.1.1.4.1.cmml" xref="A2.SS2.p2.19.m7.1.1.4">subscript</csymbol><ci id="A2.SS2.p2.19.m7.1.1.4.2.cmml" xref="A2.SS2.p2.19.m7.1.1.4.2">𝑥</ci><ci id="A2.SS2.p2.19.m7.1.1.4.3.cmml" xref="A2.SS2.p2.19.m7.1.1.4.3">𝑡</ci></apply></apply><apply id="A2.SS2.p2.19.m7.1.1c.cmml" xref="A2.SS2.p2.19.m7.1.1"><eq id="A2.SS2.p2.19.m7.1.1.5.cmml" xref="A2.SS2.p2.19.m7.1.1.5"></eq><share href="https://arxiv.org/html/2502.10178v1#A2.SS2.p2.19.m7.1.1.4.cmml" id="A2.SS2.p2.19.m7.1.1d.cmml" xref="A2.SS2.p2.19.m7.1.1"></share><cn id="A2.SS2.p2.19.m7.1.1.6.cmml" type="integer" xref="A2.SS2.p2.19.m7.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.19.m7.1c">x_{1}=x_{t}=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.19.m7.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>, the other case follows in the same way. In fact, note that</p> <table class="ltx_equation ltx_eqn_table" id="A2.E25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{n_{01}+\beta+\beta\epsilon}{n_{00}+n_{01}+2\beta+\beta\epsilon}=\frac{n_% {01}+\beta}{n_{00}+n_{01}+2\beta}\cdot\frac{1+\frac{\beta\epsilon}{n_{01}+% \beta}}{1+\frac{\beta\epsilon}{n_{00}+n_{01}+2\beta}}." class="ltx_Math" display="block" id="A2.E25.m1.1"><semantics id="A2.E25.m1.1a"><mrow id="A2.E25.m1.1.1.1" xref="A2.E25.m1.1.1.1.1.cmml"><mrow id="A2.E25.m1.1.1.1.1" xref="A2.E25.m1.1.1.1.1.cmml"><mfrac id="A2.E25.m1.1.1.1.1.2" xref="A2.E25.m1.1.1.1.1.2.cmml"><mrow id="A2.E25.m1.1.1.1.1.2.2" xref="A2.E25.m1.1.1.1.1.2.2.cmml"><msub id="A2.E25.m1.1.1.1.1.2.2.2" xref="A2.E25.m1.1.1.1.1.2.2.2.cmml"><mi id="A2.E25.m1.1.1.1.1.2.2.2.2" xref="A2.E25.m1.1.1.1.1.2.2.2.2.cmml">n</mi><mn id="A2.E25.m1.1.1.1.1.2.2.2.3" xref="A2.E25.m1.1.1.1.1.2.2.2.3.cmml">01</mn></msub><mo id="A2.E25.m1.1.1.1.1.2.2.1" xref="A2.E25.m1.1.1.1.1.2.2.1.cmml">+</mo><mi id="A2.E25.m1.1.1.1.1.2.2.3" xref="A2.E25.m1.1.1.1.1.2.2.3.cmml">β</mi><mo id="A2.E25.m1.1.1.1.1.2.2.1a" xref="A2.E25.m1.1.1.1.1.2.2.1.cmml">+</mo><mrow id="A2.E25.m1.1.1.1.1.2.2.4" xref="A2.E25.m1.1.1.1.1.2.2.4.cmml"><mi id="A2.E25.m1.1.1.1.1.2.2.4.2" xref="A2.E25.m1.1.1.1.1.2.2.4.2.cmml">β</mi><mo id="A2.E25.m1.1.1.1.1.2.2.4.1" xref="A2.E25.m1.1.1.1.1.2.2.4.1.cmml">⁢</mo><mi id="A2.E25.m1.1.1.1.1.2.2.4.3" xref="A2.E25.m1.1.1.1.1.2.2.4.3.cmml">ϵ</mi></mrow></mrow><mrow id="A2.E25.m1.1.1.1.1.2.3" xref="A2.E25.m1.1.1.1.1.2.3.cmml"><msub id="A2.E25.m1.1.1.1.1.2.3.2" xref="A2.E25.m1.1.1.1.1.2.3.2.cmml"><mi id="A2.E25.m1.1.1.1.1.2.3.2.2" xref="A2.E25.m1.1.1.1.1.2.3.2.2.cmml">n</mi><mn id="A2.E25.m1.1.1.1.1.2.3.2.3" xref="A2.E25.m1.1.1.1.1.2.3.2.3.cmml">00</mn></msub><mo id="A2.E25.m1.1.1.1.1.2.3.1" xref="A2.E25.m1.1.1.1.1.2.3.1.cmml">+</mo><msub 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xref="A2.E25.m1.1.1.1.1.2.3.5.3.cmml">ϵ</mi></mrow></mrow></mfrac><mo id="A2.E25.m1.1.1.1.1.1" xref="A2.E25.m1.1.1.1.1.1.cmml">=</mo><mrow id="A2.E25.m1.1.1.1.1.3" xref="A2.E25.m1.1.1.1.1.3.cmml"><mfrac id="A2.E25.m1.1.1.1.1.3.2" xref="A2.E25.m1.1.1.1.1.3.2.cmml"><mrow id="A2.E25.m1.1.1.1.1.3.2.2" xref="A2.E25.m1.1.1.1.1.3.2.2.cmml"><msub id="A2.E25.m1.1.1.1.1.3.2.2.2" xref="A2.E25.m1.1.1.1.1.3.2.2.2.cmml"><mi id="A2.E25.m1.1.1.1.1.3.2.2.2.2" xref="A2.E25.m1.1.1.1.1.3.2.2.2.2.cmml">n</mi><mn id="A2.E25.m1.1.1.1.1.3.2.2.2.3" xref="A2.E25.m1.1.1.1.1.3.2.2.2.3.cmml">01</mn></msub><mo id="A2.E25.m1.1.1.1.1.3.2.2.1" xref="A2.E25.m1.1.1.1.1.3.2.2.1.cmml">+</mo><mi id="A2.E25.m1.1.1.1.1.3.2.2.3" xref="A2.E25.m1.1.1.1.1.3.2.2.3.cmml">β</mi></mrow><mrow id="A2.E25.m1.1.1.1.1.3.2.3" xref="A2.E25.m1.1.1.1.1.3.2.3.cmml"><msub id="A2.E25.m1.1.1.1.1.3.2.3.2" xref="A2.E25.m1.1.1.1.1.3.2.3.2.cmml"><mi id="A2.E25.m1.1.1.1.1.3.2.3.2.2" xref="A2.E25.m1.1.1.1.1.3.2.3.2.2.cmml">n</mi><mn 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xref="A2.E25.m1.1.1.1.1.3.3.3.3.3.4.2">2</cn><ci id="A2.E25.m1.1.1.1.1.3.3.3.3.3.4.3.cmml" xref="A2.E25.m1.1.1.1.1.3.3.3.3.3.4.3">𝛽</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E25.m1.1c">\frac{n_{01}+\beta+\beta\epsilon}{n_{00}+n_{01}+2\beta+\beta\epsilon}=\frac{n_% {01}+\beta}{n_{00}+n_{01}+2\beta}\cdot\frac{1+\frac{\beta\epsilon}{n_{01}+% \beta}}{1+\frac{\beta\epsilon}{n_{00}+n_{01}+2\beta}}.</annotation><annotation encoding="application/x-llamapun" id="A2.E25.m1.1d">divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β + italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + italic_β italic_ϵ end_ARG = divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG ⋅ divide start_ARG 1 + divide start_ARG italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG end_ARG start_ARG 1 + divide start_ARG italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.23">Now, since</p> <table class="ltx_equation ltx_eqn_table" id="A2.E26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="1\leq 1+\frac{\beta\epsilon}{n_{01}+\beta}\leq 1+\epsilon" class="ltx_Math" display="block" id="A2.E26.m1.1"><semantics id="A2.E26.m1.1a"><mrow id="A2.E26.m1.1.1" xref="A2.E26.m1.1.1.cmml"><mn id="A2.E26.m1.1.1.2" xref="A2.E26.m1.1.1.2.cmml">1</mn><mo id="A2.E26.m1.1.1.3" xref="A2.E26.m1.1.1.3.cmml">≤</mo><mrow id="A2.E26.m1.1.1.4" xref="A2.E26.m1.1.1.4.cmml"><mn id="A2.E26.m1.1.1.4.2" xref="A2.E26.m1.1.1.4.2.cmml">1</mn><mo id="A2.E26.m1.1.1.4.1" xref="A2.E26.m1.1.1.4.1.cmml">+</mo><mfrac id="A2.E26.m1.1.1.4.3" xref="A2.E26.m1.1.1.4.3.cmml"><mrow id="A2.E26.m1.1.1.4.3.2" xref="A2.E26.m1.1.1.4.3.2.cmml"><mi id="A2.E26.m1.1.1.4.3.2.2" xref="A2.E26.m1.1.1.4.3.2.2.cmml">β</mi><mo id="A2.E26.m1.1.1.4.3.2.1" xref="A2.E26.m1.1.1.4.3.2.1.cmml">⁢</mo><mi id="A2.E26.m1.1.1.4.3.2.3" xref="A2.E26.m1.1.1.4.3.2.3.cmml">ϵ</mi></mrow><mrow id="A2.E26.m1.1.1.4.3.3" xref="A2.E26.m1.1.1.4.3.3.cmml"><msub id="A2.E26.m1.1.1.4.3.3.2" xref="A2.E26.m1.1.1.4.3.3.2.cmml"><mi id="A2.E26.m1.1.1.4.3.3.2.2" xref="A2.E26.m1.1.1.4.3.3.2.2.cmml">n</mi><mn id="A2.E26.m1.1.1.4.3.3.2.3" xref="A2.E26.m1.1.1.4.3.3.2.3.cmml">01</mn></msub><mo id="A2.E26.m1.1.1.4.3.3.1" xref="A2.E26.m1.1.1.4.3.3.1.cmml">+</mo><mi id="A2.E26.m1.1.1.4.3.3.3" xref="A2.E26.m1.1.1.4.3.3.3.cmml">β</mi></mrow></mfrac></mrow><mo id="A2.E26.m1.1.1.5" xref="A2.E26.m1.1.1.5.cmml">≤</mo><mrow id="A2.E26.m1.1.1.6" xref="A2.E26.m1.1.1.6.cmml"><mn id="A2.E26.m1.1.1.6.2" xref="A2.E26.m1.1.1.6.2.cmml">1</mn><mo id="A2.E26.m1.1.1.6.1" xref="A2.E26.m1.1.1.6.1.cmml">+</mo><mi id="A2.E26.m1.1.1.6.3" xref="A2.E26.m1.1.1.6.3.cmml">ϵ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E26.m1.1b"><apply id="A2.E26.m1.1.1.cmml" xref="A2.E26.m1.1.1"><and id="A2.E26.m1.1.1a.cmml" xref="A2.E26.m1.1.1"></and><apply id="A2.E26.m1.1.1b.cmml" xref="A2.E26.m1.1.1"><leq id="A2.E26.m1.1.1.3.cmml" xref="A2.E26.m1.1.1.3"></leq><cn id="A2.E26.m1.1.1.2.cmml" type="integer" xref="A2.E26.m1.1.1.2">1</cn><apply id="A2.E26.m1.1.1.4.cmml" xref="A2.E26.m1.1.1.4"><plus id="A2.E26.m1.1.1.4.1.cmml" xref="A2.E26.m1.1.1.4.1"></plus><cn id="A2.E26.m1.1.1.4.2.cmml" type="integer" xref="A2.E26.m1.1.1.4.2">1</cn><apply id="A2.E26.m1.1.1.4.3.cmml" xref="A2.E26.m1.1.1.4.3"><divide id="A2.E26.m1.1.1.4.3.1.cmml" xref="A2.E26.m1.1.1.4.3"></divide><apply id="A2.E26.m1.1.1.4.3.2.cmml" xref="A2.E26.m1.1.1.4.3.2"><times id="A2.E26.m1.1.1.4.3.2.1.cmml" xref="A2.E26.m1.1.1.4.3.2.1"></times><ci id="A2.E26.m1.1.1.4.3.2.2.cmml" xref="A2.E26.m1.1.1.4.3.2.2">𝛽</ci><ci id="A2.E26.m1.1.1.4.3.2.3.cmml" xref="A2.E26.m1.1.1.4.3.2.3">italic-ϵ</ci></apply><apply id="A2.E26.m1.1.1.4.3.3.cmml" xref="A2.E26.m1.1.1.4.3.3"><plus id="A2.E26.m1.1.1.4.3.3.1.cmml" xref="A2.E26.m1.1.1.4.3.3.1"></plus><apply id="A2.E26.m1.1.1.4.3.3.2.cmml" xref="A2.E26.m1.1.1.4.3.3.2"><csymbol cd="ambiguous" id="A2.E26.m1.1.1.4.3.3.2.1.cmml" xref="A2.E26.m1.1.1.4.3.3.2">subscript</csymbol><ci id="A2.E26.m1.1.1.4.3.3.2.2.cmml" xref="A2.E26.m1.1.1.4.3.3.2.2">𝑛</ci><cn id="A2.E26.m1.1.1.4.3.3.2.3.cmml" type="integer" xref="A2.E26.m1.1.1.4.3.3.2.3">01</cn></apply><ci id="A2.E26.m1.1.1.4.3.3.3.cmml" xref="A2.E26.m1.1.1.4.3.3.3">𝛽</ci></apply></apply></apply></apply><apply id="A2.E26.m1.1.1c.cmml" xref="A2.E26.m1.1.1"><leq id="A2.E26.m1.1.1.5.cmml" xref="A2.E26.m1.1.1.5"></leq><share href="https://arxiv.org/html/2502.10178v1#A2.E26.m1.1.1.4.cmml" id="A2.E26.m1.1.1d.cmml" xref="A2.E26.m1.1.1"></share><apply id="A2.E26.m1.1.1.6.cmml" xref="A2.E26.m1.1.1.6"><plus id="A2.E26.m1.1.1.6.1.cmml" xref="A2.E26.m1.1.1.6.1"></plus><cn id="A2.E26.m1.1.1.6.2.cmml" type="integer" xref="A2.E26.m1.1.1.6.2">1</cn><ci id="A2.E26.m1.1.1.6.3.cmml" xref="A2.E26.m1.1.1.6.3">italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E26.m1.1c">1\leq 1+\frac{\beta\epsilon}{n_{01}+\beta}\leq 1+\epsilon</annotation><annotation encoding="application/x-llamapun" id="A2.E26.m1.1d">1 ≤ 1 + divide start_ARG italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG ≤ 1 + italic_ϵ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.24">and</p> <table class="ltx_equation ltx_eqn_table" id="A2.E27"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="1\leq 1+\frac{\beta\epsilon}{n_{00}+n_{01}+2\beta}\leq 1+\epsilon" class="ltx_Math" display="block" id="A2.E27.m1.1"><semantics id="A2.E27.m1.1a"><mrow id="A2.E27.m1.1.1" xref="A2.E27.m1.1.1.cmml"><mn id="A2.E27.m1.1.1.2" xref="A2.E27.m1.1.1.2.cmml">1</mn><mo id="A2.E27.m1.1.1.3" xref="A2.E27.m1.1.1.3.cmml">≤</mo><mrow id="A2.E27.m1.1.1.4" xref="A2.E27.m1.1.1.4.cmml"><mn id="A2.E27.m1.1.1.4.2" xref="A2.E27.m1.1.1.4.2.cmml">1</mn><mo id="A2.E27.m1.1.1.4.1" xref="A2.E27.m1.1.1.4.1.cmml">+</mo><mfrac id="A2.E27.m1.1.1.4.3" xref="A2.E27.m1.1.1.4.3.cmml"><mrow id="A2.E27.m1.1.1.4.3.2" xref="A2.E27.m1.1.1.4.3.2.cmml"><mi id="A2.E27.m1.1.1.4.3.2.2" xref="A2.E27.m1.1.1.4.3.2.2.cmml">β</mi><mo id="A2.E27.m1.1.1.4.3.2.1" xref="A2.E27.m1.1.1.4.3.2.1.cmml">⁢</mo><mi id="A2.E27.m1.1.1.4.3.2.3" xref="A2.E27.m1.1.1.4.3.2.3.cmml">ϵ</mi></mrow><mrow id="A2.E27.m1.1.1.4.3.3" xref="A2.E27.m1.1.1.4.3.3.cmml"><msub id="A2.E27.m1.1.1.4.3.3.2" xref="A2.E27.m1.1.1.4.3.3.2.cmml"><mi id="A2.E27.m1.1.1.4.3.3.2.2" xref="A2.E27.m1.1.1.4.3.3.2.2.cmml">n</mi><mn id="A2.E27.m1.1.1.4.3.3.2.3" xref="A2.E27.m1.1.1.4.3.3.2.3.cmml">00</mn></msub><mo id="A2.E27.m1.1.1.4.3.3.1" xref="A2.E27.m1.1.1.4.3.3.1.cmml">+</mo><msub id="A2.E27.m1.1.1.4.3.3.3" xref="A2.E27.m1.1.1.4.3.3.3.cmml"><mi id="A2.E27.m1.1.1.4.3.3.3.2" xref="A2.E27.m1.1.1.4.3.3.3.2.cmml">n</mi><mn id="A2.E27.m1.1.1.4.3.3.3.3" xref="A2.E27.m1.1.1.4.3.3.3.3.cmml">01</mn></msub><mo id="A2.E27.m1.1.1.4.3.3.1a" xref="A2.E27.m1.1.1.4.3.3.1.cmml">+</mo><mrow id="A2.E27.m1.1.1.4.3.3.4" xref="A2.E27.m1.1.1.4.3.3.4.cmml"><mn id="A2.E27.m1.1.1.4.3.3.4.2" xref="A2.E27.m1.1.1.4.3.3.4.2.cmml">2</mn><mo id="A2.E27.m1.1.1.4.3.3.4.1" xref="A2.E27.m1.1.1.4.3.3.4.1.cmml">⁢</mo><mi id="A2.E27.m1.1.1.4.3.3.4.3" xref="A2.E27.m1.1.1.4.3.3.4.3.cmml">β</mi></mrow></mrow></mfrac></mrow><mo id="A2.E27.m1.1.1.5" xref="A2.E27.m1.1.1.5.cmml">≤</mo><mrow id="A2.E27.m1.1.1.6" xref="A2.E27.m1.1.1.6.cmml"><mn id="A2.E27.m1.1.1.6.2" xref="A2.E27.m1.1.1.6.2.cmml">1</mn><mo id="A2.E27.m1.1.1.6.1" xref="A2.E27.m1.1.1.6.1.cmml">+</mo><mi id="A2.E27.m1.1.1.6.3" xref="A2.E27.m1.1.1.6.3.cmml">ϵ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E27.m1.1b"><apply id="A2.E27.m1.1.1.cmml" xref="A2.E27.m1.1.1"><and id="A2.E27.m1.1.1a.cmml" xref="A2.E27.m1.1.1"></and><apply id="A2.E27.m1.1.1b.cmml" xref="A2.E27.m1.1.1"><leq id="A2.E27.m1.1.1.3.cmml" xref="A2.E27.m1.1.1.3"></leq><cn id="A2.E27.m1.1.1.2.cmml" type="integer" xref="A2.E27.m1.1.1.2">1</cn><apply id="A2.E27.m1.1.1.4.cmml" xref="A2.E27.m1.1.1.4"><plus id="A2.E27.m1.1.1.4.1.cmml" xref="A2.E27.m1.1.1.4.1"></plus><cn id="A2.E27.m1.1.1.4.2.cmml" type="integer" xref="A2.E27.m1.1.1.4.2">1</cn><apply id="A2.E27.m1.1.1.4.3.cmml" xref="A2.E27.m1.1.1.4.3"><divide id="A2.E27.m1.1.1.4.3.1.cmml" xref="A2.E27.m1.1.1.4.3"></divide><apply id="A2.E27.m1.1.1.4.3.2.cmml" xref="A2.E27.m1.1.1.4.3.2"><times id="A2.E27.m1.1.1.4.3.2.1.cmml" xref="A2.E27.m1.1.1.4.3.2.1"></times><ci id="A2.E27.m1.1.1.4.3.2.2.cmml" xref="A2.E27.m1.1.1.4.3.2.2">𝛽</ci><ci id="A2.E27.m1.1.1.4.3.2.3.cmml" xref="A2.E27.m1.1.1.4.3.2.3">italic-ϵ</ci></apply><apply id="A2.E27.m1.1.1.4.3.3.cmml" xref="A2.E27.m1.1.1.4.3.3"><plus id="A2.E27.m1.1.1.4.3.3.1.cmml" xref="A2.E27.m1.1.1.4.3.3.1"></plus><apply id="A2.E27.m1.1.1.4.3.3.2.cmml" xref="A2.E27.m1.1.1.4.3.3.2"><csymbol cd="ambiguous" id="A2.E27.m1.1.1.4.3.3.2.1.cmml" xref="A2.E27.m1.1.1.4.3.3.2">subscript</csymbol><ci id="A2.E27.m1.1.1.4.3.3.2.2.cmml" xref="A2.E27.m1.1.1.4.3.3.2.2">𝑛</ci><cn id="A2.E27.m1.1.1.4.3.3.2.3.cmml" type="integer" xref="A2.E27.m1.1.1.4.3.3.2.3">00</cn></apply><apply id="A2.E27.m1.1.1.4.3.3.3.cmml" xref="A2.E27.m1.1.1.4.3.3.3"><csymbol cd="ambiguous" id="A2.E27.m1.1.1.4.3.3.3.1.cmml" xref="A2.E27.m1.1.1.4.3.3.3">subscript</csymbol><ci id="A2.E27.m1.1.1.4.3.3.3.2.cmml" xref="A2.E27.m1.1.1.4.3.3.3.2">𝑛</ci><cn id="A2.E27.m1.1.1.4.3.3.3.3.cmml" type="integer" xref="A2.E27.m1.1.1.4.3.3.3.3">01</cn></apply><apply id="A2.E27.m1.1.1.4.3.3.4.cmml" xref="A2.E27.m1.1.1.4.3.3.4"><times id="A2.E27.m1.1.1.4.3.3.4.1.cmml" xref="A2.E27.m1.1.1.4.3.3.4.1"></times><cn id="A2.E27.m1.1.1.4.3.3.4.2.cmml" type="integer" xref="A2.E27.m1.1.1.4.3.3.4.2">2</cn><ci id="A2.E27.m1.1.1.4.3.3.4.3.cmml" xref="A2.E27.m1.1.1.4.3.3.4.3">𝛽</ci></apply></apply></apply></apply></apply><apply id="A2.E27.m1.1.1c.cmml" xref="A2.E27.m1.1.1"><leq id="A2.E27.m1.1.1.5.cmml" xref="A2.E27.m1.1.1.5"></leq><share href="https://arxiv.org/html/2502.10178v1#A2.E27.m1.1.1.4.cmml" id="A2.E27.m1.1.1d.cmml" xref="A2.E27.m1.1.1"></share><apply id="A2.E27.m1.1.1.6.cmml" xref="A2.E27.m1.1.1.6"><plus id="A2.E27.m1.1.1.6.1.cmml" xref="A2.E27.m1.1.1.6.1"></plus><cn id="A2.E27.m1.1.1.6.2.cmml" type="integer" xref="A2.E27.m1.1.1.6.2">1</cn><ci id="A2.E27.m1.1.1.6.3.cmml" xref="A2.E27.m1.1.1.6.3">italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E27.m1.1c">1\leq 1+\frac{\beta\epsilon}{n_{00}+n_{01}+2\beta}\leq 1+\epsilon</annotation><annotation encoding="application/x-llamapun" id="A2.E27.m1.1d">1 ≤ 1 + divide start_ARG italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG ≤ 1 + italic_ϵ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.25">we have that</p> <table class="ltx_equation ltx_eqn_table" id="A2.E28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}\leq\frac{n_{01}+\beta+\beta\epsilon}% {n_{00}+n_{01}+2\beta+\beta\epsilon}\cdot\left(1+\epsilon\right)" class="ltx_Math" display="block" id="A2.E28.m1.1"><semantics id="A2.E28.m1.1a"><mrow id="A2.E28.m1.1.1" xref="A2.E28.m1.1.1.cmml"><mfrac id="A2.E28.m1.1.1.3" xref="A2.E28.m1.1.1.3.cmml"><mrow id="A2.E28.m1.1.1.3.2" xref="A2.E28.m1.1.1.3.2.cmml"><msub id="A2.E28.m1.1.1.3.2.2" xref="A2.E28.m1.1.1.3.2.2.cmml"><mi id="A2.E28.m1.1.1.3.2.2.2" xref="A2.E28.m1.1.1.3.2.2.2.cmml">n</mi><mn id="A2.E28.m1.1.1.3.2.2.3" xref="A2.E28.m1.1.1.3.2.2.3.cmml">01</mn></msub><mo id="A2.E28.m1.1.1.3.2.1" xref="A2.E28.m1.1.1.3.2.1.cmml">+</mo><mi id="A2.E28.m1.1.1.3.2.3" xref="A2.E28.m1.1.1.3.2.3.cmml">β</mi></mrow><mrow id="A2.E28.m1.1.1.3.3" xref="A2.E28.m1.1.1.3.3.cmml"><msub id="A2.E28.m1.1.1.3.3.2" xref="A2.E28.m1.1.1.3.3.2.cmml"><mi id="A2.E28.m1.1.1.3.3.2.2" xref="A2.E28.m1.1.1.3.3.2.2.cmml">n</mi><mn id="A2.E28.m1.1.1.3.3.2.3" xref="A2.E28.m1.1.1.3.3.2.3.cmml">00</mn></msub><mo id="A2.E28.m1.1.1.3.3.1" xref="A2.E28.m1.1.1.3.3.1.cmml">+</mo><msub id="A2.E28.m1.1.1.3.3.3" xref="A2.E28.m1.1.1.3.3.3.cmml"><mi id="A2.E28.m1.1.1.3.3.3.2" xref="A2.E28.m1.1.1.3.3.3.2.cmml">n</mi><mn id="A2.E28.m1.1.1.3.3.3.3" xref="A2.E28.m1.1.1.3.3.3.3.cmml">01</mn></msub><mo id="A2.E28.m1.1.1.3.3.1a" xref="A2.E28.m1.1.1.3.3.1.cmml">+</mo><mrow id="A2.E28.m1.1.1.3.3.4" xref="A2.E28.m1.1.1.3.3.4.cmml"><mn id="A2.E28.m1.1.1.3.3.4.2" xref="A2.E28.m1.1.1.3.3.4.2.cmml">2</mn><mo id="A2.E28.m1.1.1.3.3.4.1" xref="A2.E28.m1.1.1.3.3.4.1.cmml">⁢</mo><mi id="A2.E28.m1.1.1.3.3.4.3" xref="A2.E28.m1.1.1.3.3.4.3.cmml">β</mi></mrow></mrow></mfrac><mo id="A2.E28.m1.1.1.2" xref="A2.E28.m1.1.1.2.cmml">≤</mo><mrow id="A2.E28.m1.1.1.1" xref="A2.E28.m1.1.1.1.cmml"><mfrac id="A2.E28.m1.1.1.1.3" xref="A2.E28.m1.1.1.1.3.cmml"><mrow id="A2.E28.m1.1.1.1.3.2" xref="A2.E28.m1.1.1.1.3.2.cmml"><msub id="A2.E28.m1.1.1.1.3.2.2" xref="A2.E28.m1.1.1.1.3.2.2.cmml"><mi id="A2.E28.m1.1.1.1.3.2.2.2" xref="A2.E28.m1.1.1.1.3.2.2.2.cmml">n</mi><mn id="A2.E28.m1.1.1.1.3.2.2.3" xref="A2.E28.m1.1.1.1.3.2.2.3.cmml">01</mn></msub><mo id="A2.E28.m1.1.1.1.3.2.1" xref="A2.E28.m1.1.1.1.3.2.1.cmml">+</mo><mi id="A2.E28.m1.1.1.1.3.2.3" xref="A2.E28.m1.1.1.1.3.2.3.cmml">β</mi><mo id="A2.E28.m1.1.1.1.3.2.1a" xref="A2.E28.m1.1.1.1.3.2.1.cmml">+</mo><mrow id="A2.E28.m1.1.1.1.3.2.4" xref="A2.E28.m1.1.1.1.3.2.4.cmml"><mi id="A2.E28.m1.1.1.1.3.2.4.2" xref="A2.E28.m1.1.1.1.3.2.4.2.cmml">β</mi><mo id="A2.E28.m1.1.1.1.3.2.4.1" xref="A2.E28.m1.1.1.1.3.2.4.1.cmml">⁢</mo><mi id="A2.E28.m1.1.1.1.3.2.4.3" xref="A2.E28.m1.1.1.1.3.2.4.3.cmml">ϵ</mi></mrow></mrow><mrow id="A2.E28.m1.1.1.1.3.3" xref="A2.E28.m1.1.1.1.3.3.cmml"><msub id="A2.E28.m1.1.1.1.3.3.2" xref="A2.E28.m1.1.1.1.3.3.2.cmml"><mi id="A2.E28.m1.1.1.1.3.3.2.2" xref="A2.E28.m1.1.1.1.3.3.2.2.cmml">n</mi><mn id="A2.E28.m1.1.1.1.3.3.2.3" xref="A2.E28.m1.1.1.1.3.3.2.3.cmml">00</mn></msub><mo id="A2.E28.m1.1.1.1.3.3.1" xref="A2.E28.m1.1.1.1.3.3.1.cmml">+</mo><msub id="A2.E28.m1.1.1.1.3.3.3" xref="A2.E28.m1.1.1.1.3.3.3.cmml"><mi id="A2.E28.m1.1.1.1.3.3.3.2" xref="A2.E28.m1.1.1.1.3.3.3.2.cmml">n</mi><mn id="A2.E28.m1.1.1.1.3.3.3.3" xref="A2.E28.m1.1.1.1.3.3.3.3.cmml">01</mn></msub><mo id="A2.E28.m1.1.1.1.3.3.1a" xref="A2.E28.m1.1.1.1.3.3.1.cmml">+</mo><mrow id="A2.E28.m1.1.1.1.3.3.4" xref="A2.E28.m1.1.1.1.3.3.4.cmml"><mn id="A2.E28.m1.1.1.1.3.3.4.2" 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xref="A2.E28.m1.1.1.1.1.1.1.1"></plus><cn id="A2.E28.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="A2.E28.m1.1.1.1.1.1.1.2">1</cn><ci id="A2.E28.m1.1.1.1.1.1.1.3.cmml" xref="A2.E28.m1.1.1.1.1.1.1.3">italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E28.m1.1c">\frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}\leq\frac{n_{01}+\beta+\beta\epsilon}% {n_{00}+n_{01}+2\beta+\beta\epsilon}\cdot\left(1+\epsilon\right)</annotation><annotation encoding="application/x-llamapun" id="A2.E28.m1.1d">divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG ≤ divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β + italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + italic_β italic_ϵ end_ARG ⋅ ( 1 + italic_ϵ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.26">but we also have</p> <table class="ltx_equation ltx_eqn_table" id="A2.E29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{n_{00}+n_{01}+2\beta+\beta\epsilon}{n_{00}+n_{01}+2\beta}\leq 1+\frac{% \beta\epsilon}{n_{00}+n_{01}+2\beta}\leq 1+\epsilon," class="ltx_Math" display="block" id="A2.E29.m1.1"><semantics id="A2.E29.m1.1a"><mrow id="A2.E29.m1.1.1.1" xref="A2.E29.m1.1.1.1.1.cmml"><mrow id="A2.E29.m1.1.1.1.1" xref="A2.E29.m1.1.1.1.1.cmml"><mfrac id="A2.E29.m1.1.1.1.1.2" xref="A2.E29.m1.1.1.1.1.2.cmml"><mrow 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end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG ≤ 1 + divide start_ARG italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG ≤ 1 + italic_ϵ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.27">so that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx15"> <tbody id="A2.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D_{\mathrm{KL}}\left(\mathbb{P}_{\beta}^{(1)}\left(\cdot\mid x_{% 1}^{t}\right)\|\mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)\right)" class="ltx_math_unparsed" display="inline" id="A2.Ex42.m1.1"><semantics id="A2.Ex42.m1.1a"><mrow id="A2.Ex42.m1.1b"><msub id="A2.Ex42.m1.1.2"><mi id="A2.Ex42.m1.1.2.2">D</mi><mi id="A2.Ex42.m1.1.2.3">KL</mi></msub><mrow id="A2.Ex42.m1.1.3"><mo id="A2.Ex42.m1.1.3.1">(</mo><msubsup id="A2.Ex42.m1.1.3.2"><mi id="A2.Ex42.m1.1.3.2.2.2">ℙ</mi><mi id="A2.Ex42.m1.1.3.2.2.3">β</mi><mrow id="A2.Ex42.m1.1.1.1.3"><mo id="A2.Ex42.m1.1.1.1.3.1" stretchy="false">(</mo><mn id="A2.Ex42.m1.1.1.1.1">1</mn><mo id="A2.Ex42.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msubsup><mrow id="A2.Ex42.m1.1.3.3"><mo id="A2.Ex42.m1.1.3.3.1">(</mo><mo id="A2.Ex42.m1.1.3.3.2" lspace="0em" rspace="0em">⋅</mo><mo id="A2.Ex42.m1.1.3.3.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="A2.Ex42.m1.1.3.3.4"><mi id="A2.Ex42.m1.1.3.3.4.2.2">x</mi><mn id="A2.Ex42.m1.1.3.3.4.2.3">1</mn><mi id="A2.Ex42.m1.1.3.3.4.3">t</mi></msubsup><mo id="A2.Ex42.m1.1.3.3.5">)</mo></mrow><mo id="A2.Ex42.m1.1.3.4" lspace="0em" rspace="0.167em">∥</mo><msub id="A2.Ex42.m1.1.3.5"><mi id="A2.Ex42.m1.1.3.5.2">ℙ</mi><mi id="A2.Ex42.m1.1.3.5.3">𝜽</mi></msub><mrow id="A2.Ex42.m1.1.3.6"><mo id="A2.Ex42.m1.1.3.6.1">(</mo><mo id="A2.Ex42.m1.1.3.6.2" lspace="0em" rspace="0em">⋅</mo><mo id="A2.Ex42.m1.1.3.6.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="A2.Ex42.m1.1.3.6.4"><mi id="A2.Ex42.m1.1.3.6.4.2.2">x</mi><mn id="A2.Ex42.m1.1.3.6.4.2.3">1</mn><mi id="A2.Ex42.m1.1.3.6.4.3">t</mi></msubsup><mo id="A2.Ex42.m1.1.3.6.5">)</mo></mrow><mo id="A2.Ex42.m1.1.3.7">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A2.Ex42.m1.1c">\displaystyle D_{\mathrm{KL}}\left(\mathbb{P}_{\beta}^{(1)}\left(\cdot\mid x_{% 1}^{t}\right)\|\mathbb{P}_{\bm{\theta}}\left(\cdot\mid x_{1}^{t}\right)\right)</annotation><annotation encoding="application/x-llamapun" id="A2.Ex42.m1.1d">italic_D start_POSTSUBSCRIPT roman_KL end_POSTSUBSCRIPT ( blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) ∥ blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( ⋅ ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbb{P}_{\beta}^{(1)}\left(x_{t+1}=0\mid x_{1}^{t}\right)\log% \frac{\mathbb{P}_{\beta}^{(1)}\left(x_{t+1}=0\mid x_{1}^{t}\right)}{\mathbb{P}% _{\bm{\theta}}\left(x_{t+1}=0\mid x_{1}^{t}\right)}" class="ltx_Math" display="inline" id="A2.Ex42.m2.5"><semantics id="A2.Ex42.m2.5a"><mrow id="A2.Ex42.m2.5.5" xref="A2.Ex42.m2.5.5.cmml"><mi id="A2.Ex42.m2.5.5.3" 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end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 0 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) roman_log divide start_ARG blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 0 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG start_ARG blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 0 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.E30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) roman_log divide start_ARG blackboard_P start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG start_ARG blackboard_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = 1 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td> </tr></tbody> <tbody id="A2.Ex43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{n_{00}+\beta}{n_{00}+n_{01}+2\beta}\log\frac{\frac{n_{00}+% \beta}{n_{00}+n_{01}+2\beta}}{\frac{n_{00}+\beta}{n_{00}+n_{01}+2\beta+\beta% \epsilon}}" class="ltx_Math" display="inline" id="A2.Ex43.m1.1"><semantics id="A2.Ex43.m1.1a"><mrow id="A2.Ex43.m1.1.1" xref="A2.Ex43.m1.1.1.cmml"><mi id="A2.Ex43.m1.1.1.2" xref="A2.Ex43.m1.1.1.2.cmml"></mi><mo id="A2.Ex43.m1.1.1.1" xref="A2.Ex43.m1.1.1.1.cmml">=</mo><mrow id="A2.Ex43.m1.1.1.3" xref="A2.Ex43.m1.1.1.3.cmml"><mstyle displaystyle="true" id="A2.Ex43.m1.1.1.3.2" xref="A2.Ex43.m1.1.1.3.2.cmml"><mfrac id="A2.Ex43.m1.1.1.3.2a" xref="A2.Ex43.m1.1.1.3.2.cmml"><mrow id="A2.Ex43.m1.1.1.3.2.2" xref="A2.Ex43.m1.1.1.3.2.2.cmml"><msub 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xref="A2.Ex43.m1.1.1.3.3.2.3.3.5.3">italic-ϵ</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex43.m1.1c">\displaystyle=\frac{n_{00}+\beta}{n_{00}+n_{01}+2\beta}\log\frac{\frac{n_{00}+% \beta}{n_{00}+n_{01}+2\beta}}{\frac{n_{00}+\beta}{n_{00}+n_{01}+2\beta+\beta% \epsilon}}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex43.m1.1d">= divide start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG roman_log divide start_ARG divide start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG end_ARG start_ARG divide start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + italic_β italic_ϵ end_ARG end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.E31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hskip 110.00017pt+\frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}\log% \frac{\frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}}{\frac{n_{01}+\beta+\beta% \epsilon}{n_{00}+n_{01}+2\beta+\beta\epsilon}}" class="ltx_Math" display="inline" id="A2.E31.m1.1"><semantics id="A2.E31.m1.1a"><mrow id="A2.E31.m1.1.1" xref="A2.E31.m1.1.1.cmml"><mo id="A2.E31.m1.1.1a" xref="A2.E31.m1.1.1.cmml">+</mo><mrow id="A2.E31.m1.1.1.2" xref="A2.E31.m1.1.1.2.cmml"><mstyle displaystyle="true" id="A2.E31.m1.1.1.2.2" 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id="A2.E31.m1.1.1.2.3.2.3.3.4.3.cmml" xref="A2.E31.m1.1.1.2.3.2.3.3.4.3">𝛽</ci></apply><apply id="A2.E31.m1.1.1.2.3.2.3.3.5.cmml" xref="A2.E31.m1.1.1.2.3.2.3.3.5"><times id="A2.E31.m1.1.1.2.3.2.3.3.5.1.cmml" xref="A2.E31.m1.1.1.2.3.2.3.3.5.1"></times><ci id="A2.E31.m1.1.1.2.3.2.3.3.5.2.cmml" xref="A2.E31.m1.1.1.2.3.2.3.3.5.2">𝛽</ci><ci id="A2.E31.m1.1.1.2.3.2.3.3.5.3.cmml" xref="A2.E31.m1.1.1.2.3.2.3.3.5.3">italic-ϵ</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E31.m1.1c">\displaystyle\hskip 110.00017pt+\frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}\log% \frac{\frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}}{\frac{n_{01}+\beta+\beta% \epsilon}{n_{00}+n_{01}+2\beta+\beta\epsilon}}</annotation><annotation encoding="application/x-llamapun" id="A2.E31.m1.1d">+ divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG roman_log divide start_ARG divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG end_ARG start_ARG divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β + italic_β italic_ϵ end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β + italic_β italic_ϵ end_ARG end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(31)</span></td> </tr></tbody> <tbody id="A2.E32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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xref="A2.E32.m1.4.4.2.2.3.3.4.3">𝛽</ci></apply></apply></apply><apply id="A2.E32.m1.4.4.2.2.1.2.cmml" xref="A2.E32.m1.4.4.2.2.1.1"><log id="A2.E32.m1.2.2.cmml" xref="A2.E32.m1.2.2"></log><apply id="A2.E32.m1.4.4.2.2.1.1.1.1.cmml" xref="A2.E32.m1.4.4.2.2.1.1.1.1"><plus id="A2.E32.m1.4.4.2.2.1.1.1.1.1.cmml" xref="A2.E32.m1.4.4.2.2.1.1.1.1.1"></plus><cn id="A2.E32.m1.4.4.2.2.1.1.1.1.2.cmml" type="integer" xref="A2.E32.m1.4.4.2.2.1.1.1.1.2">1</cn><ci id="A2.E32.m1.4.4.2.2.1.1.1.1.3.cmml" xref="A2.E32.m1.4.4.2.2.1.1.1.1.3">italic-ϵ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E32.m1.4c">\displaystyle\leq\frac{n_{00}+\beta}{n_{00}+n_{01}+2\beta}\log(1+\epsilon)+% \frac{n_{01}+\beta}{n_{00}+n_{01}+2\beta}\log(1+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="A2.E32.m1.4d">≤ divide start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG roman_log ( 1 + italic_ϵ ) + divide start_ARG italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + italic_β end_ARG start_ARG italic_n start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT + 2 italic_β end_ARG roman_log ( 1 + italic_ϵ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(32)</span></td> </tr></tbody> <tbody id="A2.E33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\log(1+\epsilon)" class="ltx_Math" display="inline" id="A2.E33.m1.2"><semantics id="A2.E33.m1.2a"><mrow id="A2.E33.m1.2.2" xref="A2.E33.m1.2.2.cmml"><mi id="A2.E33.m1.2.2.3" xref="A2.E33.m1.2.2.3.cmml"></mi><mo id="A2.E33.m1.2.2.2" xref="A2.E33.m1.2.2.2.cmml">≤</mo><mrow id="A2.E33.m1.2.2.1.1" xref="A2.E33.m1.2.2.1.2.cmml"><mi id="A2.E33.m1.1.1" xref="A2.E33.m1.1.1.cmml">log</mi><mo id="A2.E33.m1.2.2.1.1a" xref="A2.E33.m1.2.2.1.2.cmml">⁡</mo><mrow id="A2.E33.m1.2.2.1.1.1" xref="A2.E33.m1.2.2.1.2.cmml"><mo id="A2.E33.m1.2.2.1.1.1.2" stretchy="false" xref="A2.E33.m1.2.2.1.2.cmml">(</mo><mrow id="A2.E33.m1.2.2.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.cmml"><mn id="A2.E33.m1.2.2.1.1.1.1.2" xref="A2.E33.m1.2.2.1.1.1.1.2.cmml">1</mn><mo id="A2.E33.m1.2.2.1.1.1.1.1" xref="A2.E33.m1.2.2.1.1.1.1.1.cmml">+</mo><mi id="A2.E33.m1.2.2.1.1.1.1.3" xref="A2.E33.m1.2.2.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="A2.E33.m1.2.2.1.1.1.3" stretchy="false" xref="A2.E33.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E33.m1.2b"><apply id="A2.E33.m1.2.2.cmml" xref="A2.E33.m1.2.2"><leq id="A2.E33.m1.2.2.2.cmml" xref="A2.E33.m1.2.2.2"></leq><csymbol cd="latexml" id="A2.E33.m1.2.2.3.cmml" xref="A2.E33.m1.2.2.3">absent</csymbol><apply id="A2.E33.m1.2.2.1.2.cmml" xref="A2.E33.m1.2.2.1.1"><log id="A2.E33.m1.1.1.cmml" xref="A2.E33.m1.1.1"></log><apply id="A2.E33.m1.2.2.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1"><plus id="A2.E33.m1.2.2.1.1.1.1.1.cmml" xref="A2.E33.m1.2.2.1.1.1.1.1"></plus><cn id="A2.E33.m1.2.2.1.1.1.1.2.cmml" type="integer" xref="A2.E33.m1.2.2.1.1.1.1.2">1</cn><ci id="A2.E33.m1.2.2.1.1.1.1.3.cmml" xref="A2.E33.m1.2.2.1.1.1.1.3">italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E33.m1.2c">\displaystyle\leq\log(1+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="A2.E33.m1.2d">≤ roman_log ( 1 + italic_ϵ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(33)</span></td> </tr></tbody> <tbody id="A2.E34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\epsilon" class="ltx_Math" display="inline" id="A2.E34.m1.1"><semantics id="A2.E34.m1.1a"><mrow id="A2.E34.m1.1.1" xref="A2.E34.m1.1.1.cmml"><mi id="A2.E34.m1.1.1.2" xref="A2.E34.m1.1.1.2.cmml"></mi><mo id="A2.E34.m1.1.1.1" xref="A2.E34.m1.1.1.1.cmml">≤</mo><mi id="A2.E34.m1.1.1.3" xref="A2.E34.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.E34.m1.1b"><apply id="A2.E34.m1.1.1.cmml" xref="A2.E34.m1.1.1"><leq id="A2.E34.m1.1.1.1.cmml" xref="A2.E34.m1.1.1.1"></leq><csymbol cd="latexml" id="A2.E34.m1.1.1.2.cmml" xref="A2.E34.m1.1.1.2">absent</csymbol><ci id="A2.E34.m1.1.1.3.cmml" xref="A2.E34.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E34.m1.1c">\displaystyle\leq\epsilon</annotation><annotation encoding="application/x-llamapun" id="A2.E34.m1.1d">≤ italic_ϵ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(34)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.SS2.p2.28">concluding the proof. ∎</p> </div> </section> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>Proof of Thm. <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#Thmtheorem2" title="Theorem 2. ‣ 4.3 Higher order processes (𝑘>1) ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">2</span></a> </h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.7">Consider a recurrent model of the form <math alttext="H_{t}=h(H_{t-1},x_{t})" class="ltx_Math" display="inline" id="A3.p1.1.m1.2"><semantics id="A3.p1.1.m1.2a"><mrow id="A3.p1.1.m1.2.2" xref="A3.p1.1.m1.2.2.cmml"><msub id="A3.p1.1.m1.2.2.4" xref="A3.p1.1.m1.2.2.4.cmml"><mi id="A3.p1.1.m1.2.2.4.2" xref="A3.p1.1.m1.2.2.4.2.cmml">H</mi><mi id="A3.p1.1.m1.2.2.4.3" xref="A3.p1.1.m1.2.2.4.3.cmml">t</mi></msub><mo id="A3.p1.1.m1.2.2.3" xref="A3.p1.1.m1.2.2.3.cmml">=</mo><mrow id="A3.p1.1.m1.2.2.2" xref="A3.p1.1.m1.2.2.2.cmml"><mi id="A3.p1.1.m1.2.2.2.4" xref="A3.p1.1.m1.2.2.2.4.cmml">h</mi><mo id="A3.p1.1.m1.2.2.2.3" xref="A3.p1.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="A3.p1.1.m1.2.2.2.2.2" xref="A3.p1.1.m1.2.2.2.2.3.cmml"><mo id="A3.p1.1.m1.2.2.2.2.2.3" stretchy="false" xref="A3.p1.1.m1.2.2.2.2.3.cmml">(</mo><msub id="A3.p1.1.m1.1.1.1.1.1.1" xref="A3.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="A3.p1.1.m1.1.1.1.1.1.1.2" xref="A3.p1.1.m1.1.1.1.1.1.1.2.cmml">H</mi><mrow id="A3.p1.1.m1.1.1.1.1.1.1.3" xref="A3.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="A3.p1.1.m1.1.1.1.1.1.1.3.2" xref="A3.p1.1.m1.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="A3.p1.1.m1.1.1.1.1.1.1.3.1" xref="A3.p1.1.m1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="A3.p1.1.m1.1.1.1.1.1.1.3.3" xref="A3.p1.1.m1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A3.p1.1.m1.2.2.2.2.2.4" xref="A3.p1.1.m1.2.2.2.2.3.cmml">,</mo><msub id="A3.p1.1.m1.2.2.2.2.2.2" xref="A3.p1.1.m1.2.2.2.2.2.2.cmml"><mi id="A3.p1.1.m1.2.2.2.2.2.2.2" xref="A3.p1.1.m1.2.2.2.2.2.2.2.cmml">x</mi><mi id="A3.p1.1.m1.2.2.2.2.2.2.3" xref="A3.p1.1.m1.2.2.2.2.2.2.3.cmml">t</mi></msub><mo id="A3.p1.1.m1.2.2.2.2.2.5" stretchy="false" xref="A3.p1.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.1.m1.2b"><apply id="A3.p1.1.m1.2.2.cmml" xref="A3.p1.1.m1.2.2"><eq id="A3.p1.1.m1.2.2.3.cmml" xref="A3.p1.1.m1.2.2.3"></eq><apply 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id="A3.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="A3.p1.1.m1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="A3.p1.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A3.p1.1.m1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A3.p1.1.m1.2.2.2.2.2.2.cmml" xref="A3.p1.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.p1.1.m1.2.2.2.2.2.2.1.cmml" xref="A3.p1.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="A3.p1.1.m1.2.2.2.2.2.2.2.cmml" xref="A3.p1.1.m1.2.2.2.2.2.2.2">𝑥</ci><ci id="A3.p1.1.m1.2.2.2.2.2.2.3.cmml" xref="A3.p1.1.m1.2.2.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.1.m1.2c">H_{t}=h(H_{t-1},x_{t})</annotation><annotation encoding="application/x-llamapun" id="A3.p1.1.m1.2d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_h ( italic_H start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="y_{t}=g(H_{t})" 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stretchy="false" xref="A3.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.2.m2.1b"><apply id="A3.p1.2.m2.1.1.cmml" xref="A3.p1.2.m2.1.1"><eq id="A3.p1.2.m2.1.1.2.cmml" xref="A3.p1.2.m2.1.1.2"></eq><apply id="A3.p1.2.m2.1.1.3.cmml" xref="A3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A3.p1.2.m2.1.1.3.1.cmml" xref="A3.p1.2.m2.1.1.3">subscript</csymbol><ci id="A3.p1.2.m2.1.1.3.2.cmml" xref="A3.p1.2.m2.1.1.3.2">𝑦</ci><ci id="A3.p1.2.m2.1.1.3.3.cmml" xref="A3.p1.2.m2.1.1.3.3">𝑡</ci></apply><apply id="A3.p1.2.m2.1.1.1.cmml" xref="A3.p1.2.m2.1.1.1"><times id="A3.p1.2.m2.1.1.1.2.cmml" xref="A3.p1.2.m2.1.1.1.2"></times><ci id="A3.p1.2.m2.1.1.1.3.cmml" xref="A3.p1.2.m2.1.1.1.3">𝑔</ci><apply id="A3.p1.2.m2.1.1.1.1.1.1.cmml" xref="A3.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="A3.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="A3.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="A3.p1.2.m2.1.1.1.1.1.1.2">𝐻</ci><ci id="A3.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="A3.p1.2.m2.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.2.m2.1c">y_{t}=g(H_{t})</annotation><annotation encoding="application/x-llamapun" id="A3.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_g ( italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> for each <math alttext="t\geq 1" class="ltx_Math" display="inline" id="A3.p1.3.m3.1"><semantics id="A3.p1.3.m3.1a"><mrow id="A3.p1.3.m3.1.1" xref="A3.p1.3.m3.1.1.cmml"><mi id="A3.p1.3.m3.1.1.2" xref="A3.p1.3.m3.1.1.2.cmml">t</mi><mo id="A3.p1.3.m3.1.1.1" xref="A3.p1.3.m3.1.1.1.cmml">≥</mo><mn id="A3.p1.3.m3.1.1.3" xref="A3.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.3.m3.1b"><apply id="A3.p1.3.m3.1.1.cmml" xref="A3.p1.3.m3.1.1"><geq id="A3.p1.3.m3.1.1.1.cmml" xref="A3.p1.3.m3.1.1.1"></geq><ci id="A3.p1.3.m3.1.1.2.cmml" xref="A3.p1.3.m3.1.1.2">𝑡</ci><cn id="A3.p1.3.m3.1.1.3.cmml" type="integer" xref="A3.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.3.m3.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="A3.p1.3.m3.1d">italic_t ≥ 1</annotation></semantics></math> where <math alttext="H_{t}\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A3.p1.4.m4.1"><semantics id="A3.p1.4.m4.1a"><mrow id="A3.p1.4.m4.1.1" xref="A3.p1.4.m4.1.1.cmml"><msub id="A3.p1.4.m4.1.1.2" xref="A3.p1.4.m4.1.1.2.cmml"><mi id="A3.p1.4.m4.1.1.2.2" xref="A3.p1.4.m4.1.1.2.2.cmml">H</mi><mi id="A3.p1.4.m4.1.1.2.3" xref="A3.p1.4.m4.1.1.2.3.cmml">t</mi></msub><mo id="A3.p1.4.m4.1.1.1" xref="A3.p1.4.m4.1.1.1.cmml">∈</mo><msup id="A3.p1.4.m4.1.1.3" xref="A3.p1.4.m4.1.1.3.cmml"><mi id="A3.p1.4.m4.1.1.3.2" xref="A3.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A3.p1.4.m4.1.1.3.3" xref="A3.p1.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.4.m4.1b"><apply id="A3.p1.4.m4.1.1.cmml" xref="A3.p1.4.m4.1.1"><in id="A3.p1.4.m4.1.1.1.cmml" xref="A3.p1.4.m4.1.1.1"></in><apply id="A3.p1.4.m4.1.1.2.cmml" xref="A3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="A3.p1.4.m4.1.1.2.1.cmml" xref="A3.p1.4.m4.1.1.2">subscript</csymbol><ci id="A3.p1.4.m4.1.1.2.2.cmml" xref="A3.p1.4.m4.1.1.2.2">𝐻</ci><ci id="A3.p1.4.m4.1.1.2.3.cmml" xref="A3.p1.4.m4.1.1.2.3">𝑡</ci></apply><apply id="A3.p1.4.m4.1.1.3.cmml" xref="A3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="A3.p1.4.m4.1.1.3.1.cmml" xref="A3.p1.4.m4.1.1.3">superscript</csymbol><ci id="A3.p1.4.m4.1.1.3.2.cmml" xref="A3.p1.4.m4.1.1.3.2">ℝ</ci><ci id="A3.p1.4.m4.1.1.3.3.cmml" xref="A3.p1.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.4.m4.1c">H_{t}\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.4.m4.1d">italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and the model has a bit precision of <math alttext="p" class="ltx_Math" display="inline" id="A3.p1.5.m5.1"><semantics id="A3.p1.5.m5.1a"><mi id="A3.p1.5.m5.1.1" xref="A3.p1.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.p1.5.m5.1b"><ci id="A3.p1.5.m5.1.1.cmml" xref="A3.p1.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.p1.5.m5.1d">italic_p</annotation></semantics></math>. In this proof, we will assume that the state space of the underlying Markov chain is <math alttext="\{0,1\}" class="ltx_Math" display="inline" id="A3.p1.6.m6.2"><semantics id="A3.p1.6.m6.2a"><mrow id="A3.p1.6.m6.2.3.2" xref="A3.p1.6.m6.2.3.1.cmml"><mo id="A3.p1.6.m6.2.3.2.1" stretchy="false" xref="A3.p1.6.m6.2.3.1.cmml">{</mo><mn id="A3.p1.6.m6.1.1" xref="A3.p1.6.m6.1.1.cmml">0</mn><mo id="A3.p1.6.m6.2.3.2.2" xref="A3.p1.6.m6.2.3.1.cmml">,</mo><mn id="A3.p1.6.m6.2.2" xref="A3.p1.6.m6.2.2.cmml">1</mn><mo id="A3.p1.6.m6.2.3.2.3" stretchy="false" xref="A3.p1.6.m6.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.6.m6.2b"><set id="A3.p1.6.m6.2.3.1.cmml" xref="A3.p1.6.m6.2.3.2"><cn id="A3.p1.6.m6.1.1.cmml" type="integer" xref="A3.p1.6.m6.1.1">0</cn><cn id="A3.p1.6.m6.2.2.cmml" type="integer" xref="A3.p1.6.m6.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.6.m6.2c">\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.6.m6.2d">{ 0 , 1 }</annotation></semantics></math>. By the recurrent architecture, the predicted distribution over the next token <math alttext="x_{t+k+1}" class="ltx_Math" display="inline" id="A3.p1.7.m7.1"><semantics id="A3.p1.7.m7.1a"><msub id="A3.p1.7.m7.1.1" xref="A3.p1.7.m7.1.1.cmml"><mi id="A3.p1.7.m7.1.1.2" xref="A3.p1.7.m7.1.1.2.cmml">x</mi><mrow id="A3.p1.7.m7.1.1.3" xref="A3.p1.7.m7.1.1.3.cmml"><mi id="A3.p1.7.m7.1.1.3.2" xref="A3.p1.7.m7.1.1.3.2.cmml">t</mi><mo id="A3.p1.7.m7.1.1.3.1" xref="A3.p1.7.m7.1.1.3.1.cmml">+</mo><mi id="A3.p1.7.m7.1.1.3.3" xref="A3.p1.7.m7.1.1.3.3.cmml">k</mi><mo id="A3.p1.7.m7.1.1.3.1a" xref="A3.p1.7.m7.1.1.3.1.cmml">+</mo><mn id="A3.p1.7.m7.1.1.3.4" xref="A3.p1.7.m7.1.1.3.4.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.p1.7.m7.1b"><apply id="A3.p1.7.m7.1.1.cmml" xref="A3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A3.p1.7.m7.1.1.1.cmml" xref="A3.p1.7.m7.1.1">subscript</csymbol><ci id="A3.p1.7.m7.1.1.2.cmml" xref="A3.p1.7.m7.1.1.2">𝑥</ci><apply id="A3.p1.7.m7.1.1.3.cmml" xref="A3.p1.7.m7.1.1.3"><plus id="A3.p1.7.m7.1.1.3.1.cmml" xref="A3.p1.7.m7.1.1.3.1"></plus><ci id="A3.p1.7.m7.1.1.3.2.cmml" xref="A3.p1.7.m7.1.1.3.2">𝑡</ci><ci id="A3.p1.7.m7.1.1.3.3.cmml" xref="A3.p1.7.m7.1.1.3.3">𝑘</ci><cn id="A3.p1.7.m7.1.1.3.4.cmml" type="integer" xref="A3.p1.7.m7.1.1.3.4">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.7.m7.1c">x_{t+k+1}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.7.m7.1d">italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is of the form,</p> <table class="ltx_equation ltx_eqn_table" id="A3.E35"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="y_{t+k}=\mathbb{P}_{\bm{\theta}}\left(x_{t+k+1}=z\mid x_{1}^{t+k}\right)=g(z,H% _{t+k})." class="ltx_Math" display="block" id="A3.E35.m1.2"><semantics id="A3.E35.m1.2a"><mrow id="A3.E35.m1.2.2.1" xref="A3.E35.m1.2.2.1.1.cmml"><mrow id="A3.E35.m1.2.2.1.1" xref="A3.E35.m1.2.2.1.1.cmml"><msub id="A3.E35.m1.2.2.1.1.4" xref="A3.E35.m1.2.2.1.1.4.cmml"><mi id="A3.E35.m1.2.2.1.1.4.2" xref="A3.E35.m1.2.2.1.1.4.2.cmml">y</mi><mrow id="A3.E35.m1.2.2.1.1.4.3" xref="A3.E35.m1.2.2.1.1.4.3.cmml"><mi id="A3.E35.m1.2.2.1.1.4.3.2" xref="A3.E35.m1.2.2.1.1.4.3.2.cmml">t</mi><mo id="A3.E35.m1.2.2.1.1.4.3.1" xref="A3.E35.m1.2.2.1.1.4.3.1.cmml">+</mo><mi id="A3.E35.m1.2.2.1.1.4.3.3" xref="A3.E35.m1.2.2.1.1.4.3.3.cmml">k</mi></mrow></msub><mo id="A3.E35.m1.2.2.1.1.5" xref="A3.E35.m1.2.2.1.1.5.cmml">=</mo><mrow id="A3.E35.m1.2.2.1.1.1" xref="A3.E35.m1.2.2.1.1.1.cmml"><msub id="A3.E35.m1.2.2.1.1.1.3" xref="A3.E35.m1.2.2.1.1.1.3.cmml"><mi id="A3.E35.m1.2.2.1.1.1.3.2" xref="A3.E35.m1.2.2.1.1.1.3.2.cmml">ℙ</mi><mi id="A3.E35.m1.2.2.1.1.1.3.3" xref="A3.E35.m1.2.2.1.1.1.3.3.cmml">𝜽</mi></msub><mo id="A3.E35.m1.2.2.1.1.1.2" xref="A3.E35.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.E35.m1.2.2.1.1.1.1.1" xref="A3.E35.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A3.E35.m1.2.2.1.1.1.1.1.2" xref="A3.E35.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E35.m1.2.2.1.1.1.1.1.1" xref="A3.E35.m1.2.2.1.1.1.1.1.1.cmml"><msub id="A3.E35.m1.2.2.1.1.1.1.1.1.2" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="A3.E35.m1.2.2.1.1.1.1.1.1.2.2" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.2.cmml">x</mi><mrow id="A3.E35.m1.2.2.1.1.1.1.1.1.2.3" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.cmml"><mi id="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.2" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.2.cmml">t</mi><mo id="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.1" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.1.cmml">+</mo><mi id="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.3" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.3.cmml">k</mi><mo id="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.1a" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.4" xref="A3.E35.m1.2.2.1.1.1.1.1.1.2.3.4.cmml">1</mn></mrow></msub><mo 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end_POSTSUPERSCRIPT ) = italic_g ( italic_z , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(35)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.8">Recall that the add-<math alttext="1" class="ltx_Math" display="inline" id="A3.p1.8.m1.1"><semantics id="A3.p1.8.m1.1a"><mn id="A3.p1.8.m1.1.1" xref="A3.p1.8.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A3.p1.8.m1.1b"><cn id="A3.p1.8.m1.1.1.cmml" type="integer" xref="A3.p1.8.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.8.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A3.p1.8.m1.1d">1</annotation></semantics></math> estimator is defined as,</p> <table class="ltx_equationgroup ltx_eqn_align 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class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.15">where <math alttext="n(z_{1}^{k})=\sum_{i=1}^{t+1}\mathbb{I}(x_{i}^{i+k-1}=z_{1}^{k})" class="ltx_Math" display="inline" id="A3.p1.9.m1.2"><semantics id="A3.p1.9.m1.2a"><mrow id="A3.p1.9.m1.2.2" xref="A3.p1.9.m1.2.2.cmml"><mrow id="A3.p1.9.m1.1.1.1" xref="A3.p1.9.m1.1.1.1.cmml"><mi id="A3.p1.9.m1.1.1.1.3" xref="A3.p1.9.m1.1.1.1.3.cmml">n</mi><mo id="A3.p1.9.m1.1.1.1.2" xref="A3.p1.9.m1.1.1.1.2.cmml">⁢</mo><mrow id="A3.p1.9.m1.1.1.1.1.1" xref="A3.p1.9.m1.1.1.1.1.1.1.cmml"><mo id="A3.p1.9.m1.1.1.1.1.1.2" stretchy="false" xref="A3.p1.9.m1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A3.p1.9.m1.1.1.1.1.1.1" xref="A3.p1.9.m1.1.1.1.1.1.1.cmml"><mi id="A3.p1.9.m1.1.1.1.1.1.1.2.2" xref="A3.p1.9.m1.1.1.1.1.1.1.2.2.cmml">z</mi><mn 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end_POSTSUPERSCRIPT blackboard_I ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i + italic_k - 1 end_POSTSUPERSCRIPT = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> indicates the number of times <math alttext="z_{1}^{k}" class="ltx_Math" display="inline" id="A3.p1.10.m2.1"><semantics id="A3.p1.10.m2.1a"><msubsup id="A3.p1.10.m2.1.1" xref="A3.p1.10.m2.1.1.cmml"><mi id="A3.p1.10.m2.1.1.2.2" xref="A3.p1.10.m2.1.1.2.2.cmml">z</mi><mn id="A3.p1.10.m2.1.1.2.3" xref="A3.p1.10.m2.1.1.2.3.cmml">1</mn><mi id="A3.p1.10.m2.1.1.3" xref="A3.p1.10.m2.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.10.m2.1b"><apply id="A3.p1.10.m2.1.1.cmml" xref="A3.p1.10.m2.1.1"><csymbol cd="ambiguous" id="A3.p1.10.m2.1.1.1.cmml" xref="A3.p1.10.m2.1.1">superscript</csymbol><apply id="A3.p1.10.m2.1.1.2.cmml" xref="A3.p1.10.m2.1.1"><csymbol cd="ambiguous" id="A3.p1.10.m2.1.1.2.1.cmml" xref="A3.p1.10.m2.1.1">subscript</csymbol><ci id="A3.p1.10.m2.1.1.2.2.cmml" xref="A3.p1.10.m2.1.1.2.2">𝑧</ci><cn id="A3.p1.10.m2.1.1.2.3.cmml" type="integer" xref="A3.p1.10.m2.1.1.2.3">1</cn></apply><ci id="A3.p1.10.m2.1.1.3.cmml" xref="A3.p1.10.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.10.m2.1c">z_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.10.m2.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> appears in the sequence. This is the optimal estimator for sequences drawn from the product-Dirichlet prior: for every <math alttext="i_{1}^{k}" class="ltx_Math" display="inline" id="A3.p1.11.m3.1"><semantics id="A3.p1.11.m3.1a"><msubsup id="A3.p1.11.m3.1.1" xref="A3.p1.11.m3.1.1.cmml"><mi id="A3.p1.11.m3.1.1.2.2" xref="A3.p1.11.m3.1.1.2.2.cmml">i</mi><mn id="A3.p1.11.m3.1.1.2.3" xref="A3.p1.11.m3.1.1.2.3.cmml">1</mn><mi id="A3.p1.11.m3.1.1.3" xref="A3.p1.11.m3.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.11.m3.1b"><apply id="A3.p1.11.m3.1.1.cmml" xref="A3.p1.11.m3.1.1"><csymbol cd="ambiguous" id="A3.p1.11.m3.1.1.1.cmml" xref="A3.p1.11.m3.1.1">superscript</csymbol><apply id="A3.p1.11.m3.1.1.2.cmml" xref="A3.p1.11.m3.1.1"><csymbol cd="ambiguous" id="A3.p1.11.m3.1.1.2.1.cmml" xref="A3.p1.11.m3.1.1">subscript</csymbol><ci id="A3.p1.11.m3.1.1.2.2.cmml" xref="A3.p1.11.m3.1.1.2.2">𝑖</ci><cn id="A3.p1.11.m3.1.1.2.3.cmml" type="integer" xref="A3.p1.11.m3.1.1.2.3">1</cn></apply><ci id="A3.p1.11.m3.1.1.3.cmml" xref="A3.p1.11.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.11.m3.1c">i_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.11.m3.1d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="P(\cdot\mid i_{1}^{k})\sim\mathrm{Dir}(1\cdot\bm{1})" class="ltx_math_unparsed" display="inline" id="A3.p1.12.m4.1"><semantics id="A3.p1.12.m4.1a"><mrow id="A3.p1.12.m4.1b"><mi id="A3.p1.12.m4.1.1">P</mi><mrow id="A3.p1.12.m4.1.2"><mo id="A3.p1.12.m4.1.2.1" stretchy="false">(</mo><mo id="A3.p1.12.m4.1.2.2" lspace="0em" rspace="0em">⋅</mo><mo id="A3.p1.12.m4.1.2.3" lspace="0em" rspace="0.167em">∣</mo><msubsup id="A3.p1.12.m4.1.2.4"><mi id="A3.p1.12.m4.1.2.4.2.2">i</mi><mn id="A3.p1.12.m4.1.2.4.2.3">1</mn><mi id="A3.p1.12.m4.1.2.4.3">k</mi></msubsup><mo id="A3.p1.12.m4.1.2.5" stretchy="false">)</mo></mrow><mo id="A3.p1.12.m4.1.3">∼</mo><mi id="A3.p1.12.m4.1.4">Dir</mi><mrow id="A3.p1.12.m4.1.5"><mo id="A3.p1.12.m4.1.5.1" stretchy="false">(</mo><mn id="A3.p1.12.m4.1.5.2">1</mn><mo id="A3.p1.12.m4.1.5.3" lspace="0.222em" rspace="0.222em">⋅</mo><mn id="A3.p1.12.m4.1.5.4">𝟏</mn><mo id="A3.p1.12.m4.1.5.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.p1.12.m4.1c">P(\cdot\mid i_{1}^{k})\sim\mathrm{Dir}(1\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="A3.p1.12.m4.1d">italic_P ( ⋅ ∣ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) ∼ roman_Dir ( 1 ⋅ bold_1 )</annotation></semantics></math>, which is the distribution we will assume for this proof. Fixing <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="A3.p1.13.m5.1"><semantics id="A3.p1.13.m5.1a"><msubsup id="A3.p1.13.m5.1.1" xref="A3.p1.13.m5.1.1.cmml"><mi id="A3.p1.13.m5.1.1.2.2" xref="A3.p1.13.m5.1.1.2.2.cmml">x</mi><mn id="A3.p1.13.m5.1.1.2.3" xref="A3.p1.13.m5.1.1.2.3.cmml">1</mn><mi id="A3.p1.13.m5.1.1.3" xref="A3.p1.13.m5.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.13.m5.1b"><apply id="A3.p1.13.m5.1.1.cmml" xref="A3.p1.13.m5.1.1"><csymbol cd="ambiguous" id="A3.p1.13.m5.1.1.1.cmml" xref="A3.p1.13.m5.1.1">superscript</csymbol><apply id="A3.p1.13.m5.1.1.2.cmml" xref="A3.p1.13.m5.1.1"><csymbol cd="ambiguous" id="A3.p1.13.m5.1.1.2.1.cmml" xref="A3.p1.13.m5.1.1">subscript</csymbol><ci id="A3.p1.13.m5.1.1.2.2.cmml" xref="A3.p1.13.m5.1.1.2.2">𝑥</ci><cn id="A3.p1.13.m5.1.1.2.3.cmml" type="integer" xref="A3.p1.13.m5.1.1.2.3">1</cn></apply><ci id="A3.p1.13.m5.1.1.3.cmml" xref="A3.p1.13.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.13.m5.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.13.m5.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, we can write the add-<math alttext="1" class="ltx_Math" display="inline" id="A3.p1.14.m6.1"><semantics id="A3.p1.14.m6.1a"><mn id="A3.p1.14.m6.1.1" xref="A3.p1.14.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A3.p1.14.m6.1b"><cn id="A3.p1.14.m6.1.1.cmml" type="integer" xref="A3.p1.14.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.14.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="A3.p1.14.m6.1d">1</annotation></semantics></math> estimator more explicitly as a function of <math alttext="x_{t+1}^{t+k}" class="ltx_Math" display="inline" id="A3.p1.15.m7.1"><semantics id="A3.p1.15.m7.1a"><msubsup id="A3.p1.15.m7.1.1" xref="A3.p1.15.m7.1.1.cmml"><mi id="A3.p1.15.m7.1.1.2.2" xref="A3.p1.15.m7.1.1.2.2.cmml">x</mi><mrow id="A3.p1.15.m7.1.1.2.3" xref="A3.p1.15.m7.1.1.2.3.cmml"><mi id="A3.p1.15.m7.1.1.2.3.2" xref="A3.p1.15.m7.1.1.2.3.2.cmml">t</mi><mo id="A3.p1.15.m7.1.1.2.3.1" xref="A3.p1.15.m7.1.1.2.3.1.cmml">+</mo><mn id="A3.p1.15.m7.1.1.2.3.3" xref="A3.p1.15.m7.1.1.2.3.3.cmml">1</mn></mrow><mrow id="A3.p1.15.m7.1.1.3" xref="A3.p1.15.m7.1.1.3.cmml"><mi id="A3.p1.15.m7.1.1.3.2" xref="A3.p1.15.m7.1.1.3.2.cmml">t</mi><mo id="A3.p1.15.m7.1.1.3.1" xref="A3.p1.15.m7.1.1.3.1.cmml">+</mo><mi id="A3.p1.15.m7.1.1.3.3" xref="A3.p1.15.m7.1.1.3.3.cmml">k</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.15.m7.1b"><apply id="A3.p1.15.m7.1.1.cmml" xref="A3.p1.15.m7.1.1"><csymbol cd="ambiguous" id="A3.p1.15.m7.1.1.1.cmml" xref="A3.p1.15.m7.1.1">superscript</csymbol><apply id="A3.p1.15.m7.1.1.2.cmml" xref="A3.p1.15.m7.1.1"><csymbol cd="ambiguous" id="A3.p1.15.m7.1.1.2.1.cmml" xref="A3.p1.15.m7.1.1">subscript</csymbol><ci id="A3.p1.15.m7.1.1.2.2.cmml" xref="A3.p1.15.m7.1.1.2.2">𝑥</ci><apply id="A3.p1.15.m7.1.1.2.3.cmml" xref="A3.p1.15.m7.1.1.2.3"><plus id="A3.p1.15.m7.1.1.2.3.1.cmml" xref="A3.p1.15.m7.1.1.2.3.1"></plus><ci id="A3.p1.15.m7.1.1.2.3.2.cmml" xref="A3.p1.15.m7.1.1.2.3.2">𝑡</ci><cn id="A3.p1.15.m7.1.1.2.3.3.cmml" type="integer" xref="A3.p1.15.m7.1.1.2.3.3">1</cn></apply></apply><apply id="A3.p1.15.m7.1.1.3.cmml" xref="A3.p1.15.m7.1.1.3"><plus id="A3.p1.15.m7.1.1.3.1.cmml" xref="A3.p1.15.m7.1.1.3.1"></plus><ci id="A3.p1.15.m7.1.1.3.2.cmml" xref="A3.p1.15.m7.1.1.3.2">𝑡</ci><ci id="A3.p1.15.m7.1.1.3.3.cmml" xref="A3.p1.15.m7.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.15.m7.1c">x_{t+1}^{t+k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.15.m7.1d">italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> as,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx17"> <tbody id="A3.E37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{P}_{1}^{(k)}\left(x_{t+k+1}=z\mid x_{1}^{t},x_{t+1}^{t+k}% =z_{1}^{k}\right)=\frac{n(z_{1}^{k},z)+1}{n(z_{1}^{k})+2}." class="ltx_Math" display="inline" id="A3.E37.m1.5"><semantics id="A3.E37.m1.5a"><mrow id="A3.E37.m1.5.5.1" xref="A3.E37.m1.5.5.1.1.cmml"><mrow id="A3.E37.m1.5.5.1.1" xref="A3.E37.m1.5.5.1.1.cmml"><mrow id="A3.E37.m1.5.5.1.1.1" xref="A3.E37.m1.5.5.1.1.1.cmml"><msubsup id="A3.E37.m1.5.5.1.1.1.3" xref="A3.E37.m1.5.5.1.1.1.3.cmml"><mi id="A3.E37.m1.5.5.1.1.1.3.2.2" xref="A3.E37.m1.5.5.1.1.1.3.2.2.cmml">ℙ</mi><mn id="A3.E37.m1.5.5.1.1.1.3.2.3" xref="A3.E37.m1.5.5.1.1.1.3.2.3.cmml">1</mn><mrow id="A3.E37.m1.1.1.1.3" xref="A3.E37.m1.5.5.1.1.1.3.cmml"><mo id="A3.E37.m1.1.1.1.3.1" stretchy="false" xref="A3.E37.m1.5.5.1.1.1.3.cmml">(</mo><mi id="A3.E37.m1.1.1.1.1" xref="A3.E37.m1.1.1.1.1.cmml">k</mi><mo id="A3.E37.m1.1.1.1.3.2" stretchy="false" xref="A3.E37.m1.5.5.1.1.1.3.cmml">)</mo></mrow></msubsup><mo id="A3.E37.m1.5.5.1.1.1.2" xref="A3.E37.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="A3.E37.m1.5.5.1.1.1.1.1" xref="A3.E37.m1.5.5.1.1.1.1.1.1.cmml"><mo id="A3.E37.m1.5.5.1.1.1.1.1.2" xref="A3.E37.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E37.m1.5.5.1.1.1.1.1.1" xref="A3.E37.m1.5.5.1.1.1.1.1.1.cmml"><msub id="A3.E37.m1.5.5.1.1.1.1.1.1.4" xref="A3.E37.m1.5.5.1.1.1.1.1.1.4.cmml"><mi id="A3.E37.m1.5.5.1.1.1.1.1.1.4.2" xref="A3.E37.m1.5.5.1.1.1.1.1.1.4.2.cmml">x</mi><mrow id="A3.E37.m1.5.5.1.1.1.1.1.1.4.3" xref="A3.E37.m1.5.5.1.1.1.1.1.1.4.3.cmml"><mi id="A3.E37.m1.5.5.1.1.1.1.1.1.4.3.2" xref="A3.E37.m1.5.5.1.1.1.1.1.1.4.3.2.cmml">t</mi><mo id="A3.E37.m1.5.5.1.1.1.1.1.1.4.3.1" 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end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = italic_z ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + italic_k end_POSTSUPERSCRIPT = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = divide start_ARG italic_n ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_z ) + 1 end_ARG start_ARG italic_n ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) + 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(37)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.19">Now, fixing <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="A3.p1.16.m1.1"><semantics id="A3.p1.16.m1.1a"><msubsup id="A3.p1.16.m1.1.1" xref="A3.p1.16.m1.1.1.cmml"><mi id="A3.p1.16.m1.1.1.2.2" xref="A3.p1.16.m1.1.1.2.2.cmml">x</mi><mn id="A3.p1.16.m1.1.1.2.3" xref="A3.p1.16.m1.1.1.2.3.cmml">1</mn><mi id="A3.p1.16.m1.1.1.3" xref="A3.p1.16.m1.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.16.m1.1b"><apply id="A3.p1.16.m1.1.1.cmml" xref="A3.p1.16.m1.1.1"><csymbol cd="ambiguous" id="A3.p1.16.m1.1.1.1.cmml" xref="A3.p1.16.m1.1.1">superscript</csymbol><apply id="A3.p1.16.m1.1.1.2.cmml" xref="A3.p1.16.m1.1.1"><csymbol cd="ambiguous" id="A3.p1.16.m1.1.1.2.1.cmml" xref="A3.p1.16.m1.1.1">subscript</csymbol><ci id="A3.p1.16.m1.1.1.2.2.cmml" xref="A3.p1.16.m1.1.1.2.2">𝑥</ci><cn id="A3.p1.16.m1.1.1.2.3.cmml" type="integer" xref="A3.p1.16.m1.1.1.2.3">1</cn></apply><ci id="A3.p1.16.m1.1.1.3.cmml" xref="A3.p1.16.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.16.m1.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.16.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, correctness of the recurrent model means that, almost surely over <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.17.m2.1"><semantics id="A3.p1.17.m2.1a"><mi id="A3.p1.17.m2.1.1" xref="A3.p1.17.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.17.m2.1b"><ci id="A3.p1.17.m2.1.1.cmml" xref="A3.p1.17.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.17.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.17.m2.1d">italic_P</annotation></semantics></math> drawn from the prior, and <math alttext="x_{1}^{t}\sim P" class="ltx_Math" display="inline" id="A3.p1.18.m3.1"><semantics id="A3.p1.18.m3.1a"><mrow id="A3.p1.18.m3.1.1" xref="A3.p1.18.m3.1.1.cmml"><msubsup id="A3.p1.18.m3.1.1.2" xref="A3.p1.18.m3.1.1.2.cmml"><mi id="A3.p1.18.m3.1.1.2.2.2" xref="A3.p1.18.m3.1.1.2.2.2.cmml">x</mi><mn id="A3.p1.18.m3.1.1.2.2.3" xref="A3.p1.18.m3.1.1.2.2.3.cmml">1</mn><mi id="A3.p1.18.m3.1.1.2.3" xref="A3.p1.18.m3.1.1.2.3.cmml">t</mi></msubsup><mo id="A3.p1.18.m3.1.1.1" xref="A3.p1.18.m3.1.1.1.cmml">∼</mo><mi id="A3.p1.18.m3.1.1.3" xref="A3.p1.18.m3.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.18.m3.1b"><apply id="A3.p1.18.m3.1.1.cmml" xref="A3.p1.18.m3.1.1"><csymbol cd="latexml" id="A3.p1.18.m3.1.1.1.cmml" xref="A3.p1.18.m3.1.1.1">similar-to</csymbol><apply id="A3.p1.18.m3.1.1.2.cmml" xref="A3.p1.18.m3.1.1.2"><csymbol cd="ambiguous" id="A3.p1.18.m3.1.1.2.1.cmml" xref="A3.p1.18.m3.1.1.2">superscript</csymbol><apply id="A3.p1.18.m3.1.1.2.2.cmml" xref="A3.p1.18.m3.1.1.2"><csymbol cd="ambiguous" id="A3.p1.18.m3.1.1.2.2.1.cmml" xref="A3.p1.18.m3.1.1.2">subscript</csymbol><ci id="A3.p1.18.m3.1.1.2.2.2.cmml" xref="A3.p1.18.m3.1.1.2.2.2">𝑥</ci><cn id="A3.p1.18.m3.1.1.2.2.3.cmml" type="integer" xref="A3.p1.18.m3.1.1.2.2.3">1</cn></apply><ci id="A3.p1.18.m3.1.1.2.3.cmml" xref="A3.p1.18.m3.1.1.2.3">𝑡</ci></apply><ci id="A3.p1.18.m3.1.1.3.cmml" xref="A3.p1.18.m3.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.18.m3.1c">x_{1}^{t}\sim P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.18.m3.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ italic_P</annotation></semantics></math> and <math alttext="z_{1}^{k}\sim P(\cdot|x_{1}^{t})" class="ltx_math_unparsed" display="inline" id="A3.p1.19.m4.1"><semantics id="A3.p1.19.m4.1a"><mrow id="A3.p1.19.m4.1b"><msubsup id="A3.p1.19.m4.1.1"><mi id="A3.p1.19.m4.1.1.2.2">z</mi><mn id="A3.p1.19.m4.1.1.2.3">1</mn><mi id="A3.p1.19.m4.1.1.3">k</mi></msubsup><mo id="A3.p1.19.m4.1.2">∼</mo><mi id="A3.p1.19.m4.1.3">P</mi><mrow id="A3.p1.19.m4.1.4"><mo id="A3.p1.19.m4.1.4.1" stretchy="false">(</mo><mo id="A3.p1.19.m4.1.4.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A3.p1.19.m4.1.4.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.p1.19.m4.1.4.4"><mi id="A3.p1.19.m4.1.4.4.2.2">x</mi><mn id="A3.p1.19.m4.1.4.4.2.3">1</mn><mi id="A3.p1.19.m4.1.4.4.3">t</mi></msubsup><mo id="A3.p1.19.m4.1.4.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.p1.19.m4.1c">z_{1}^{k}\sim P(\cdot|x_{1}^{t})</annotation><annotation encoding="application/x-llamapun" id="A3.p1.19.m4.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∼ italic_P ( ⋅ | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>,</p> <table class="ltx_equationgroup 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x_{1}^{t},x_{t+1}^{t+k}=z_{1}^{k}\right)+[-\varepsilon,\varepsilon].</annotation><annotation encoding="application/x-llamapun" id="A3.E38.m1.4d">italic_g ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT ) ∈ blackboard_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + italic_k end_POSTSUPERSCRIPT = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) + [ - italic_ε , italic_ε ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(38)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.30">where <math alttext="H_{t+k}" class="ltx_Math" display="inline" id="A3.p1.20.m1.1"><semantics id="A3.p1.20.m1.1a"><msub id="A3.p1.20.m1.1.1" xref="A3.p1.20.m1.1.1.cmml"><mi id="A3.p1.20.m1.1.1.2" xref="A3.p1.20.m1.1.1.2.cmml">H</mi><mrow id="A3.p1.20.m1.1.1.3" xref="A3.p1.20.m1.1.1.3.cmml"><mi id="A3.p1.20.m1.1.1.3.2" xref="A3.p1.20.m1.1.1.3.2.cmml">t</mi><mo id="A3.p1.20.m1.1.1.3.1" xref="A3.p1.20.m1.1.1.3.1.cmml">+</mo><mi id="A3.p1.20.m1.1.1.3.3" xref="A3.p1.20.m1.1.1.3.3.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.p1.20.m1.1b"><apply id="A3.p1.20.m1.1.1.cmml" xref="A3.p1.20.m1.1.1"><csymbol cd="ambiguous" id="A3.p1.20.m1.1.1.1.cmml" xref="A3.p1.20.m1.1.1">subscript</csymbol><ci id="A3.p1.20.m1.1.1.2.cmml" xref="A3.p1.20.m1.1.1.2">𝐻</ci><apply id="A3.p1.20.m1.1.1.3.cmml" xref="A3.p1.20.m1.1.1.3"><plus id="A3.p1.20.m1.1.1.3.1.cmml" xref="A3.p1.20.m1.1.1.3.1"></plus><ci id="A3.p1.20.m1.1.1.3.2.cmml" xref="A3.p1.20.m1.1.1.3.2">𝑡</ci><ci id="A3.p1.20.m1.1.1.3.3.cmml" xref="A3.p1.20.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.20.m1.1c">H_{t+k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.20.m1.1d">italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is a function of <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="A3.p1.21.m2.1"><semantics id="A3.p1.21.m2.1a"><msubsup id="A3.p1.21.m2.1.1" xref="A3.p1.21.m2.1.1.cmml"><mi id="A3.p1.21.m2.1.1.2.2" xref="A3.p1.21.m2.1.1.2.2.cmml">x</mi><mn id="A3.p1.21.m2.1.1.2.3" xref="A3.p1.21.m2.1.1.2.3.cmml">1</mn><mi id="A3.p1.21.m2.1.1.3" xref="A3.p1.21.m2.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.21.m2.1b"><apply id="A3.p1.21.m2.1.1.cmml" xref="A3.p1.21.m2.1.1"><csymbol cd="ambiguous" id="A3.p1.21.m2.1.1.1.cmml" xref="A3.p1.21.m2.1.1">superscript</csymbol><apply id="A3.p1.21.m2.1.1.2.cmml" xref="A3.p1.21.m2.1.1"><csymbol cd="ambiguous" id="A3.p1.21.m2.1.1.2.1.cmml" xref="A3.p1.21.m2.1.1">subscript</csymbol><ci id="A3.p1.21.m2.1.1.2.2.cmml" xref="A3.p1.21.m2.1.1.2.2">𝑥</ci><cn id="A3.p1.21.m2.1.1.2.3.cmml" type="integer" xref="A3.p1.21.m2.1.1.2.3">1</cn></apply><ci id="A3.p1.21.m2.1.1.3.cmml" xref="A3.p1.21.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.21.m2.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.21.m2.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="z_{1}^{k}" class="ltx_Math" display="inline" id="A3.p1.22.m3.1"><semantics id="A3.p1.22.m3.1a"><msubsup id="A3.p1.22.m3.1.1" xref="A3.p1.22.m3.1.1.cmml"><mi id="A3.p1.22.m3.1.1.2.2" xref="A3.p1.22.m3.1.1.2.2.cmml">z</mi><mn id="A3.p1.22.m3.1.1.2.3" xref="A3.p1.22.m3.1.1.2.3.cmml">1</mn><mi id="A3.p1.22.m3.1.1.3" xref="A3.p1.22.m3.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.22.m3.1b"><apply id="A3.p1.22.m3.1.1.cmml" xref="A3.p1.22.m3.1.1"><csymbol cd="ambiguous" id="A3.p1.22.m3.1.1.1.cmml" xref="A3.p1.22.m3.1.1">superscript</csymbol><apply id="A3.p1.22.m3.1.1.2.cmml" xref="A3.p1.22.m3.1.1"><csymbol cd="ambiguous" id="A3.p1.22.m3.1.1.2.1.cmml" xref="A3.p1.22.m3.1.1">subscript</csymbol><ci id="A3.p1.22.m3.1.1.2.2.cmml" xref="A3.p1.22.m3.1.1.2.2">𝑧</ci><cn id="A3.p1.22.m3.1.1.2.3.cmml" type="integer" xref="A3.p1.22.m3.1.1.2.3">1</cn></apply><ci id="A3.p1.22.m3.1.1.3.cmml" xref="A3.p1.22.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.22.m3.1c">z_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.22.m3.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. As <math alttext="t\to\infty" class="ltx_Math" display="inline" id="A3.p1.23.m4.1"><semantics id="A3.p1.23.m4.1a"><mrow id="A3.p1.23.m4.1.1" xref="A3.p1.23.m4.1.1.cmml"><mi id="A3.p1.23.m4.1.1.2" xref="A3.p1.23.m4.1.1.2.cmml">t</mi><mo id="A3.p1.23.m4.1.1.1" stretchy="false" xref="A3.p1.23.m4.1.1.1.cmml">→</mo><mi id="A3.p1.23.m4.1.1.3" mathvariant="normal" xref="A3.p1.23.m4.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.23.m4.1b"><apply id="A3.p1.23.m4.1.1.cmml" xref="A3.p1.23.m4.1.1"><ci id="A3.p1.23.m4.1.1.1.cmml" xref="A3.p1.23.m4.1.1.1">→</ci><ci id="A3.p1.23.m4.1.1.2.cmml" xref="A3.p1.23.m4.1.1.2">𝑡</ci><infinity id="A3.p1.23.m4.1.1.3.cmml" xref="A3.p1.23.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.23.m4.1c">t\to\infty</annotation><annotation encoding="application/x-llamapun" id="A3.p1.23.m4.1d">italic_t → ∞</annotation></semantics></math>, under the randomness of the draw of <math alttext="x_{1}^{t}\sim P" class="ltx_Math" display="inline" id="A3.p1.24.m5.1"><semantics id="A3.p1.24.m5.1a"><mrow id="A3.p1.24.m5.1.1" xref="A3.p1.24.m5.1.1.cmml"><msubsup id="A3.p1.24.m5.1.1.2" xref="A3.p1.24.m5.1.1.2.cmml"><mi id="A3.p1.24.m5.1.1.2.2.2" xref="A3.p1.24.m5.1.1.2.2.2.cmml">x</mi><mn id="A3.p1.24.m5.1.1.2.2.3" xref="A3.p1.24.m5.1.1.2.2.3.cmml">1</mn><mi id="A3.p1.24.m5.1.1.2.3" xref="A3.p1.24.m5.1.1.2.3.cmml">t</mi></msubsup><mo id="A3.p1.24.m5.1.1.1" xref="A3.p1.24.m5.1.1.1.cmml">∼</mo><mi id="A3.p1.24.m5.1.1.3" xref="A3.p1.24.m5.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.24.m5.1b"><apply id="A3.p1.24.m5.1.1.cmml" xref="A3.p1.24.m5.1.1"><csymbol cd="latexml" id="A3.p1.24.m5.1.1.1.cmml" xref="A3.p1.24.m5.1.1.1">similar-to</csymbol><apply id="A3.p1.24.m5.1.1.2.cmml" xref="A3.p1.24.m5.1.1.2"><csymbol cd="ambiguous" id="A3.p1.24.m5.1.1.2.1.cmml" xref="A3.p1.24.m5.1.1.2">superscript</csymbol><apply id="A3.p1.24.m5.1.1.2.2.cmml" xref="A3.p1.24.m5.1.1.2"><csymbol cd="ambiguous" id="A3.p1.24.m5.1.1.2.2.1.cmml" xref="A3.p1.24.m5.1.1.2">subscript</csymbol><ci id="A3.p1.24.m5.1.1.2.2.2.cmml" xref="A3.p1.24.m5.1.1.2.2.2">𝑥</ci><cn id="A3.p1.24.m5.1.1.2.2.3.cmml" type="integer" xref="A3.p1.24.m5.1.1.2.2.3">1</cn></apply><ci id="A3.p1.24.m5.1.1.2.3.cmml" xref="A3.p1.24.m5.1.1.2.3">𝑡</ci></apply><ci id="A3.p1.24.m5.1.1.3.cmml" xref="A3.p1.24.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.24.m5.1c">x_{1}^{t}\sim P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.24.m5.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∼ italic_P</annotation></semantics></math>, by the strong law of large numbers RHS converges almost surely to the conditional distribution under <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.25.m6.1"><semantics id="A3.p1.25.m6.1a"><mi id="A3.p1.25.m6.1.1" xref="A3.p1.25.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.25.m6.1b"><ci id="A3.p1.25.m6.1.1.cmml" xref="A3.p1.25.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.25.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.25.m6.1d">italic_P</annotation></semantics></math>, almost surely over the choice of <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.26.m7.1"><semantics id="A3.p1.26.m7.1a"><mi id="A3.p1.26.m7.1.1" xref="A3.p1.26.m7.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.26.m7.1b"><ci id="A3.p1.26.m7.1.1.cmml" xref="A3.p1.26.m7.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.26.m7.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.26.m7.1d">italic_P</annotation></semantics></math> from the product-Dirichlet prior. Here we use the fact that for <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.27.m8.1"><semantics id="A3.p1.27.m8.1a"><mi id="A3.p1.27.m8.1.1" xref="A3.p1.27.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.27.m8.1b"><ci id="A3.p1.27.m8.1.1.cmml" xref="A3.p1.27.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.27.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.27.m8.1d">italic_P</annotation></semantics></math> drawn from the product-Dirichlet prior, <math alttext="P(z|z_{1}^{k})>0" class="ltx_Math" display="inline" id="A3.p1.28.m9.1"><semantics id="A3.p1.28.m9.1a"><mrow id="A3.p1.28.m9.1.1" xref="A3.p1.28.m9.1.1.cmml"><mrow id="A3.p1.28.m9.1.1.1" xref="A3.p1.28.m9.1.1.1.cmml"><mi id="A3.p1.28.m9.1.1.1.3" xref="A3.p1.28.m9.1.1.1.3.cmml">P</mi><mo id="A3.p1.28.m9.1.1.1.2" xref="A3.p1.28.m9.1.1.1.2.cmml">⁢</mo><mrow id="A3.p1.28.m9.1.1.1.1.1" xref="A3.p1.28.m9.1.1.1.1.1.1.cmml"><mo id="A3.p1.28.m9.1.1.1.1.1.2" stretchy="false" xref="A3.p1.28.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.p1.28.m9.1.1.1.1.1.1" xref="A3.p1.28.m9.1.1.1.1.1.1.cmml"><mi id="A3.p1.28.m9.1.1.1.1.1.1.2" xref="A3.p1.28.m9.1.1.1.1.1.1.2.cmml">z</mi><mo fence="false" id="A3.p1.28.m9.1.1.1.1.1.1.1" xref="A3.p1.28.m9.1.1.1.1.1.1.1.cmml">|</mo><msubsup id="A3.p1.28.m9.1.1.1.1.1.1.3" xref="A3.p1.28.m9.1.1.1.1.1.1.3.cmml"><mi id="A3.p1.28.m9.1.1.1.1.1.1.3.2.2" xref="A3.p1.28.m9.1.1.1.1.1.1.3.2.2.cmml">z</mi><mn id="A3.p1.28.m9.1.1.1.1.1.1.3.2.3" xref="A3.p1.28.m9.1.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="A3.p1.28.m9.1.1.1.1.1.1.3.3" xref="A3.p1.28.m9.1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A3.p1.28.m9.1.1.1.1.1.3" stretchy="false" xref="A3.p1.28.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.p1.28.m9.1.1.2" xref="A3.p1.28.m9.1.1.2.cmml">></mo><mn id="A3.p1.28.m9.1.1.3" xref="A3.p1.28.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.28.m9.1b"><apply id="A3.p1.28.m9.1.1.cmml" xref="A3.p1.28.m9.1.1"><gt id="A3.p1.28.m9.1.1.2.cmml" xref="A3.p1.28.m9.1.1.2"></gt><apply id="A3.p1.28.m9.1.1.1.cmml" xref="A3.p1.28.m9.1.1.1"><times id="A3.p1.28.m9.1.1.1.2.cmml" xref="A3.p1.28.m9.1.1.1.2"></times><ci id="A3.p1.28.m9.1.1.1.3.cmml" xref="A3.p1.28.m9.1.1.1.3">𝑃</ci><apply id="A3.p1.28.m9.1.1.1.1.1.1.cmml" xref="A3.p1.28.m9.1.1.1.1.1"><csymbol cd="latexml" id="A3.p1.28.m9.1.1.1.1.1.1.1.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.1">conditional</csymbol><ci id="A3.p1.28.m9.1.1.1.1.1.1.2.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.2">𝑧</ci><apply id="A3.p1.28.m9.1.1.1.1.1.1.3.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p1.28.m9.1.1.1.1.1.1.3.1.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.3">superscript</csymbol><apply id="A3.p1.28.m9.1.1.1.1.1.1.3.2.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p1.28.m9.1.1.1.1.1.1.3.2.1.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.3">subscript</csymbol><ci id="A3.p1.28.m9.1.1.1.1.1.1.3.2.2.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.3.2.2">𝑧</ci><cn id="A3.p1.28.m9.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.p1.28.m9.1.1.1.1.1.1.3.2.3">1</cn></apply><ci id="A3.p1.28.m9.1.1.1.1.1.1.3.3.cmml" xref="A3.p1.28.m9.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply><cn id="A3.p1.28.m9.1.1.3.cmml" type="integer" xref="A3.p1.28.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.28.m9.1c">P(z|z_{1}^{k})>0</annotation><annotation encoding="application/x-llamapun" id="A3.p1.28.m9.1d">italic_P ( italic_z | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) > 0</annotation></semantics></math> almost surely, and so the resulting distributions are exponentially mixing and ergodic. Namely, for each <math alttext="z_{1}^{k}\in\{0,1\}^{k}" class="ltx_Math" display="inline" id="A3.p1.29.m10.2"><semantics id="A3.p1.29.m10.2a"><mrow id="A3.p1.29.m10.2.3" xref="A3.p1.29.m10.2.3.cmml"><msubsup id="A3.p1.29.m10.2.3.2" xref="A3.p1.29.m10.2.3.2.cmml"><mi id="A3.p1.29.m10.2.3.2.2.2" xref="A3.p1.29.m10.2.3.2.2.2.cmml">z</mi><mn id="A3.p1.29.m10.2.3.2.2.3" xref="A3.p1.29.m10.2.3.2.2.3.cmml">1</mn><mi id="A3.p1.29.m10.2.3.2.3" xref="A3.p1.29.m10.2.3.2.3.cmml">k</mi></msubsup><mo id="A3.p1.29.m10.2.3.1" xref="A3.p1.29.m10.2.3.1.cmml">∈</mo><msup id="A3.p1.29.m10.2.3.3" xref="A3.p1.29.m10.2.3.3.cmml"><mrow id="A3.p1.29.m10.2.3.3.2.2" xref="A3.p1.29.m10.2.3.3.2.1.cmml"><mo id="A3.p1.29.m10.2.3.3.2.2.1" stretchy="false" xref="A3.p1.29.m10.2.3.3.2.1.cmml">{</mo><mn id="A3.p1.29.m10.1.1" xref="A3.p1.29.m10.1.1.cmml">0</mn><mo id="A3.p1.29.m10.2.3.3.2.2.2" xref="A3.p1.29.m10.2.3.3.2.1.cmml">,</mo><mn id="A3.p1.29.m10.2.2" xref="A3.p1.29.m10.2.2.cmml">1</mn><mo id="A3.p1.29.m10.2.3.3.2.2.3" stretchy="false" xref="A3.p1.29.m10.2.3.3.2.1.cmml">}</mo></mrow><mi id="A3.p1.29.m10.2.3.3.3" xref="A3.p1.29.m10.2.3.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.29.m10.2b"><apply id="A3.p1.29.m10.2.3.cmml" xref="A3.p1.29.m10.2.3"><in id="A3.p1.29.m10.2.3.1.cmml" xref="A3.p1.29.m10.2.3.1"></in><apply id="A3.p1.29.m10.2.3.2.cmml" xref="A3.p1.29.m10.2.3.2"><csymbol cd="ambiguous" id="A3.p1.29.m10.2.3.2.1.cmml" xref="A3.p1.29.m10.2.3.2">superscript</csymbol><apply id="A3.p1.29.m10.2.3.2.2.cmml" xref="A3.p1.29.m10.2.3.2"><csymbol cd="ambiguous" id="A3.p1.29.m10.2.3.2.2.1.cmml" xref="A3.p1.29.m10.2.3.2">subscript</csymbol><ci id="A3.p1.29.m10.2.3.2.2.2.cmml" xref="A3.p1.29.m10.2.3.2.2.2">𝑧</ci><cn id="A3.p1.29.m10.2.3.2.2.3.cmml" type="integer" xref="A3.p1.29.m10.2.3.2.2.3">1</cn></apply><ci id="A3.p1.29.m10.2.3.2.3.cmml" xref="A3.p1.29.m10.2.3.2.3">𝑘</ci></apply><apply id="A3.p1.29.m10.2.3.3.cmml" xref="A3.p1.29.m10.2.3.3"><csymbol cd="ambiguous" id="A3.p1.29.m10.2.3.3.1.cmml" xref="A3.p1.29.m10.2.3.3">superscript</csymbol><set id="A3.p1.29.m10.2.3.3.2.1.cmml" xref="A3.p1.29.m10.2.3.3.2.2"><cn id="A3.p1.29.m10.1.1.cmml" type="integer" xref="A3.p1.29.m10.1.1">0</cn><cn id="A3.p1.29.m10.2.2.cmml" type="integer" xref="A3.p1.29.m10.2.2">1</cn></set><ci id="A3.p1.29.m10.2.3.3.3.cmml" xref="A3.p1.29.m10.2.3.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.29.m10.2c">z_{1}^{k}\in\{0,1\}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.29.m10.2d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>, almost surely over <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.30.m11.1"><semantics id="A3.p1.30.m11.1a"><mi id="A3.p1.30.m11.1.1" xref="A3.p1.30.m11.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.30.m11.1b"><ci id="A3.p1.30.m11.1.1.cmml" xref="A3.p1.30.m11.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.30.m11.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.30.m11.1d">italic_P</annotation></semantics></math> drawn from the product-Dirichlet prior,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx19"> <tbody id="A3.E39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Pr\left(\limsup_{t\to\infty}\left|\mathbb{P}_{1}^{(k)}\left(x_{t% +k+1}=0\mid x_{1}^{t},x_{t+1}^{t+k}=z_{1}^{k}\right)-P(0|z_{1}^{k})\right|>% \gamma)\right)=0" class="ltx_math_unparsed" display="inline" id="A3.E39.m1.1"><semantics id="A3.E39.m1.1a"><mrow id="A3.E39.m1.1b"><mi id="A3.E39.m1.1.2">Pr</mi><mrow id="A3.E39.m1.1.3"><mo id="A3.E39.m1.1.3.1">(</mo><munder id="A3.E39.m1.1.3.2"><mo id="A3.E39.m1.1.3.2.2" lspace="0em" movablelimits="false" rspace="0em">lim sup</mo><mrow id="A3.E39.m1.1.3.2.3"><mi id="A3.E39.m1.1.3.2.3.2">t</mi><mo id="A3.E39.m1.1.3.2.3.1" stretchy="false">→</mo><mi id="A3.E39.m1.1.3.2.3.3" mathvariant="normal">∞</mi></mrow></munder><mo fence="false" id="A3.E39.m1.1.3.3" rspace="0.167em">|</mo><msubsup id="A3.E39.m1.1.3.4"><mi id="A3.E39.m1.1.3.4.2.2">ℙ</mi><mn id="A3.E39.m1.1.3.4.2.3">1</mn><mrow id="A3.E39.m1.1.1.1.3"><mo id="A3.E39.m1.1.1.1.3.1" stretchy="false">(</mo><mi id="A3.E39.m1.1.1.1.1">k</mi><mo id="A3.E39.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msubsup><mrow id="A3.E39.m1.1.3.5"><mo id="A3.E39.m1.1.3.5.1">(</mo><msub id="A3.E39.m1.1.3.5.2"><mi id="A3.E39.m1.1.3.5.2.2">x</mi><mrow id="A3.E39.m1.1.3.5.2.3"><mi id="A3.E39.m1.1.3.5.2.3.2">t</mi><mo id="A3.E39.m1.1.3.5.2.3.1">+</mo><mi id="A3.E39.m1.1.3.5.2.3.3">k</mi><mo id="A3.E39.m1.1.3.5.2.3.1a">+</mo><mn id="A3.E39.m1.1.3.5.2.3.4">1</mn></mrow></msub><mo id="A3.E39.m1.1.3.5.3">=</mo><mn id="A3.E39.m1.1.3.5.4">0</mn><mo id="A3.E39.m1.1.3.5.5" lspace="0em" rspace="0.167em">∣</mo><msubsup id="A3.E39.m1.1.3.5.6"><mi id="A3.E39.m1.1.3.5.6.2.2">x</mi><mn id="A3.E39.m1.1.3.5.6.2.3">1</mn><mi id="A3.E39.m1.1.3.5.6.3">t</mi></msubsup><mo id="A3.E39.m1.1.3.5.7">,</mo><msubsup id="A3.E39.m1.1.3.5.8"><mi id="A3.E39.m1.1.3.5.8.2.2">x</mi><mrow id="A3.E39.m1.1.3.5.8.2.3"><mi id="A3.E39.m1.1.3.5.8.2.3.2">t</mi><mo id="A3.E39.m1.1.3.5.8.2.3.1">+</mo><mn id="A3.E39.m1.1.3.5.8.2.3.3">1</mn></mrow><mrow id="A3.E39.m1.1.3.5.8.3"><mi id="A3.E39.m1.1.3.5.8.3.2">t</mi><mo id="A3.E39.m1.1.3.5.8.3.1">+</mo><mi id="A3.E39.m1.1.3.5.8.3.3">k</mi></mrow></msubsup><mo id="A3.E39.m1.1.3.5.9">=</mo><msubsup id="A3.E39.m1.1.3.5.10"><mi id="A3.E39.m1.1.3.5.10.2.2">z</mi><mn id="A3.E39.m1.1.3.5.10.2.3">1</mn><mi id="A3.E39.m1.1.3.5.10.3">k</mi></msubsup><mo id="A3.E39.m1.1.3.5.11">)</mo></mrow><mo id="A3.E39.m1.1.3.6">−</mo><mi id="A3.E39.m1.1.3.7">P</mi><mrow id="A3.E39.m1.1.3.8"><mo id="A3.E39.m1.1.3.8.1" stretchy="false">(</mo><mn id="A3.E39.m1.1.3.8.2">0</mn><mo fence="false" id="A3.E39.m1.1.3.8.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.E39.m1.1.3.8.4"><mi id="A3.E39.m1.1.3.8.4.2.2">z</mi><mn id="A3.E39.m1.1.3.8.4.2.3">1</mn><mi id="A3.E39.m1.1.3.8.4.3">k</mi></msubsup><mo id="A3.E39.m1.1.3.8.5" stretchy="false">)</mo></mrow><mo fence="false" id="A3.E39.m1.1.3.9">|</mo><mo id="A3.E39.m1.1.3.10" lspace="0.167em">></mo><mi id="A3.E39.m1.1.3.11">γ</mi><mo id="A3.E39.m1.1.3.12" stretchy="false">)</mo></mrow><mo id="A3.E39.m1.1.4">)</mo><mo id="A3.E39.m1.1.5">=</mo><mn id="A3.E39.m1.1.6">0</mn></mrow><annotation encoding="application/x-tex" id="A3.E39.m1.1c">\displaystyle\Pr\left(\limsup_{t\to\infty}\left|\mathbb{P}_{1}^{(k)}\left(x_{t% +k+1}=0\mid x_{1}^{t},x_{t+1}^{t+k}=z_{1}^{k}\right)-P(0|z_{1}^{k})\right|>% \gamma)\right)=0</annotation><annotation encoding="application/x-llamapun" id="A3.E39.m1.1d">roman_Pr ( lim sup start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT | blackboard_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 ∣ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t + italic_k end_POSTSUPERSCRIPT = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) - italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) | > italic_γ ) ) = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(39)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.32">for any <math alttext="\gamma>0" class="ltx_Math" display="inline" id="A3.p1.31.m1.1"><semantics id="A3.p1.31.m1.1a"><mrow id="A3.p1.31.m1.1.1" xref="A3.p1.31.m1.1.1.cmml"><mi id="A3.p1.31.m1.1.1.2" xref="A3.p1.31.m1.1.1.2.cmml">γ</mi><mo id="A3.p1.31.m1.1.1.1" xref="A3.p1.31.m1.1.1.1.cmml">></mo><mn id="A3.p1.31.m1.1.1.3" xref="A3.p1.31.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.31.m1.1b"><apply id="A3.p1.31.m1.1.1.cmml" xref="A3.p1.31.m1.1.1"><gt id="A3.p1.31.m1.1.1.1.cmml" xref="A3.p1.31.m1.1.1.1"></gt><ci id="A3.p1.31.m1.1.1.2.cmml" xref="A3.p1.31.m1.1.1.2">𝛾</ci><cn id="A3.p1.31.m1.1.1.3.cmml" type="integer" xref="A3.p1.31.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.31.m1.1c">\gamma>0</annotation><annotation encoding="application/x-llamapun" id="A3.p1.31.m1.1d">italic_γ > 0</annotation></semantics></math>. Therefore, a necessary condition to satisfy <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A3.E38" title="In Appendix C Proof of Thm. 2 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">38</span></a> is, for each <math alttext="z_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A3.p1.32.m2.1"><semantics id="A3.p1.32.m2.1a"><mrow id="A3.p1.32.m2.1.1" xref="A3.p1.32.m2.1.1.cmml"><msubsup id="A3.p1.32.m2.1.1.2" xref="A3.p1.32.m2.1.1.2.cmml"><mi id="A3.p1.32.m2.1.1.2.2.2" xref="A3.p1.32.m2.1.1.2.2.2.cmml">z</mi><mn id="A3.p1.32.m2.1.1.2.2.3" xref="A3.p1.32.m2.1.1.2.2.3.cmml">1</mn><mi id="A3.p1.32.m2.1.1.2.3" xref="A3.p1.32.m2.1.1.2.3.cmml">k</mi></msubsup><mo id="A3.p1.32.m2.1.1.1" xref="A3.p1.32.m2.1.1.1.cmml">∈</mo><msup id="A3.p1.32.m2.1.1.3" xref="A3.p1.32.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.p1.32.m2.1.1.3.2" xref="A3.p1.32.m2.1.1.3.2.cmml">𝒳</mi><mi id="A3.p1.32.m2.1.1.3.3" xref="A3.p1.32.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.32.m2.1b"><apply id="A3.p1.32.m2.1.1.cmml" xref="A3.p1.32.m2.1.1"><in id="A3.p1.32.m2.1.1.1.cmml" xref="A3.p1.32.m2.1.1.1"></in><apply id="A3.p1.32.m2.1.1.2.cmml" xref="A3.p1.32.m2.1.1.2"><csymbol cd="ambiguous" id="A3.p1.32.m2.1.1.2.1.cmml" xref="A3.p1.32.m2.1.1.2">superscript</csymbol><apply id="A3.p1.32.m2.1.1.2.2.cmml" xref="A3.p1.32.m2.1.1.2"><csymbol cd="ambiguous" id="A3.p1.32.m2.1.1.2.2.1.cmml" xref="A3.p1.32.m2.1.1.2">subscript</csymbol><ci id="A3.p1.32.m2.1.1.2.2.2.cmml" xref="A3.p1.32.m2.1.1.2.2.2">𝑧</ci><cn id="A3.p1.32.m2.1.1.2.2.3.cmml" type="integer" xref="A3.p1.32.m2.1.1.2.2.3">1</cn></apply><ci id="A3.p1.32.m2.1.1.2.3.cmml" xref="A3.p1.32.m2.1.1.2.3">𝑘</ci></apply><apply id="A3.p1.32.m2.1.1.3.cmml" xref="A3.p1.32.m2.1.1.3"><csymbol cd="ambiguous" id="A3.p1.32.m2.1.1.3.1.cmml" xref="A3.p1.32.m2.1.1.3">superscript</csymbol><ci id="A3.p1.32.m2.1.1.3.2.cmml" xref="A3.p1.32.m2.1.1.3.2">𝒳</ci><ci id="A3.p1.32.m2.1.1.3.3.cmml" xref="A3.p1.32.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.32.m2.1c">z_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.32.m2.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx20"> <tbody id="A3.E40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g(x_{t+k+1}=0,H_{t+k})\in P(0|z_{1}^{k})+[-\varepsilon-\eta_{P}(% t),\varepsilon+\eta_{P}(t)]." class="ltx_Math" display="inline" 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start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(40)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.46">for some <math alttext="\eta_{P}(t)" class="ltx_Math" display="inline" id="A3.p1.33.m1.1"><semantics id="A3.p1.33.m1.1a"><mrow id="A3.p1.33.m1.1.2" xref="A3.p1.33.m1.1.2.cmml"><msub id="A3.p1.33.m1.1.2.2" xref="A3.p1.33.m1.1.2.2.cmml"><mi id="A3.p1.33.m1.1.2.2.2" xref="A3.p1.33.m1.1.2.2.2.cmml">η</mi><mi id="A3.p1.33.m1.1.2.2.3" xref="A3.p1.33.m1.1.2.2.3.cmml">P</mi></msub><mo id="A3.p1.33.m1.1.2.1" xref="A3.p1.33.m1.1.2.1.cmml">⁢</mo><mrow id="A3.p1.33.m1.1.2.3.2" xref="A3.p1.33.m1.1.2.cmml"><mo id="A3.p1.33.m1.1.2.3.2.1" stretchy="false" xref="A3.p1.33.m1.1.2.cmml">(</mo><mi id="A3.p1.33.m1.1.1" xref="A3.p1.33.m1.1.1.cmml">t</mi><mo id="A3.p1.33.m1.1.2.3.2.2" stretchy="false" xref="A3.p1.33.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.33.m1.1b"><apply id="A3.p1.33.m1.1.2.cmml" xref="A3.p1.33.m1.1.2"><times id="A3.p1.33.m1.1.2.1.cmml" xref="A3.p1.33.m1.1.2.1"></times><apply id="A3.p1.33.m1.1.2.2.cmml" xref="A3.p1.33.m1.1.2.2"><csymbol cd="ambiguous" id="A3.p1.33.m1.1.2.2.1.cmml" xref="A3.p1.33.m1.1.2.2">subscript</csymbol><ci id="A3.p1.33.m1.1.2.2.2.cmml" xref="A3.p1.33.m1.1.2.2.2">𝜂</ci><ci id="A3.p1.33.m1.1.2.2.3.cmml" xref="A3.p1.33.m1.1.2.2.3">𝑃</ci></apply><ci id="A3.p1.33.m1.1.1.cmml" xref="A3.p1.33.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.33.m1.1c">\eta_{P}(t)</annotation><annotation encoding="application/x-llamapun" id="A3.p1.33.m1.1d">italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, which is a function of <math alttext="P" 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xref="A3.p1.35.m3.1.2.2.1.3.2.cmml">t</mi><mo id="A3.p1.35.m3.1.2.2.1.3.1" stretchy="false" xref="A3.p1.35.m3.1.2.2.1.3.1.cmml">→</mo><mi id="A3.p1.35.m3.1.2.2.1.3.3" mathvariant="normal" xref="A3.p1.35.m3.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="A3.p1.35.m3.1.2.2.2" xref="A3.p1.35.m3.1.2.2.2.cmml"><msub id="A3.p1.35.m3.1.2.2.2.2" xref="A3.p1.35.m3.1.2.2.2.2.cmml"><mi id="A3.p1.35.m3.1.2.2.2.2.2" xref="A3.p1.35.m3.1.2.2.2.2.2.cmml">η</mi><mi id="A3.p1.35.m3.1.2.2.2.2.3" xref="A3.p1.35.m3.1.2.2.2.2.3.cmml">P</mi></msub><mo id="A3.p1.35.m3.1.2.2.2.1" xref="A3.p1.35.m3.1.2.2.2.1.cmml">⁢</mo><mrow id="A3.p1.35.m3.1.2.2.2.3.2" xref="A3.p1.35.m3.1.2.2.2.cmml"><mo id="A3.p1.35.m3.1.2.2.2.3.2.1" stretchy="false" xref="A3.p1.35.m3.1.2.2.2.cmml">(</mo><mi id="A3.p1.35.m3.1.1" xref="A3.p1.35.m3.1.1.cmml">t</mi><mo id="A3.p1.35.m3.1.2.2.2.3.2.2" stretchy="false" xref="A3.p1.35.m3.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A3.p1.35.m3.1.2.1" xref="A3.p1.35.m3.1.2.1.cmml">=</mo><mn id="A3.p1.35.m3.1.2.3" xref="A3.p1.35.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.35.m3.1b"><apply id="A3.p1.35.m3.1.2.cmml" xref="A3.p1.35.m3.1.2"><eq id="A3.p1.35.m3.1.2.1.cmml" xref="A3.p1.35.m3.1.2.1"></eq><apply id="A3.p1.35.m3.1.2.2.cmml" xref="A3.p1.35.m3.1.2.2"><apply id="A3.p1.35.m3.1.2.2.1.cmml" xref="A3.p1.35.m3.1.2.2.1"><csymbol cd="ambiguous" id="A3.p1.35.m3.1.2.2.1.1.cmml" xref="A3.p1.35.m3.1.2.2.1">subscript</csymbol><csymbol cd="latexml" id="A3.p1.35.m3.1.2.2.1.2.cmml" xref="A3.p1.35.m3.1.2.2.1.2">limit-supremum</csymbol><apply id="A3.p1.35.m3.1.2.2.1.3.cmml" xref="A3.p1.35.m3.1.2.2.1.3"><ci id="A3.p1.35.m3.1.2.2.1.3.1.cmml" xref="A3.p1.35.m3.1.2.2.1.3.1">→</ci><ci id="A3.p1.35.m3.1.2.2.1.3.2.cmml" xref="A3.p1.35.m3.1.2.2.1.3.2">𝑡</ci><infinity id="A3.p1.35.m3.1.2.2.1.3.3.cmml" xref="A3.p1.35.m3.1.2.2.1.3.3"></infinity></apply></apply><apply id="A3.p1.35.m3.1.2.2.2.cmml" xref="A3.p1.35.m3.1.2.2.2"><times id="A3.p1.35.m3.1.2.2.2.1.cmml" xref="A3.p1.35.m3.1.2.2.2.1"></times><apply id="A3.p1.35.m3.1.2.2.2.2.cmml" xref="A3.p1.35.m3.1.2.2.2.2"><csymbol cd="ambiguous" id="A3.p1.35.m3.1.2.2.2.2.1.cmml" xref="A3.p1.35.m3.1.2.2.2.2">subscript</csymbol><ci id="A3.p1.35.m3.1.2.2.2.2.2.cmml" xref="A3.p1.35.m3.1.2.2.2.2.2">𝜂</ci><ci id="A3.p1.35.m3.1.2.2.2.2.3.cmml" xref="A3.p1.35.m3.1.2.2.2.2.3">𝑃</ci></apply><ci id="A3.p1.35.m3.1.1.cmml" xref="A3.p1.35.m3.1.1">𝑡</ci></apply></apply><cn id="A3.p1.35.m3.1.2.3.cmml" type="integer" xref="A3.p1.35.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.35.m3.1c">\limsup_{t\to\infty}\eta_{P}(t)=0</annotation><annotation encoding="application/x-llamapun" id="A3.p1.35.m3.1d">lim sup start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) = 0</annotation></semantics></math> almost surely over <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.36.m4.1"><semantics id="A3.p1.36.m4.1a"><mi id="A3.p1.36.m4.1.1" xref="A3.p1.36.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.36.m4.1b"><ci id="A3.p1.36.m4.1.1.cmml" xref="A3.p1.36.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.36.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.36.m4.1d">italic_P</annotation></semantics></math> drawn from the prior; note that <math alttext="H_{t+k}" class="ltx_Math" display="inline" id="A3.p1.37.m5.1"><semantics id="A3.p1.37.m5.1a"><msub id="A3.p1.37.m5.1.1" xref="A3.p1.37.m5.1.1.cmml"><mi id="A3.p1.37.m5.1.1.2" xref="A3.p1.37.m5.1.1.2.cmml">H</mi><mrow id="A3.p1.37.m5.1.1.3" xref="A3.p1.37.m5.1.1.3.cmml"><mi id="A3.p1.37.m5.1.1.3.2" xref="A3.p1.37.m5.1.1.3.2.cmml">t</mi><mo id="A3.p1.37.m5.1.1.3.1" xref="A3.p1.37.m5.1.1.3.1.cmml">+</mo><mi id="A3.p1.37.m5.1.1.3.3" xref="A3.p1.37.m5.1.1.3.3.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.p1.37.m5.1b"><apply id="A3.p1.37.m5.1.1.cmml" xref="A3.p1.37.m5.1.1"><csymbol cd="ambiguous" id="A3.p1.37.m5.1.1.1.cmml" xref="A3.p1.37.m5.1.1">subscript</csymbol><ci id="A3.p1.37.m5.1.1.2.cmml" xref="A3.p1.37.m5.1.1.2">𝐻</ci><apply id="A3.p1.37.m5.1.1.3.cmml" xref="A3.p1.37.m5.1.1.3"><plus id="A3.p1.37.m5.1.1.3.1.cmml" xref="A3.p1.37.m5.1.1.3.1"></plus><ci id="A3.p1.37.m5.1.1.3.2.cmml" xref="A3.p1.37.m5.1.1.3.2">𝑡</ci><ci id="A3.p1.37.m5.1.1.3.3.cmml" xref="A3.p1.37.m5.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.37.m5.1c">H_{t+k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.37.m5.1d">italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is implicitly a function of <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="A3.p1.38.m6.1"><semantics id="A3.p1.38.m6.1a"><msubsup id="A3.p1.38.m6.1.1" xref="A3.p1.38.m6.1.1.cmml"><mi id="A3.p1.38.m6.1.1.2.2" xref="A3.p1.38.m6.1.1.2.2.cmml">x</mi><mn id="A3.p1.38.m6.1.1.2.3" xref="A3.p1.38.m6.1.1.2.3.cmml">1</mn><mi id="A3.p1.38.m6.1.1.3" xref="A3.p1.38.m6.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.38.m6.1b"><apply id="A3.p1.38.m6.1.1.cmml" xref="A3.p1.38.m6.1.1"><csymbol cd="ambiguous" id="A3.p1.38.m6.1.1.1.cmml" xref="A3.p1.38.m6.1.1">superscript</csymbol><apply id="A3.p1.38.m6.1.1.2.cmml" xref="A3.p1.38.m6.1.1"><csymbol cd="ambiguous" id="A3.p1.38.m6.1.1.2.1.cmml" xref="A3.p1.38.m6.1.1">subscript</csymbol><ci id="A3.p1.38.m6.1.1.2.2.cmml" xref="A3.p1.38.m6.1.1.2.2">𝑥</ci><cn id="A3.p1.38.m6.1.1.2.3.cmml" type="integer" xref="A3.p1.38.m6.1.1.2.3">1</cn></apply><ci id="A3.p1.38.m6.1.1.3.cmml" xref="A3.p1.38.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.38.m6.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.38.m6.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="z_{1}^{k}" class="ltx_Math" display="inline" id="A3.p1.39.m7.1"><semantics id="A3.p1.39.m7.1a"><msubsup id="A3.p1.39.m7.1.1" xref="A3.p1.39.m7.1.1.cmml"><mi id="A3.p1.39.m7.1.1.2.2" xref="A3.p1.39.m7.1.1.2.2.cmml">z</mi><mn id="A3.p1.39.m7.1.1.2.3" xref="A3.p1.39.m7.1.1.2.3.cmml">1</mn><mi id="A3.p1.39.m7.1.1.3" xref="A3.p1.39.m7.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.39.m7.1b"><apply id="A3.p1.39.m7.1.1.cmml" xref="A3.p1.39.m7.1.1"><csymbol cd="ambiguous" id="A3.p1.39.m7.1.1.1.cmml" xref="A3.p1.39.m7.1.1">superscript</csymbol><apply id="A3.p1.39.m7.1.1.2.cmml" xref="A3.p1.39.m7.1.1"><csymbol cd="ambiguous" id="A3.p1.39.m7.1.1.2.1.cmml" xref="A3.p1.39.m7.1.1">subscript</csymbol><ci id="A3.p1.39.m7.1.1.2.2.cmml" xref="A3.p1.39.m7.1.1.2.2">𝑧</ci><cn id="A3.p1.39.m7.1.1.2.3.cmml" type="integer" xref="A3.p1.39.m7.1.1.2.3">1</cn></apply><ci id="A3.p1.39.m7.1.1.3.cmml" xref="A3.p1.39.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.39.m7.1c">z_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.39.m7.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Divide the interval <math alttext="[0,1]" class="ltx_Math" display="inline" id="A3.p1.40.m8.2"><semantics id="A3.p1.40.m8.2a"><mrow id="A3.p1.40.m8.2.3.2" xref="A3.p1.40.m8.2.3.1.cmml"><mo id="A3.p1.40.m8.2.3.2.1" stretchy="false" xref="A3.p1.40.m8.2.3.1.cmml">[</mo><mn id="A3.p1.40.m8.1.1" xref="A3.p1.40.m8.1.1.cmml">0</mn><mo id="A3.p1.40.m8.2.3.2.2" xref="A3.p1.40.m8.2.3.1.cmml">,</mo><mn id="A3.p1.40.m8.2.2" xref="A3.p1.40.m8.2.2.cmml">1</mn><mo id="A3.p1.40.m8.2.3.2.3" stretchy="false" xref="A3.p1.40.m8.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.40.m8.2b"><interval closure="closed" id="A3.p1.40.m8.2.3.1.cmml" xref="A3.p1.40.m8.2.3.2"><cn id="A3.p1.40.m8.1.1.cmml" type="integer" xref="A3.p1.40.m8.1.1">0</cn><cn id="A3.p1.40.m8.2.2.cmml" type="integer" xref="A3.p1.40.m8.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.40.m8.2c">[0,1]</annotation><annotation encoding="application/x-llamapun" id="A3.p1.40.m8.2d">[ 0 , 1 ]</annotation></semantics></math> into <math alttext="1/\varepsilon" class="ltx_Math" display="inline" id="A3.p1.41.m9.1"><semantics id="A3.p1.41.m9.1a"><mrow id="A3.p1.41.m9.1.1" xref="A3.p1.41.m9.1.1.cmml"><mn id="A3.p1.41.m9.1.1.2" xref="A3.p1.41.m9.1.1.2.cmml">1</mn><mo id="A3.p1.41.m9.1.1.1" xref="A3.p1.41.m9.1.1.1.cmml">/</mo><mi id="A3.p1.41.m9.1.1.3" xref="A3.p1.41.m9.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.41.m9.1b"><apply id="A3.p1.41.m9.1.1.cmml" xref="A3.p1.41.m9.1.1"><divide id="A3.p1.41.m9.1.1.1.cmml" xref="A3.p1.41.m9.1.1.1"></divide><cn id="A3.p1.41.m9.1.1.2.cmml" type="integer" xref="A3.p1.41.m9.1.1.2">1</cn><ci id="A3.p1.41.m9.1.1.3.cmml" xref="A3.p1.41.m9.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.41.m9.1c">1/\varepsilon</annotation><annotation encoding="application/x-llamapun" id="A3.p1.41.m9.1d">1 / italic_ε</annotation></semantics></math> disjoint intervals of size <math alttext="\varepsilon" class="ltx_Math" display="inline" id="A3.p1.42.m10.1"><semantics id="A3.p1.42.m10.1a"><mi id="A3.p1.42.m10.1.1" xref="A3.p1.42.m10.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="A3.p1.42.m10.1b"><ci id="A3.p1.42.m10.1.1.cmml" xref="A3.p1.42.m10.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.42.m10.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="A3.p1.42.m10.1d">italic_ε</annotation></semantics></math> each. Recall that <math alttext="P(\cdot|z_{1}^{k})\sim\rho=\mathrm{Dir}(1\cdot\bm{1})" class="ltx_math_unparsed" display="inline" id="A3.p1.43.m11.1"><semantics id="A3.p1.43.m11.1a"><mrow id="A3.p1.43.m11.1b"><mi id="A3.p1.43.m11.1.1">P</mi><mrow id="A3.p1.43.m11.1.2"><mo id="A3.p1.43.m11.1.2.1" stretchy="false">(</mo><mo id="A3.p1.43.m11.1.2.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A3.p1.43.m11.1.2.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.p1.43.m11.1.2.4"><mi id="A3.p1.43.m11.1.2.4.2.2">z</mi><mn id="A3.p1.43.m11.1.2.4.2.3">1</mn><mi id="A3.p1.43.m11.1.2.4.3">k</mi></msubsup><mo id="A3.p1.43.m11.1.2.5" stretchy="false">)</mo></mrow><mo id="A3.p1.43.m11.1.3">∼</mo><mi id="A3.p1.43.m11.1.4">ρ</mi><mo id="A3.p1.43.m11.1.5">=</mo><mi id="A3.p1.43.m11.1.6">Dir</mi><mrow id="A3.p1.43.m11.1.7"><mo id="A3.p1.43.m11.1.7.1" stretchy="false">(</mo><mn id="A3.p1.43.m11.1.7.2">1</mn><mo id="A3.p1.43.m11.1.7.3" lspace="0.222em" rspace="0.222em">⋅</mo><mn id="A3.p1.43.m11.1.7.4">𝟏</mn><mo id="A3.p1.43.m11.1.7.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.p1.43.m11.1c">P(\cdot|z_{1}^{k})\sim\rho=\mathrm{Dir}(1\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="A3.p1.43.m11.1d">italic_P ( ⋅ | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) ∼ italic_ρ = roman_Dir ( 1 ⋅ bold_1 )</annotation></semantics></math>, which implies that the random variable <math alttext="P(0|z_{1}^{k})" class="ltx_Math" display="inline" id="A3.p1.44.m12.1"><semantics id="A3.p1.44.m12.1a"><mrow id="A3.p1.44.m12.1.1" xref="A3.p1.44.m12.1.1.cmml"><mi id="A3.p1.44.m12.1.1.3" xref="A3.p1.44.m12.1.1.3.cmml">P</mi><mo id="A3.p1.44.m12.1.1.2" xref="A3.p1.44.m12.1.1.2.cmml">⁢</mo><mrow id="A3.p1.44.m12.1.1.1.1" xref="A3.p1.44.m12.1.1.1.1.1.cmml"><mo id="A3.p1.44.m12.1.1.1.1.2" stretchy="false" xref="A3.p1.44.m12.1.1.1.1.1.cmml">(</mo><mrow id="A3.p1.44.m12.1.1.1.1.1" xref="A3.p1.44.m12.1.1.1.1.1.cmml"><mn id="A3.p1.44.m12.1.1.1.1.1.2" xref="A3.p1.44.m12.1.1.1.1.1.2.cmml">0</mn><mo fence="false" id="A3.p1.44.m12.1.1.1.1.1.1" xref="A3.p1.44.m12.1.1.1.1.1.1.cmml">|</mo><msubsup id="A3.p1.44.m12.1.1.1.1.1.3" xref="A3.p1.44.m12.1.1.1.1.1.3.cmml"><mi id="A3.p1.44.m12.1.1.1.1.1.3.2.2" xref="A3.p1.44.m12.1.1.1.1.1.3.2.2.cmml">z</mi><mn id="A3.p1.44.m12.1.1.1.1.1.3.2.3" xref="A3.p1.44.m12.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="A3.p1.44.m12.1.1.1.1.1.3.3" xref="A3.p1.44.m12.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A3.p1.44.m12.1.1.1.1.3" stretchy="false" xref="A3.p1.44.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.44.m12.1b"><apply id="A3.p1.44.m12.1.1.cmml" xref="A3.p1.44.m12.1.1"><times id="A3.p1.44.m12.1.1.2.cmml" xref="A3.p1.44.m12.1.1.2"></times><ci id="A3.p1.44.m12.1.1.3.cmml" xref="A3.p1.44.m12.1.1.3">𝑃</ci><apply id="A3.p1.44.m12.1.1.1.1.1.cmml" xref="A3.p1.44.m12.1.1.1.1"><csymbol cd="latexml" id="A3.p1.44.m12.1.1.1.1.1.1.cmml" xref="A3.p1.44.m12.1.1.1.1.1.1">conditional</csymbol><cn id="A3.p1.44.m12.1.1.1.1.1.2.cmml" type="integer" xref="A3.p1.44.m12.1.1.1.1.1.2">0</cn><apply id="A3.p1.44.m12.1.1.1.1.1.3.cmml" xref="A3.p1.44.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p1.44.m12.1.1.1.1.1.3.1.cmml" xref="A3.p1.44.m12.1.1.1.1.1.3">superscript</csymbol><apply id="A3.p1.44.m12.1.1.1.1.1.3.2.cmml" xref="A3.p1.44.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p1.44.m12.1.1.1.1.1.3.2.1.cmml" xref="A3.p1.44.m12.1.1.1.1.1.3">subscript</csymbol><ci id="A3.p1.44.m12.1.1.1.1.1.3.2.2.cmml" xref="A3.p1.44.m12.1.1.1.1.1.3.2.2">𝑧</ci><cn id="A3.p1.44.m12.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.p1.44.m12.1.1.1.1.1.3.2.3">1</cn></apply><ci id="A3.p1.44.m12.1.1.1.1.1.3.3.cmml" xref="A3.p1.44.m12.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.44.m12.1c">P(0|z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.p1.44.m12.1d">italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> for each fixed <math alttext="z_{1}^{k}" class="ltx_Math" display="inline" id="A3.p1.45.m13.1"><semantics id="A3.p1.45.m13.1a"><msubsup id="A3.p1.45.m13.1.1" xref="A3.p1.45.m13.1.1.cmml"><mi id="A3.p1.45.m13.1.1.2.2" xref="A3.p1.45.m13.1.1.2.2.cmml">z</mi><mn id="A3.p1.45.m13.1.1.2.3" xref="A3.p1.45.m13.1.1.2.3.cmml">1</mn><mi id="A3.p1.45.m13.1.1.3" xref="A3.p1.45.m13.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p1.45.m13.1b"><apply id="A3.p1.45.m13.1.1.cmml" xref="A3.p1.45.m13.1.1"><csymbol cd="ambiguous" id="A3.p1.45.m13.1.1.1.cmml" xref="A3.p1.45.m13.1.1">superscript</csymbol><apply id="A3.p1.45.m13.1.1.2.cmml" xref="A3.p1.45.m13.1.1"><csymbol cd="ambiguous" id="A3.p1.45.m13.1.1.2.1.cmml" xref="A3.p1.45.m13.1.1">subscript</csymbol><ci id="A3.p1.45.m13.1.1.2.2.cmml" xref="A3.p1.45.m13.1.1.2.2">𝑧</ci><cn id="A3.p1.45.m13.1.1.2.3.cmml" type="integer" xref="A3.p1.45.m13.1.1.2.3">1</cn></apply><ci id="A3.p1.45.m13.1.1.3.cmml" xref="A3.p1.45.m13.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.45.m13.1c">z_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.45.m13.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> (randomness is over <math alttext="P" class="ltx_Math" display="inline" id="A3.p1.46.m14.1"><semantics id="A3.p1.46.m14.1a"><mi id="A3.p1.46.m14.1.1" xref="A3.p1.46.m14.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p1.46.m14.1b"><ci id="A3.p1.46.m14.1.1.cmml" xref="A3.p1.46.m14.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.46.m14.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p1.46.m14.1d">italic_P</annotation></semantics></math>) is distributed as,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx21"> <tbody id="A3.E41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Pr_{\rho}\left[P(0|z_{1}^{k})=\cdot\middle|z_{1}^{k}\right]=% \mathrm{Unif}([0,1])." class="ltx_math_unparsed" display="inline" id="A3.E41.m1.1"><semantics id="A3.E41.m1.1a"><mrow id="A3.E41.m1.1b"><munder id="A3.E41.m1.1.1"><mi id="A3.E41.m1.1.1.2">Pr</mi><mi id="A3.E41.m1.1.1.3">ρ</mi></munder><mrow id="A3.E41.m1.1.2"><mo id="A3.E41.m1.1.2.1">[</mo><mi id="A3.E41.m1.1.2.2">P</mi><mrow id="A3.E41.m1.1.2.3"><mo id="A3.E41.m1.1.2.3.1" stretchy="false">(</mo><mn id="A3.E41.m1.1.2.3.2">0</mn><mo fence="false" id="A3.E41.m1.1.2.3.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.E41.m1.1.2.3.4"><mi id="A3.E41.m1.1.2.3.4.2.2">z</mi><mn id="A3.E41.m1.1.2.3.4.2.3">1</mn><mi id="A3.E41.m1.1.2.3.4.3">k</mi></msubsup><mo id="A3.E41.m1.1.2.3.5" stretchy="false">)</mo></mrow><mo id="A3.E41.m1.1.2.4" rspace="0em">=</mo><mo id="A3.E41.m1.1.2.5" lspace="0em">⋅</mo><mo id="A3.E41.m1.1.2.6" lspace="0.055em" rspace="0em" stretchy="true">|</mo><msubsup id="A3.E41.m1.1.2.7"><mi id="A3.E41.m1.1.2.7.2.2">z</mi><mn id="A3.E41.m1.1.2.7.2.3">1</mn><mi id="A3.E41.m1.1.2.7.3">k</mi></msubsup><mo id="A3.E41.m1.1.2.8">]</mo></mrow><mo id="A3.E41.m1.1.3">=</mo><mi id="A3.E41.m1.1.4">Unif</mi><mrow id="A3.E41.m1.1.5"><mo id="A3.E41.m1.1.5.1" stretchy="false">(</mo><mrow id="A3.E41.m1.1.5.2"><mo id="A3.E41.m1.1.5.2.1" stretchy="false">[</mo><mn id="A3.E41.m1.1.5.2.2">0</mn><mo id="A3.E41.m1.1.5.2.3">,</mo><mn id="A3.E41.m1.1.5.2.4">1</mn><mo id="A3.E41.m1.1.5.2.5" stretchy="false">]</mo></mrow><mo id="A3.E41.m1.1.5.3" stretchy="false">)</mo></mrow><mo id="A3.E41.m1.1.6" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="A3.E41.m1.1c">\displaystyle\Pr_{\rho}\left[P(0|z_{1}^{k})=\cdot\middle|z_{1}^{k}\right]=% \mathrm{Unif}([0,1]).</annotation><annotation encoding="application/x-llamapun" id="A3.E41.m1.1d">roman_Pr start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT [ italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = ⋅ | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ] = roman_Unif ( [ 0 , 1 ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(41)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.50">Consider the buckets <math alttext="B_{\varepsilon}=\{[0,\varepsilon),[\varepsilon,2\varepsilon),\cdots,[1-% \varepsilon,1]\}" class="ltx_Math" display="inline" id="A3.p1.47.m1.8"><semantics id="A3.p1.47.m1.8a"><mrow id="A3.p1.47.m1.8.8" xref="A3.p1.47.m1.8.8.cmml"><msub id="A3.p1.47.m1.8.8.5" xref="A3.p1.47.m1.8.8.5.cmml"><mi id="A3.p1.47.m1.8.8.5.2" xref="A3.p1.47.m1.8.8.5.2.cmml">B</mi><mi id="A3.p1.47.m1.8.8.5.3" xref="A3.p1.47.m1.8.8.5.3.cmml">ε</mi></msub><mo id="A3.p1.47.m1.8.8.4" xref="A3.p1.47.m1.8.8.4.cmml">=</mo><mrow id="A3.p1.47.m1.8.8.3.3" xref="A3.p1.47.m1.8.8.3.4.cmml"><mo id="A3.p1.47.m1.8.8.3.3.4" stretchy="false" xref="A3.p1.47.m1.8.8.3.4.cmml">{</mo><mrow id="A3.p1.47.m1.6.6.1.1.1.2" xref="A3.p1.47.m1.6.6.1.1.1.1.cmml"><mo id="A3.p1.47.m1.6.6.1.1.1.2.1" stretchy="false" xref="A3.p1.47.m1.6.6.1.1.1.1.cmml">[</mo><mn id="A3.p1.47.m1.1.1" xref="A3.p1.47.m1.1.1.cmml">0</mn><mo id="A3.p1.47.m1.6.6.1.1.1.2.2" xref="A3.p1.47.m1.6.6.1.1.1.1.cmml">,</mo><mi id="A3.p1.47.m1.2.2" xref="A3.p1.47.m1.2.2.cmml">ε</mi><mo id="A3.p1.47.m1.6.6.1.1.1.2.3" stretchy="false" xref="A3.p1.47.m1.6.6.1.1.1.1.cmml">)</mo></mrow><mo id="A3.p1.47.m1.8.8.3.3.5" xref="A3.p1.47.m1.8.8.3.4.cmml">,</mo><mrow id="A3.p1.47.m1.7.7.2.2.2.1" xref="A3.p1.47.m1.7.7.2.2.2.2.cmml"><mo id="A3.p1.47.m1.7.7.2.2.2.1.2" stretchy="false" xref="A3.p1.47.m1.7.7.2.2.2.2.cmml">[</mo><mi id="A3.p1.47.m1.3.3" xref="A3.p1.47.m1.3.3.cmml">ε</mi><mo id="A3.p1.47.m1.7.7.2.2.2.1.3" xref="A3.p1.47.m1.7.7.2.2.2.2.cmml">,</mo><mrow id="A3.p1.47.m1.7.7.2.2.2.1.1" xref="A3.p1.47.m1.7.7.2.2.2.1.1.cmml"><mn id="A3.p1.47.m1.7.7.2.2.2.1.1.2" xref="A3.p1.47.m1.7.7.2.2.2.1.1.2.cmml">2</mn><mo id="A3.p1.47.m1.7.7.2.2.2.1.1.1" xref="A3.p1.47.m1.7.7.2.2.2.1.1.1.cmml">⁢</mo><mi id="A3.p1.47.m1.7.7.2.2.2.1.1.3" xref="A3.p1.47.m1.7.7.2.2.2.1.1.3.cmml">ε</mi></mrow><mo id="A3.p1.47.m1.7.7.2.2.2.1.4" stretchy="false" xref="A3.p1.47.m1.7.7.2.2.2.2.cmml">)</mo></mrow><mo id="A3.p1.47.m1.8.8.3.3.6" xref="A3.p1.47.m1.8.8.3.4.cmml">,</mo><mi id="A3.p1.47.m1.5.5" mathvariant="normal" xref="A3.p1.47.m1.5.5.cmml">⋯</mi><mo id="A3.p1.47.m1.8.8.3.3.7" xref="A3.p1.47.m1.8.8.3.4.cmml">,</mo><mrow id="A3.p1.47.m1.8.8.3.3.3.1" xref="A3.p1.47.m1.8.8.3.3.3.2.cmml"><mo id="A3.p1.47.m1.8.8.3.3.3.1.2" stretchy="false" xref="A3.p1.47.m1.8.8.3.3.3.2.cmml">[</mo><mrow id="A3.p1.47.m1.8.8.3.3.3.1.1" xref="A3.p1.47.m1.8.8.3.3.3.1.1.cmml"><mn id="A3.p1.47.m1.8.8.3.3.3.1.1.2" xref="A3.p1.47.m1.8.8.3.3.3.1.1.2.cmml">1</mn><mo id="A3.p1.47.m1.8.8.3.3.3.1.1.1" xref="A3.p1.47.m1.8.8.3.3.3.1.1.1.cmml">−</mo><mi id="A3.p1.47.m1.8.8.3.3.3.1.1.3" xref="A3.p1.47.m1.8.8.3.3.3.1.1.3.cmml">ε</mi></mrow><mo id="A3.p1.47.m1.8.8.3.3.3.1.3" xref="A3.p1.47.m1.8.8.3.3.3.2.cmml">,</mo><mn id="A3.p1.47.m1.4.4" xref="A3.p1.47.m1.4.4.cmml">1</mn><mo id="A3.p1.47.m1.8.8.3.3.3.1.4" stretchy="false" xref="A3.p1.47.m1.8.8.3.3.3.2.cmml">]</mo></mrow><mo id="A3.p1.47.m1.8.8.3.3.8" stretchy="false" xref="A3.p1.47.m1.8.8.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" 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id="A3.p1.47.m1.7.7.2.2.2.1.1.1.cmml" xref="A3.p1.47.m1.7.7.2.2.2.1.1.1"></times><cn id="A3.p1.47.m1.7.7.2.2.2.1.1.2.cmml" type="integer" xref="A3.p1.47.m1.7.7.2.2.2.1.1.2">2</cn><ci id="A3.p1.47.m1.7.7.2.2.2.1.1.3.cmml" xref="A3.p1.47.m1.7.7.2.2.2.1.1.3">𝜀</ci></apply></interval><ci id="A3.p1.47.m1.5.5.cmml" xref="A3.p1.47.m1.5.5">⋯</ci><interval closure="closed" id="A3.p1.47.m1.8.8.3.3.3.2.cmml" xref="A3.p1.47.m1.8.8.3.3.3.1"><apply id="A3.p1.47.m1.8.8.3.3.3.1.1.cmml" xref="A3.p1.47.m1.8.8.3.3.3.1.1"><minus id="A3.p1.47.m1.8.8.3.3.3.1.1.1.cmml" xref="A3.p1.47.m1.8.8.3.3.3.1.1.1"></minus><cn id="A3.p1.47.m1.8.8.3.3.3.1.1.2.cmml" type="integer" xref="A3.p1.47.m1.8.8.3.3.3.1.1.2">1</cn><ci id="A3.p1.47.m1.8.8.3.3.3.1.1.3.cmml" xref="A3.p1.47.m1.8.8.3.3.3.1.1.3">𝜀</ci></apply><cn id="A3.p1.47.m1.4.4.cmml" type="integer" xref="A3.p1.47.m1.4.4">1</cn></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.47.m1.8c">B_{\varepsilon}=\{[0,\varepsilon),[\varepsilon,2\varepsilon),\cdots,[1-% \varepsilon,1]\}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.47.m1.8d">italic_B start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT = { [ 0 , italic_ε ) , [ italic_ε , 2 italic_ε ) , ⋯ , [ 1 - italic_ε , 1 ] }</annotation></semantics></math>. Define the function <math alttext="\mathrm{round}(p):[0,1]\to\{0,\cdots,|B_{\varepsilon}|-1\}" class="ltx_Math" display="inline" id="A3.p1.48.m2.6"><semantics id="A3.p1.48.m2.6a"><mrow id="A3.p1.48.m2.6.6" xref="A3.p1.48.m2.6.6.cmml"><mrow id="A3.p1.48.m2.6.6.3" xref="A3.p1.48.m2.6.6.3.cmml"><mi id="A3.p1.48.m2.6.6.3.2" xref="A3.p1.48.m2.6.6.3.2.cmml">round</mi><mo id="A3.p1.48.m2.6.6.3.1" xref="A3.p1.48.m2.6.6.3.1.cmml">⁢</mo><mrow id="A3.p1.48.m2.6.6.3.3.2" xref="A3.p1.48.m2.6.6.3.cmml"><mo id="A3.p1.48.m2.6.6.3.3.2.1" stretchy="false" xref="A3.p1.48.m2.6.6.3.cmml">(</mo><mi id="A3.p1.48.m2.1.1" xref="A3.p1.48.m2.1.1.cmml">p</mi><mo id="A3.p1.48.m2.6.6.3.3.2.2" rspace="0.278em" stretchy="false" xref="A3.p1.48.m2.6.6.3.cmml">)</mo></mrow></mrow><mo id="A3.p1.48.m2.6.6.2" rspace="0.278em" xref="A3.p1.48.m2.6.6.2.cmml">:</mo><mrow id="A3.p1.48.m2.6.6.1" xref="A3.p1.48.m2.6.6.1.cmml"><mrow id="A3.p1.48.m2.6.6.1.3.2" xref="A3.p1.48.m2.6.6.1.3.1.cmml"><mo id="A3.p1.48.m2.6.6.1.3.2.1" stretchy="false" xref="A3.p1.48.m2.6.6.1.3.1.cmml">[</mo><mn id="A3.p1.48.m2.2.2" xref="A3.p1.48.m2.2.2.cmml">0</mn><mo id="A3.p1.48.m2.6.6.1.3.2.2" xref="A3.p1.48.m2.6.6.1.3.1.cmml">,</mo><mn id="A3.p1.48.m2.3.3" xref="A3.p1.48.m2.3.3.cmml">1</mn><mo id="A3.p1.48.m2.6.6.1.3.2.3" stretchy="false" xref="A3.p1.48.m2.6.6.1.3.1.cmml">]</mo></mrow><mo id="A3.p1.48.m2.6.6.1.2" stretchy="false" xref="A3.p1.48.m2.6.6.1.2.cmml">→</mo><mrow id="A3.p1.48.m2.6.6.1.1.1" xref="A3.p1.48.m2.6.6.1.1.2.cmml"><mo id="A3.p1.48.m2.6.6.1.1.1.2" stretchy="false" xref="A3.p1.48.m2.6.6.1.1.2.cmml">{</mo><mn id="A3.p1.48.m2.4.4" xref="A3.p1.48.m2.4.4.cmml">0</mn><mo id="A3.p1.48.m2.6.6.1.1.1.3" xref="A3.p1.48.m2.6.6.1.1.2.cmml">,</mo><mi id="A3.p1.48.m2.5.5" mathvariant="normal" xref="A3.p1.48.m2.5.5.cmml">⋯</mi><mo id="A3.p1.48.m2.6.6.1.1.1.4" xref="A3.p1.48.m2.6.6.1.1.2.cmml">,</mo><mrow id="A3.p1.48.m2.6.6.1.1.1.1" xref="A3.p1.48.m2.6.6.1.1.1.1.cmml"><mrow id="A3.p1.48.m2.6.6.1.1.1.1.1.1" xref="A3.p1.48.m2.6.6.1.1.1.1.1.2.cmml"><mo id="A3.p1.48.m2.6.6.1.1.1.1.1.1.2" stretchy="false" xref="A3.p1.48.m2.6.6.1.1.1.1.1.2.1.cmml">|</mo><msub id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.cmml"><mi id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.2" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.2.cmml">B</mi><mi id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.3" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.3.cmml">ε</mi></msub><mo id="A3.p1.48.m2.6.6.1.1.1.1.1.1.3" stretchy="false" xref="A3.p1.48.m2.6.6.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="A3.p1.48.m2.6.6.1.1.1.1.2" xref="A3.p1.48.m2.6.6.1.1.1.1.2.cmml">−</mo><mn id="A3.p1.48.m2.6.6.1.1.1.1.3" xref="A3.p1.48.m2.6.6.1.1.1.1.3.cmml">1</mn></mrow><mo id="A3.p1.48.m2.6.6.1.1.1.5" stretchy="false" xref="A3.p1.48.m2.6.6.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.48.m2.6b"><apply id="A3.p1.48.m2.6.6.cmml" xref="A3.p1.48.m2.6.6"><ci id="A3.p1.48.m2.6.6.2.cmml" xref="A3.p1.48.m2.6.6.2">:</ci><apply id="A3.p1.48.m2.6.6.3.cmml" xref="A3.p1.48.m2.6.6.3"><times id="A3.p1.48.m2.6.6.3.1.cmml" xref="A3.p1.48.m2.6.6.3.1"></times><ci id="A3.p1.48.m2.6.6.3.2.cmml" xref="A3.p1.48.m2.6.6.3.2">round</ci><ci id="A3.p1.48.m2.1.1.cmml" xref="A3.p1.48.m2.1.1">𝑝</ci></apply><apply id="A3.p1.48.m2.6.6.1.cmml" xref="A3.p1.48.m2.6.6.1"><ci id="A3.p1.48.m2.6.6.1.2.cmml" xref="A3.p1.48.m2.6.6.1.2">→</ci><interval closure="closed" id="A3.p1.48.m2.6.6.1.3.1.cmml" xref="A3.p1.48.m2.6.6.1.3.2"><cn id="A3.p1.48.m2.2.2.cmml" type="integer" xref="A3.p1.48.m2.2.2">0</cn><cn id="A3.p1.48.m2.3.3.cmml" type="integer" xref="A3.p1.48.m2.3.3">1</cn></interval><set id="A3.p1.48.m2.6.6.1.1.2.cmml" xref="A3.p1.48.m2.6.6.1.1.1"><cn id="A3.p1.48.m2.4.4.cmml" type="integer" xref="A3.p1.48.m2.4.4">0</cn><ci id="A3.p1.48.m2.5.5.cmml" xref="A3.p1.48.m2.5.5">⋯</ci><apply id="A3.p1.48.m2.6.6.1.1.1.1.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1"><minus id="A3.p1.48.m2.6.6.1.1.1.1.2.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.2"></minus><apply id="A3.p1.48.m2.6.6.1.1.1.1.1.2.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1"><abs id="A3.p1.48.m2.6.6.1.1.1.1.1.2.1.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.2"></abs><apply id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.1.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.2.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.2">𝐵</ci><ci id="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.3.cmml" xref="A3.p1.48.m2.6.6.1.1.1.1.1.1.1.3">𝜀</ci></apply></apply><cn id="A3.p1.48.m2.6.6.1.1.1.1.3.cmml" type="integer" xref="A3.p1.48.m2.6.6.1.1.1.1.3">1</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.48.m2.6c">\mathrm{round}(p):[0,1]\to\{0,\cdots,|B_{\varepsilon}|-1\}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.48.m2.6d">roman_round ( italic_p ) : [ 0 , 1 ] → { 0 , ⋯ , | italic_B start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT | - 1 }</annotation></semantics></math> to return the index of the bucket in <math alttext="B_{\varepsilon}" class="ltx_Math" display="inline" id="A3.p1.49.m3.1"><semantics id="A3.p1.49.m3.1a"><msub id="A3.p1.49.m3.1.1" xref="A3.p1.49.m3.1.1.cmml"><mi id="A3.p1.49.m3.1.1.2" xref="A3.p1.49.m3.1.1.2.cmml">B</mi><mi id="A3.p1.49.m3.1.1.3" xref="A3.p1.49.m3.1.1.3.cmml">ε</mi></msub><annotation-xml encoding="MathML-Content" id="A3.p1.49.m3.1b"><apply id="A3.p1.49.m3.1.1.cmml" xref="A3.p1.49.m3.1.1"><csymbol cd="ambiguous" id="A3.p1.49.m3.1.1.1.cmml" xref="A3.p1.49.m3.1.1">subscript</csymbol><ci id="A3.p1.49.m3.1.1.2.cmml" xref="A3.p1.49.m3.1.1.2">𝐵</ci><ci id="A3.p1.49.m3.1.1.3.cmml" xref="A3.p1.49.m3.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.49.m3.1c">B_{\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.49.m3.1d">italic_B start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="p" class="ltx_Math" display="inline" id="A3.p1.50.m4.1"><semantics id="A3.p1.50.m4.1a"><mi id="A3.p1.50.m4.1.1" xref="A3.p1.50.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.p1.50.m4.1b"><ci id="A3.p1.50.m4.1.1.cmml" xref="A3.p1.50.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.50.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.p1.50.m4.1d">italic_p</annotation></semantics></math> falls in that bucket.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmlemma1.1.1.1">Lemma 1</span></span><span class="ltx_text ltx_font_bold" id="Thmlemma1.2.2">.</span> </h6> <div class="ltx_para" id="Thmlemma1.p1"> <p class="ltx_p" id="Thmlemma1.p1.1"><span class="ltx_text ltx_font_italic" id="Thmlemma1.p1.1.1">Consider any function <math alttext="f(z_{1}^{k}):\mathcal{X}^{k}\to\{0,\cdots,|B_{\varepsilon}|-1\}" class="ltx_Math" display="inline" id="Thmlemma1.p1.1.1.m1.4"><semantics id="Thmlemma1.p1.1.1.m1.4a"><mrow id="Thmlemma1.p1.1.1.m1.4.4" xref="Thmlemma1.p1.1.1.m1.4.4.cmml"><mrow id="Thmlemma1.p1.1.1.m1.3.3.1" xref="Thmlemma1.p1.1.1.m1.3.3.1.cmml"><mi id="Thmlemma1.p1.1.1.m1.3.3.1.3" xref="Thmlemma1.p1.1.1.m1.3.3.1.3.cmml">f</mi><mo id="Thmlemma1.p1.1.1.m1.3.3.1.2" xref="Thmlemma1.p1.1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="Thmlemma1.p1.1.1.m1.3.3.1.1.1" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.2" stretchy="false" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.cmml">(</mo><msubsup id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.2" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.2.cmml">z</mi><mn id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.3" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.3.cmml">1</mn><mi id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.3" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.3.cmml">k</mi></msubsup><mo id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.3" rspace="0.278em" stretchy="false" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma1.p1.1.1.m1.4.4.3" rspace="0.278em" xref="Thmlemma1.p1.1.1.m1.4.4.3.cmml">:</mo><mrow id="Thmlemma1.p1.1.1.m1.4.4.2" xref="Thmlemma1.p1.1.1.m1.4.4.2.cmml"><msup id="Thmlemma1.p1.1.1.m1.4.4.2.3" xref="Thmlemma1.p1.1.1.m1.4.4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmlemma1.p1.1.1.m1.4.4.2.3.2" xref="Thmlemma1.p1.1.1.m1.4.4.2.3.2.cmml">𝒳</mi><mi id="Thmlemma1.p1.1.1.m1.4.4.2.3.3" xref="Thmlemma1.p1.1.1.m1.4.4.2.3.3.cmml">k</mi></msup><mo id="Thmlemma1.p1.1.1.m1.4.4.2.2" stretchy="false" xref="Thmlemma1.p1.1.1.m1.4.4.2.2.cmml">→</mo><mrow id="Thmlemma1.p1.1.1.m1.4.4.2.1.1" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.2.cmml"><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.2" stretchy="false" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.2.cmml">{</mo><mn id="Thmlemma1.p1.1.1.m1.1.1" xref="Thmlemma1.p1.1.1.m1.1.1.cmml">0</mn><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.3" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.2.cmml">,</mo><mi id="Thmlemma1.p1.1.1.m1.2.2" mathvariant="normal" xref="Thmlemma1.p1.1.1.m1.2.2.cmml">⋯</mi><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.4" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.2.cmml">,</mo><mrow id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.cmml"><mrow id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.2.cmml"><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.2" stretchy="false" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.2.1.cmml">|</mo><msub id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.cmml"><mi id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.2" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.2.cmml">B</mi><mi id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.3" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.3.cmml">ε</mi></msub><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.3" stretchy="false" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.2" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.2.cmml">−</mo><mn id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.3" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.3.cmml">1</mn></mrow><mo id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.5" stretchy="false" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.1.1.m1.4b"><apply id="Thmlemma1.p1.1.1.m1.4.4.cmml" xref="Thmlemma1.p1.1.1.m1.4.4"><ci id="Thmlemma1.p1.1.1.m1.4.4.3.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.3">:</ci><apply id="Thmlemma1.p1.1.1.m1.3.3.1.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1"><times id="Thmlemma1.p1.1.1.m1.3.3.1.2.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.2"></times><ci id="Thmlemma1.p1.1.1.m1.3.3.1.3.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.3">𝑓</ci><apply id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1">superscript</csymbol><apply id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.1.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1">subscript</csymbol><ci id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.2.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.2">𝑧</ci><cn id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.3.cmml" type="integer" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.2.3">1</cn></apply><ci id="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.3.cmml" xref="Thmlemma1.p1.1.1.m1.3.3.1.1.1.1.3">𝑘</ci></apply></apply><apply id="Thmlemma1.p1.1.1.m1.4.4.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2"><ci id="Thmlemma1.p1.1.1.m1.4.4.2.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.2">→</ci><apply id="Thmlemma1.p1.1.1.m1.4.4.2.3.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.3"><csymbol cd="ambiguous" id="Thmlemma1.p1.1.1.m1.4.4.2.3.1.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.3">superscript</csymbol><ci id="Thmlemma1.p1.1.1.m1.4.4.2.3.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.3.2">𝒳</ci><ci id="Thmlemma1.p1.1.1.m1.4.4.2.3.3.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.3.3">𝑘</ci></apply><set id="Thmlemma1.p1.1.1.m1.4.4.2.1.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1"><cn id="Thmlemma1.p1.1.1.m1.1.1.cmml" type="integer" xref="Thmlemma1.p1.1.1.m1.1.1">0</cn><ci id="Thmlemma1.p1.1.1.m1.2.2.cmml" xref="Thmlemma1.p1.1.1.m1.2.2">⋯</ci><apply id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1"><minus id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.2"></minus><apply id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1"><abs id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.2.1.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.2"></abs><apply id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.1.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.2.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.2">𝐵</ci><ci id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.3.cmml" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.1.1.1.3">𝜀</ci></apply></apply><cn id="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.3.cmml" type="integer" xref="Thmlemma1.p1.1.1.m1.4.4.2.1.1.1.3">1</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.1.1.m1.4c">f(z_{1}^{k}):\mathcal{X}^{k}\to\{0,\cdots,|B_{\varepsilon}|-1\}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.1.1.m1.4d">italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT → { 0 , ⋯ , | italic_B start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT | - 1 }</annotation></semantics></math> such that, pointwise,</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx22"> <tbody id="A3.E42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|\mathrm{round}(P(0|z_{1}^{k}))-f(z_{1}^{k})|\leq r." class="ltx_math_unparsed" display="inline" id="A3.E42.m1.1"><semantics id="A3.E42.m1.1a"><mrow id="A3.E42.m1.1b"><mo fence="false" id="A3.E42.m1.1.1" rspace="0.167em" stretchy="false">|</mo><mi id="A3.E42.m1.1.2">round</mi><mrow id="A3.E42.m1.1.3"><mo id="A3.E42.m1.1.3.1" stretchy="false">(</mo><mi id="A3.E42.m1.1.3.2">P</mi><mrow id="A3.E42.m1.1.3.3"><mo id="A3.E42.m1.1.3.3.1" stretchy="false">(</mo><mn id="A3.E42.m1.1.3.3.2">0</mn><mo fence="false" id="A3.E42.m1.1.3.3.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.E42.m1.1.3.3.4"><mi id="A3.E42.m1.1.3.3.4.2.2">z</mi><mn id="A3.E42.m1.1.3.3.4.2.3">1</mn><mi id="A3.E42.m1.1.3.3.4.3">k</mi></msubsup><mo id="A3.E42.m1.1.3.3.5" stretchy="false">)</mo></mrow><mo id="A3.E42.m1.1.3.4" stretchy="false">)</mo></mrow><mo id="A3.E42.m1.1.4">−</mo><mi id="A3.E42.m1.1.5">f</mi><mrow id="A3.E42.m1.1.6"><mo id="A3.E42.m1.1.6.1" stretchy="false">(</mo><msubsup id="A3.E42.m1.1.6.2"><mi id="A3.E42.m1.1.6.2.2.2">z</mi><mn id="A3.E42.m1.1.6.2.2.3">1</mn><mi id="A3.E42.m1.1.6.2.3">k</mi></msubsup><mo id="A3.E42.m1.1.6.3" stretchy="false">)</mo></mrow><mo fence="false" id="A3.E42.m1.1.7" stretchy="false">|</mo><mo id="A3.E42.m1.1.8" lspace="0.167em">≤</mo><mi id="A3.E42.m1.1.9">r</mi><mo id="A3.E42.m1.1.10" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="A3.E42.m1.1c">\displaystyle|\mathrm{round}(P(0|z_{1}^{k}))-f(z_{1}^{k})|\leq r.</annotation><annotation encoding="application/x-llamapun" id="A3.E42.m1.1d">| roman_round ( italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) ) - italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) | ≤ italic_r .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(42)</span></td> </tr></tbody> </table> <p class="ltx_p" id="Thmlemma1.p1.2"><span class="ltx_text ltx_font_italic" id="Thmlemma1.p1.2.1">Then, when <math alttext="P(0|z_{1}^{k})\overset{\text{i.i.d.}}{\sim}\mathrm{Dir}(1\cdot\bm{1})" class="ltx_Math" display="inline" id="Thmlemma1.p1.2.1.m1.2"><semantics id="Thmlemma1.p1.2.1.m1.2a"><mrow id="Thmlemma1.p1.2.1.m1.2.2" xref="Thmlemma1.p1.2.1.m1.2.2.cmml"><mi id="Thmlemma1.p1.2.1.m1.2.2.4" xref="Thmlemma1.p1.2.1.m1.2.2.4.cmml">P</mi><mo id="Thmlemma1.p1.2.1.m1.2.2.3" xref="Thmlemma1.p1.2.1.m1.2.2.3.cmml">⁢</mo><mrow id="Thmlemma1.p1.2.1.m1.1.1.1.1" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.cmml"><mo id="Thmlemma1.p1.2.1.m1.1.1.1.1.2" stretchy="false" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="Thmlemma1.p1.2.1.m1.1.1.1.1.1" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.cmml"><mn id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.2" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.2.cmml">0</mn><mo fence="false" id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.1" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.1.cmml">|</mo><msubsup id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.cmml"><mi id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.2" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.2.cmml">z</mi><mn id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.3" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.3" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="Thmlemma1.p1.2.1.m1.1.1.1.1.3" stretchy="false" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmlemma1.p1.2.1.m1.2.2.3a" xref="Thmlemma1.p1.2.1.m1.2.2.3.cmml">⁢</mo><mover accent="true" id="Thmlemma1.p1.2.1.m1.2.2.5" xref="Thmlemma1.p1.2.1.m1.2.2.5.cmml"><mo id="Thmlemma1.p1.2.1.m1.2.2.5.2" xref="Thmlemma1.p1.2.1.m1.2.2.5.2.cmml">∼</mo><mtext class="ltx_mathvariant_italic" id="Thmlemma1.p1.2.1.m1.2.2.5.1" xref="Thmlemma1.p1.2.1.m1.2.2.5.1a.cmml">i.i.d.</mtext></mover><mo id="Thmlemma1.p1.2.1.m1.2.2.3b" xref="Thmlemma1.p1.2.1.m1.2.2.3.cmml">⁢</mo><mi id="Thmlemma1.p1.2.1.m1.2.2.6" xref="Thmlemma1.p1.2.1.m1.2.2.6.cmml">Dir</mi><mo id="Thmlemma1.p1.2.1.m1.2.2.3c" xref="Thmlemma1.p1.2.1.m1.2.2.3.cmml">⁢</mo><mrow id="Thmlemma1.p1.2.1.m1.2.2.2.1" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.cmml"><mo id="Thmlemma1.p1.2.1.m1.2.2.2.1.2" stretchy="false" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.cmml">(</mo><mrow id="Thmlemma1.p1.2.1.m1.2.2.2.1.1" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.cmml"><mn id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.2" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.2.cmml">1</mn><mo id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.1" lspace="0.222em" rspace="0.222em" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.1.cmml">⋅</mo><mn id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.3" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.3.cmml">𝟏</mn></mrow><mo id="Thmlemma1.p1.2.1.m1.2.2.2.1.3" stretchy="false" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.2.1.m1.2b"><apply id="Thmlemma1.p1.2.1.m1.2.2.cmml" xref="Thmlemma1.p1.2.1.m1.2.2"><times id="Thmlemma1.p1.2.1.m1.2.2.3.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.3"></times><ci id="Thmlemma1.p1.2.1.m1.2.2.4.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.4">𝑃</ci><apply id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1"><csymbol cd="latexml" id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.1.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.1">conditional</csymbol><cn id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.2">0</cn><apply id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.1.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3">superscript</csymbol><apply id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.1.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.2.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.2">𝑧</ci><cn id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.2.3">1</cn></apply><ci id="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.3.cmml" xref="Thmlemma1.p1.2.1.m1.1.1.1.1.1.3.3">𝑘</ci></apply></apply><apply id="Thmlemma1.p1.2.1.m1.2.2.5.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.5"><ci id="Thmlemma1.p1.2.1.m1.2.2.5.1a.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.5.1"><mtext class="ltx_mathvariant_italic" id="Thmlemma1.p1.2.1.m1.2.2.5.1.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.5.1">i.i.d.</mtext></ci><csymbol cd="latexml" id="Thmlemma1.p1.2.1.m1.2.2.5.2.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.5.2">similar-to</csymbol></apply><ci id="Thmlemma1.p1.2.1.m1.2.2.6.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.6">Dir</ci><apply id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.2.1"><ci id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.1.cmml" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.1">⋅</ci><cn id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.2.cmml" type="integer" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.2">1</cn><cn id="Thmlemma1.p1.2.1.m1.2.2.2.1.1.3.cmml" type="integer" xref="Thmlemma1.p1.2.1.m1.2.2.2.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.2.1.m1.2c">P(0|z_{1}^{k})\overset{\text{i.i.d.}}{\sim}\mathrm{Dir}(1\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.2.1.m1.2d">italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) overi.i.d. start_ARG ∼ end_ARG roman_Dir ( 1 ⋅ bold_1 )</annotation></semantics></math>,</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx23"> <tbody id="A3.E43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\text{Shannon}}(\{f(z_{1}^{k}):z_{1}^{k}\in\{0,1\}^{k}\})\geq 2% ^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-\log(2r+1)\right)" class="ltx_Math" display="inline" id="A3.E43.m1.6"><semantics id="A3.E43.m1.6a"><mrow id="A3.E43.m1.6.6" xref="A3.E43.m1.6.6.cmml"><mrow id="A3.E43.m1.5.5.1" xref="A3.E43.m1.5.5.1.cmml"><msub id="A3.E43.m1.5.5.1.3" xref="A3.E43.m1.5.5.1.3.cmml"><mi id="A3.E43.m1.5.5.1.3.2" xref="A3.E43.m1.5.5.1.3.2.cmml">H</mi><mtext class="ltx_mathvariant_italic" id="A3.E43.m1.5.5.1.3.3" 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xref="A3.E43.m1.6.6.2.1.1.1.3.1.1.1.2.2">2</cn><ci id="A3.E43.m1.6.6.2.1.1.1.3.1.1.1.2.3.cmml" xref="A3.E43.m1.6.6.2.1.1.1.3.1.1.1.2.3">𝑟</ci></apply><cn id="A3.E43.m1.6.6.2.1.1.1.3.1.1.1.3.cmml" type="integer" xref="A3.E43.m1.6.6.2.1.1.1.3.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E43.m1.6c">\displaystyle H_{\text{Shannon}}(\{f(z_{1}^{k}):z_{1}^{k}\in\{0,1\}^{k}\})\geq 2% ^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-\log(2r+1)\right)</annotation><annotation encoding="application/x-llamapun" id="A3.E43.m1.6d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) ≥ 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( ( 1 - 3 italic_ε ) roman_log ( 1 / italic_ε ) - roman_log ( 2 italic_r + 1 ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(43)</span></td> </tr></tbody> </table> <p class="ltx_p" id="Thmlemma1.p1.4"><span class="ltx_text ltx_font_italic" id="Thmlemma1.p1.4.2">where the randomness is over the draw of <math alttext="P" class="ltx_Math" display="inline" id="Thmlemma1.p1.3.1.m1.1"><semantics id="Thmlemma1.p1.3.1.m1.1a"><mi id="Thmlemma1.p1.3.1.m1.1.1" xref="Thmlemma1.p1.3.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.3.1.m1.1b"><ci id="Thmlemma1.p1.3.1.m1.1.1.cmml" xref="Thmlemma1.p1.3.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.3.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.3.1.m1.1d">italic_P</annotation></semantics></math> and <math alttext="H_{\text{Shannon}}" class="ltx_Math" display="inline" id="Thmlemma1.p1.4.2.m2.1"><semantics id="Thmlemma1.p1.4.2.m2.1a"><msub id="Thmlemma1.p1.4.2.m2.1.1" xref="Thmlemma1.p1.4.2.m2.1.1.cmml"><mi id="Thmlemma1.p1.4.2.m2.1.1.2" xref="Thmlemma1.p1.4.2.m2.1.1.2.cmml">H</mi><mtext class="ltx_mathvariant_italic" id="Thmlemma1.p1.4.2.m2.1.1.3" xref="Thmlemma1.p1.4.2.m2.1.1.3a.cmml">Shannon</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.4.2.m2.1b"><apply id="Thmlemma1.p1.4.2.m2.1.1.cmml" xref="Thmlemma1.p1.4.2.m2.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.4.2.m2.1.1.1.cmml" xref="Thmlemma1.p1.4.2.m2.1.1">subscript</csymbol><ci id="Thmlemma1.p1.4.2.m2.1.1.2.cmml" xref="Thmlemma1.p1.4.2.m2.1.1.2">𝐻</ci><ci id="Thmlemma1.p1.4.2.m2.1.1.3a.cmml" xref="Thmlemma1.p1.4.2.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="Thmlemma1.p1.4.2.m2.1.1.3.cmml" mathsize="70%" xref="Thmlemma1.p1.4.2.m2.1.1.3">Shannon</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.4.2.m2.1c">H_{\text{Shannon}}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.4.2.m2.1d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT</annotation></semantics></math> is the discrete Shannon entropy.</span></p> </div> </div> <div class="ltx_proof" id="A3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A3.1.p1"> <p class="ltx_p" id="A3.1.p1.2">Recall that <math alttext="P(0|z_{1}^{k})\overset{\text{i.i.d.}}{\sim}\text{Unif}([0,1])" class="ltx_Math" display="inline" id="A3.1.p1.1.m1.4"><semantics id="A3.1.p1.1.m1.4a"><mrow id="A3.1.p1.1.m1.4.4" xref="A3.1.p1.1.m1.4.4.cmml"><mi id="A3.1.p1.1.m1.4.4.4" xref="A3.1.p1.1.m1.4.4.4.cmml">P</mi><mo id="A3.1.p1.1.m1.4.4.3" xref="A3.1.p1.1.m1.4.4.3.cmml">⁢</mo><mrow id="A3.1.p1.1.m1.3.3.1.1" 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xref="A3.1.p1.1.m1.4.4.5.2.cmml">∼</mo><mtext id="A3.1.p1.1.m1.4.4.5.1" xref="A3.1.p1.1.m1.4.4.5.1a.cmml">i.i.d.</mtext></mover><mo id="A3.1.p1.1.m1.4.4.3b" xref="A3.1.p1.1.m1.4.4.3.cmml">⁢</mo><mtext id="A3.1.p1.1.m1.4.4.6" xref="A3.1.p1.1.m1.4.4.6a.cmml">Unif</mtext><mo id="A3.1.p1.1.m1.4.4.3c" xref="A3.1.p1.1.m1.4.4.3.cmml">⁢</mo><mrow id="A3.1.p1.1.m1.4.4.2.1" xref="A3.1.p1.1.m1.4.4.cmml"><mo id="A3.1.p1.1.m1.4.4.2.1.2" stretchy="false" xref="A3.1.p1.1.m1.4.4.cmml">(</mo><mrow id="A3.1.p1.1.m1.4.4.2.1.1.2" xref="A3.1.p1.1.m1.4.4.2.1.1.1.cmml"><mo id="A3.1.p1.1.m1.4.4.2.1.1.2.1" stretchy="false" xref="A3.1.p1.1.m1.4.4.2.1.1.1.cmml">[</mo><mn id="A3.1.p1.1.m1.1.1" xref="A3.1.p1.1.m1.1.1.cmml">0</mn><mo id="A3.1.p1.1.m1.4.4.2.1.1.2.2" xref="A3.1.p1.1.m1.4.4.2.1.1.1.cmml">,</mo><mn id="A3.1.p1.1.m1.2.2" xref="A3.1.p1.1.m1.2.2.cmml">1</mn><mo id="A3.1.p1.1.m1.4.4.2.1.1.2.3" stretchy="false" xref="A3.1.p1.1.m1.4.4.2.1.1.1.cmml">]</mo></mrow><mo id="A3.1.p1.1.m1.4.4.2.1.3" stretchy="false" xref="A3.1.p1.1.m1.4.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.1.m1.4b"><apply id="A3.1.p1.1.m1.4.4.cmml" xref="A3.1.p1.1.m1.4.4"><times id="A3.1.p1.1.m1.4.4.3.cmml" xref="A3.1.p1.1.m1.4.4.3"></times><ci id="A3.1.p1.1.m1.4.4.4.cmml" xref="A3.1.p1.1.m1.4.4.4">𝑃</ci><apply id="A3.1.p1.1.m1.3.3.1.1.1.cmml" xref="A3.1.p1.1.m1.3.3.1.1"><csymbol cd="latexml" id="A3.1.p1.1.m1.3.3.1.1.1.1.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.1">conditional</csymbol><cn id="A3.1.p1.1.m1.3.3.1.1.1.2.cmml" type="integer" xref="A3.1.p1.1.m1.3.3.1.1.1.2">0</cn><apply id="A3.1.p1.1.m1.3.3.1.1.1.3.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.1.m1.3.3.1.1.1.3.1.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.3">superscript</csymbol><apply id="A3.1.p1.1.m1.3.3.1.1.1.3.2.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.1.m1.3.3.1.1.1.3.2.1.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.1.m1.3.3.1.1.1.3.2.2.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.3.2.2">𝑧</ci><cn id="A3.1.p1.1.m1.3.3.1.1.1.3.2.3.cmml" type="integer" xref="A3.1.p1.1.m1.3.3.1.1.1.3.2.3">1</cn></apply><ci id="A3.1.p1.1.m1.3.3.1.1.1.3.3.cmml" xref="A3.1.p1.1.m1.3.3.1.1.1.3.3">𝑘</ci></apply></apply><apply id="A3.1.p1.1.m1.4.4.5.cmml" xref="A3.1.p1.1.m1.4.4.5"><ci id="A3.1.p1.1.m1.4.4.5.1a.cmml" xref="A3.1.p1.1.m1.4.4.5.1"><mtext id="A3.1.p1.1.m1.4.4.5.1.cmml" xref="A3.1.p1.1.m1.4.4.5.1">i.i.d.</mtext></ci><csymbol cd="latexml" id="A3.1.p1.1.m1.4.4.5.2.cmml" xref="A3.1.p1.1.m1.4.4.5.2">similar-to</csymbol></apply><ci id="A3.1.p1.1.m1.4.4.6a.cmml" xref="A3.1.p1.1.m1.4.4.6"><mtext id="A3.1.p1.1.m1.4.4.6.cmml" xref="A3.1.p1.1.m1.4.4.6">Unif</mtext></ci><interval closure="closed" id="A3.1.p1.1.m1.4.4.2.1.1.1.cmml" xref="A3.1.p1.1.m1.4.4.2.1.1.2"><cn id="A3.1.p1.1.m1.1.1.cmml" type="integer" xref="A3.1.p1.1.m1.1.1">0</cn><cn id="A3.1.p1.1.m1.2.2.cmml" type="integer" xref="A3.1.p1.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.1.m1.4c">P(0|z_{1}^{k})\overset{\text{i.i.d.}}{\sim}\text{Unif}([0,1])</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.1.m1.4d">italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) overi.i.d. start_ARG ∼ end_ARG Unif ( [ 0 , 1 ] )</annotation></semantics></math> across <math alttext="z_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A3.1.p1.2.m2.1"><semantics id="A3.1.p1.2.m2.1a"><mrow id="A3.1.p1.2.m2.1.1" xref="A3.1.p1.2.m2.1.1.cmml"><msubsup id="A3.1.p1.2.m2.1.1.2" xref="A3.1.p1.2.m2.1.1.2.cmml"><mi id="A3.1.p1.2.m2.1.1.2.2.2" xref="A3.1.p1.2.m2.1.1.2.2.2.cmml">z</mi><mn id="A3.1.p1.2.m2.1.1.2.2.3" xref="A3.1.p1.2.m2.1.1.2.2.3.cmml">1</mn><mi id="A3.1.p1.2.m2.1.1.2.3" xref="A3.1.p1.2.m2.1.1.2.3.cmml">k</mi></msubsup><mo id="A3.1.p1.2.m2.1.1.1" xref="A3.1.p1.2.m2.1.1.1.cmml">∈</mo><msup id="A3.1.p1.2.m2.1.1.3" xref="A3.1.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.1.p1.2.m2.1.1.3.2" xref="A3.1.p1.2.m2.1.1.3.2.cmml">𝒳</mi><mi id="A3.1.p1.2.m2.1.1.3.3" xref="A3.1.p1.2.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.2.m2.1b"><apply id="A3.1.p1.2.m2.1.1.cmml" xref="A3.1.p1.2.m2.1.1"><in id="A3.1.p1.2.m2.1.1.1.cmml" xref="A3.1.p1.2.m2.1.1.1"></in><apply id="A3.1.p1.2.m2.1.1.2.cmml" xref="A3.1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.2.m2.1.1.2.1.cmml" xref="A3.1.p1.2.m2.1.1.2">superscript</csymbol><apply id="A3.1.p1.2.m2.1.1.2.2.cmml" xref="A3.1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.2.m2.1.1.2.2.1.cmml" xref="A3.1.p1.2.m2.1.1.2">subscript</csymbol><ci id="A3.1.p1.2.m2.1.1.2.2.2.cmml" xref="A3.1.p1.2.m2.1.1.2.2.2">𝑧</ci><cn id="A3.1.p1.2.m2.1.1.2.2.3.cmml" type="integer" xref="A3.1.p1.2.m2.1.1.2.2.3">1</cn></apply><ci id="A3.1.p1.2.m2.1.1.2.3.cmml" xref="A3.1.p1.2.m2.1.1.2.3">𝑘</ci></apply><apply id="A3.1.p1.2.m2.1.1.3.cmml" xref="A3.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.2.m2.1.1.3.1.cmml" xref="A3.1.p1.2.m2.1.1.3">superscript</csymbol><ci id="A3.1.p1.2.m2.1.1.3.2.cmml" xref="A3.1.p1.2.m2.1.1.3.2">𝒳</ci><ci id="A3.1.p1.2.m2.1.1.3.3.cmml" xref="A3.1.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.2.m2.1c">z_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.2.m2.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>. Then,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx24"> <tbody id="A3.E44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Pr(\mathrm{round}(P(0|z_{1}^{k}))=j)" class="ltx_Math" display="inline" id="A3.E44.m1.2"><semantics id="A3.E44.m1.2a"><mrow id="A3.E44.m1.2.2.1" xref="A3.E44.m1.2.2.2.cmml"><mi id="A3.E44.m1.1.1" xref="A3.E44.m1.1.1.cmml">Pr</mi><mo id="A3.E44.m1.2.2.1a" xref="A3.E44.m1.2.2.2.cmml">⁡</mo><mrow id="A3.E44.m1.2.2.1.1" xref="A3.E44.m1.2.2.2.cmml"><mo id="A3.E44.m1.2.2.1.1.2" stretchy="false" xref="A3.E44.m1.2.2.2.cmml">(</mo><mrow id="A3.E44.m1.2.2.1.1.1" xref="A3.E44.m1.2.2.1.1.1.cmml"><mrow id="A3.E44.m1.2.2.1.1.1.1" xref="A3.E44.m1.2.2.1.1.1.1.cmml"><mi id="A3.E44.m1.2.2.1.1.1.1.3" xref="A3.E44.m1.2.2.1.1.1.1.3.cmml">round</mi><mo id="A3.E44.m1.2.2.1.1.1.1.2" 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id="A3.E44.m2.2d">= roman_Pr ( italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) ∈ [ italic_ε ( italic_j - 1 / 2 ) , italic_ε ( italic_j + 1 / 2 ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(44)</span></td> </tr></tbody> <tbody id="A3.E45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\begin{cases}\varepsilon\quad&\text{if }1\leq j\leq|B_{% \varepsilon}|-2,\\ 3\varepsilon/2&\text{if }j=0\text{ or }j=|B_{\varepsilon}|-1.\end{cases}" class="ltx_Math" display="inline" id="A3.E45.m1.4"><semantics id="A3.E45.m1.4a"><mrow id="A3.E45.m1.4.5" 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start_ROW start_CELL 3 italic_ε / 2 end_CELL start_CELL if italic_j = 0 or italic_j = | italic_B start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT | - 1 . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(45)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.4">This implies that, by independence of the <math alttext="P(0|z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.3.m1.1"><semantics id="A3.1.p1.3.m1.1a"><mrow id="A3.1.p1.3.m1.1.1" xref="A3.1.p1.3.m1.1.1.cmml"><mi id="A3.1.p1.3.m1.1.1.3" xref="A3.1.p1.3.m1.1.1.3.cmml">P</mi><mo id="A3.1.p1.3.m1.1.1.2" xref="A3.1.p1.3.m1.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.3.m1.1.1.1.1" xref="A3.1.p1.3.m1.1.1.1.1.1.cmml"><mo id="A3.1.p1.3.m1.1.1.1.1.2" stretchy="false" xref="A3.1.p1.3.m1.1.1.1.1.1.cmml">(</mo><mrow 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id="A3.1.p1.3.m1.1c">P(0|z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.3.m1.1d">italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>’s across <math alttext="z_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A3.1.p1.4.m2.1"><semantics id="A3.1.p1.4.m2.1a"><mrow id="A3.1.p1.4.m2.1.1" xref="A3.1.p1.4.m2.1.1.cmml"><msubsup id="A3.1.p1.4.m2.1.1.2" xref="A3.1.p1.4.m2.1.1.2.cmml"><mi id="A3.1.p1.4.m2.1.1.2.2.2" xref="A3.1.p1.4.m2.1.1.2.2.2.cmml">z</mi><mn id="A3.1.p1.4.m2.1.1.2.2.3" xref="A3.1.p1.4.m2.1.1.2.2.3.cmml">1</mn><mi id="A3.1.p1.4.m2.1.1.2.3" xref="A3.1.p1.4.m2.1.1.2.3.cmml">k</mi></msubsup><mo id="A3.1.p1.4.m2.1.1.1" xref="A3.1.p1.4.m2.1.1.1.cmml">∈</mo><msup id="A3.1.p1.4.m2.1.1.3" xref="A3.1.p1.4.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.1.p1.4.m2.1.1.3.2" xref="A3.1.p1.4.m2.1.1.3.2.cmml">𝒳</mi><mi id="A3.1.p1.4.m2.1.1.3.3" xref="A3.1.p1.4.m2.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.4.m2.1b"><apply id="A3.1.p1.4.m2.1.1.cmml" xref="A3.1.p1.4.m2.1.1"><in id="A3.1.p1.4.m2.1.1.1.cmml" xref="A3.1.p1.4.m2.1.1.1"></in><apply id="A3.1.p1.4.m2.1.1.2.cmml" xref="A3.1.p1.4.m2.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.4.m2.1.1.2.1.cmml" xref="A3.1.p1.4.m2.1.1.2">superscript</csymbol><apply id="A3.1.p1.4.m2.1.1.2.2.cmml" xref="A3.1.p1.4.m2.1.1.2"><csymbol cd="ambiguous" id="A3.1.p1.4.m2.1.1.2.2.1.cmml" xref="A3.1.p1.4.m2.1.1.2">subscript</csymbol><ci id="A3.1.p1.4.m2.1.1.2.2.2.cmml" xref="A3.1.p1.4.m2.1.1.2.2.2">𝑧</ci><cn id="A3.1.p1.4.m2.1.1.2.2.3.cmml" type="integer" xref="A3.1.p1.4.m2.1.1.2.2.3">1</cn></apply><ci id="A3.1.p1.4.m2.1.1.2.3.cmml" xref="A3.1.p1.4.m2.1.1.2.3">𝑘</ci></apply><apply id="A3.1.p1.4.m2.1.1.3.cmml" xref="A3.1.p1.4.m2.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.4.m2.1.1.3.1.cmml" xref="A3.1.p1.4.m2.1.1.3">superscript</csymbol><ci id="A3.1.p1.4.m2.1.1.3.2.cmml" xref="A3.1.p1.4.m2.1.1.3.2">𝒳</ci><ci id="A3.1.p1.4.m2.1.1.3.3.cmml" xref="A3.1.p1.4.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.4.m2.1c">z_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.4.m2.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx25"> <tbody id="A3.E46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\text{Shannon}}\left(\left\{P(0|z_{1}^{k}):z_{1}^{k}\in% \mathcal{X}^{k}\right\}\right)\geq|\mathcal{X}|^{k}(1-3\varepsilon)\log(1/% \varepsilon)." class="ltx_math_unparsed" display="inline" id="A3.E46.m1.1"><semantics id="A3.E46.m1.1a"><mrow id="A3.E46.m1.1b"><msub id="A3.E46.m1.1.1"><mi id="A3.E46.m1.1.1.2">H</mi><mtext id="A3.E46.m1.1.1.3">Shannon</mtext></msub><mrow id="A3.E46.m1.1.2"><mo id="A3.E46.m1.1.2.1">(</mo><mrow id="A3.E46.m1.1.2.2"><mo id="A3.E46.m1.1.2.2.1">{</mo><mi id="A3.E46.m1.1.2.2.2">P</mi><mrow id="A3.E46.m1.1.2.2.3"><mo id="A3.E46.m1.1.2.2.3.1" stretchy="false">(</mo><mn id="A3.E46.m1.1.2.2.3.2">0</mn><mo fence="false" id="A3.E46.m1.1.2.2.3.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.E46.m1.1.2.2.3.4"><mi id="A3.E46.m1.1.2.2.3.4.2.2">z</mi><mn id="A3.E46.m1.1.2.2.3.4.2.3">1</mn><mi id="A3.E46.m1.1.2.2.3.4.3">k</mi></msubsup><mo id="A3.E46.m1.1.2.2.3.5" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="A3.E46.m1.1.2.2.4" rspace="0.278em">:</mo><msubsup id="A3.E46.m1.1.2.2.5"><mi id="A3.E46.m1.1.2.2.5.2.2">z</mi><mn id="A3.E46.m1.1.2.2.5.2.3">1</mn><mi id="A3.E46.m1.1.2.2.5.3">k</mi></msubsup><mo id="A3.E46.m1.1.2.2.6">∈</mo><msup id="A3.E46.m1.1.2.2.7"><mi class="ltx_font_mathcaligraphic" id="A3.E46.m1.1.2.2.7.2">𝒳</mi><mi id="A3.E46.m1.1.2.2.7.3">k</mi></msup><mo id="A3.E46.m1.1.2.2.8">}</mo></mrow><mo id="A3.E46.m1.1.2.3">)</mo></mrow><mo id="A3.E46.m1.1.3" rspace="0em">≥</mo><mo fence="false" id="A3.E46.m1.1.4" rspace="0.167em" stretchy="false">|</mo><mi class="ltx_font_mathcaligraphic" id="A3.E46.m1.1.5">𝒳</mi><msup id="A3.E46.m1.1.6"><mo fence="false" id="A3.E46.m1.1.6.2" rspace="0.167em" stretchy="false">|</mo><mi id="A3.E46.m1.1.6.3">k</mi></msup><mrow id="A3.E46.m1.1.7"><mo id="A3.E46.m1.1.7.1" stretchy="false">(</mo><mn id="A3.E46.m1.1.7.2">1</mn><mo id="A3.E46.m1.1.7.3">−</mo><mn id="A3.E46.m1.1.7.4">3</mn><mi id="A3.E46.m1.1.7.5">ε</mi><mo id="A3.E46.m1.1.7.6" rspace="0.167em" stretchy="false">)</mo></mrow><mi id="A3.E46.m1.1.8">log</mi><mrow id="A3.E46.m1.1.9"><mo id="A3.E46.m1.1.9.1" stretchy="false">(</mo><mn id="A3.E46.m1.1.9.2">1</mn><mo id="A3.E46.m1.1.9.3">/</mo><mi id="A3.E46.m1.1.9.4">ε</mi><mo id="A3.E46.m1.1.9.5" stretchy="false">)</mo></mrow><mo id="A3.E46.m1.1.10" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="A3.E46.m1.1c">\displaystyle H_{\text{Shannon}}\left(\left\{P(0|z_{1}^{k}):z_{1}^{k}\in% \mathcal{X}^{k}\right\}\right)\geq|\mathcal{X}|^{k}(1-3\varepsilon)\log(1/% \varepsilon).</annotation><annotation encoding="application/x-llamapun" id="A3.E46.m1.1d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) ≥ | caligraphic_X | start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( 1 - 3 italic_ε ) roman_log ( 1 / italic_ε ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(46)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.9">Let <math alttext="e(z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.5.m1.1"><semantics id="A3.1.p1.5.m1.1a"><mrow id="A3.1.p1.5.m1.1.1" xref="A3.1.p1.5.m1.1.1.cmml"><mi id="A3.1.p1.5.m1.1.1.3" xref="A3.1.p1.5.m1.1.1.3.cmml">e</mi><mo id="A3.1.p1.5.m1.1.1.2" xref="A3.1.p1.5.m1.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.5.m1.1.1.1.1" xref="A3.1.p1.5.m1.1.1.1.1.1.cmml"><mo id="A3.1.p1.5.m1.1.1.1.1.2" stretchy="false" xref="A3.1.p1.5.m1.1.1.1.1.1.cmml">(</mo><msubsup id="A3.1.p1.5.m1.1.1.1.1.1" xref="A3.1.p1.5.m1.1.1.1.1.1.cmml"><mi id="A3.1.p1.5.m1.1.1.1.1.1.2.2" xref="A3.1.p1.5.m1.1.1.1.1.1.2.2.cmml">z</mi><mn id="A3.1.p1.5.m1.1.1.1.1.1.2.3" xref="A3.1.p1.5.m1.1.1.1.1.1.2.3.cmml">1</mn><mi id="A3.1.p1.5.m1.1.1.1.1.1.3" xref="A3.1.p1.5.m1.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="A3.1.p1.5.m1.1.1.1.1.3" stretchy="false" xref="A3.1.p1.5.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.5.m1.1b"><apply id="A3.1.p1.5.m1.1.1.cmml" xref="A3.1.p1.5.m1.1.1"><times id="A3.1.p1.5.m1.1.1.2.cmml" xref="A3.1.p1.5.m1.1.1.2"></times><ci id="A3.1.p1.5.m1.1.1.3.cmml" xref="A3.1.p1.5.m1.1.1.3">𝑒</ci><apply id="A3.1.p1.5.m1.1.1.1.1.1.cmml" xref="A3.1.p1.5.m1.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.5.m1.1.1.1.1.1.1.cmml" xref="A3.1.p1.5.m1.1.1.1.1">superscript</csymbol><apply id="A3.1.p1.5.m1.1.1.1.1.1.2.cmml" xref="A3.1.p1.5.m1.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.5.m1.1.1.1.1.1.2.1.cmml" xref="A3.1.p1.5.m1.1.1.1.1">subscript</csymbol><ci id="A3.1.p1.5.m1.1.1.1.1.1.2.2.cmml" xref="A3.1.p1.5.m1.1.1.1.1.1.2.2">𝑧</ci><cn id="A3.1.p1.5.m1.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.1.p1.5.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="A3.1.p1.5.m1.1.1.1.1.1.3.cmml" xref="A3.1.p1.5.m1.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.5.m1.1c">e(z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.5.m1.1d">italic_e ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> be the random variable <math alttext="P(0|z_{1}^{k})-f(z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.6.m2.2"><semantics id="A3.1.p1.6.m2.2a"><mrow id="A3.1.p1.6.m2.2.2" xref="A3.1.p1.6.m2.2.2.cmml"><mrow id="A3.1.p1.6.m2.1.1.1" xref="A3.1.p1.6.m2.1.1.1.cmml"><mi id="A3.1.p1.6.m2.1.1.1.3" xref="A3.1.p1.6.m2.1.1.1.3.cmml">P</mi><mo id="A3.1.p1.6.m2.1.1.1.2" xref="A3.1.p1.6.m2.1.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.6.m2.1.1.1.1.1" xref="A3.1.p1.6.m2.1.1.1.1.1.1.cmml"><mo id="A3.1.p1.6.m2.1.1.1.1.1.2" stretchy="false" xref="A3.1.p1.6.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.1.p1.6.m2.1.1.1.1.1.1" xref="A3.1.p1.6.m2.1.1.1.1.1.1.cmml"><mn id="A3.1.p1.6.m2.1.1.1.1.1.1.2" xref="A3.1.p1.6.m2.1.1.1.1.1.1.2.cmml">0</mn><mo fence="false" id="A3.1.p1.6.m2.1.1.1.1.1.1.1" xref="A3.1.p1.6.m2.1.1.1.1.1.1.1.cmml">|</mo><msubsup id="A3.1.p1.6.m2.1.1.1.1.1.1.3" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.cmml"><mi id="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.2" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.2.cmml">z</mi><mn id="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.3" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="A3.1.p1.6.m2.1.1.1.1.1.1.3.3" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A3.1.p1.6.m2.1.1.1.1.1.3" stretchy="false" xref="A3.1.p1.6.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.6.m2.2.2.3" xref="A3.1.p1.6.m2.2.2.3.cmml">−</mo><mrow id="A3.1.p1.6.m2.2.2.2" xref="A3.1.p1.6.m2.2.2.2.cmml"><mi id="A3.1.p1.6.m2.2.2.2.3" xref="A3.1.p1.6.m2.2.2.2.3.cmml">f</mi><mo id="A3.1.p1.6.m2.2.2.2.2" xref="A3.1.p1.6.m2.2.2.2.2.cmml">⁢</mo><mrow id="A3.1.p1.6.m2.2.2.2.1.1" xref="A3.1.p1.6.m2.2.2.2.1.1.1.cmml"><mo id="A3.1.p1.6.m2.2.2.2.1.1.2" stretchy="false" xref="A3.1.p1.6.m2.2.2.2.1.1.1.cmml">(</mo><msubsup id="A3.1.p1.6.m2.2.2.2.1.1.1" xref="A3.1.p1.6.m2.2.2.2.1.1.1.cmml"><mi id="A3.1.p1.6.m2.2.2.2.1.1.1.2.2" xref="A3.1.p1.6.m2.2.2.2.1.1.1.2.2.cmml">z</mi><mn id="A3.1.p1.6.m2.2.2.2.1.1.1.2.3" xref="A3.1.p1.6.m2.2.2.2.1.1.1.2.3.cmml">1</mn><mi id="A3.1.p1.6.m2.2.2.2.1.1.1.3" xref="A3.1.p1.6.m2.2.2.2.1.1.1.3.cmml">k</mi></msubsup><mo id="A3.1.p1.6.m2.2.2.2.1.1.3" stretchy="false" xref="A3.1.p1.6.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.6.m2.2b"><apply id="A3.1.p1.6.m2.2.2.cmml" xref="A3.1.p1.6.m2.2.2"><minus id="A3.1.p1.6.m2.2.2.3.cmml" xref="A3.1.p1.6.m2.2.2.3"></minus><apply id="A3.1.p1.6.m2.1.1.1.cmml" xref="A3.1.p1.6.m2.1.1.1"><times id="A3.1.p1.6.m2.1.1.1.2.cmml" xref="A3.1.p1.6.m2.1.1.1.2"></times><ci id="A3.1.p1.6.m2.1.1.1.3.cmml" xref="A3.1.p1.6.m2.1.1.1.3">𝑃</ci><apply id="A3.1.p1.6.m2.1.1.1.1.1.1.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1"><csymbol cd="latexml" id="A3.1.p1.6.m2.1.1.1.1.1.1.1.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.1">conditional</csymbol><cn id="A3.1.p1.6.m2.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.1.p1.6.m2.1.1.1.1.1.1.2">0</cn><apply id="A3.1.p1.6.m2.1.1.1.1.1.1.3.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.6.m2.1.1.1.1.1.1.3.1.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3">superscript</csymbol><apply id="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.1.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.2.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.2">𝑧</ci><cn id="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.2.3">1</cn></apply><ci id="A3.1.p1.6.m2.1.1.1.1.1.1.3.3.cmml" xref="A3.1.p1.6.m2.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply><apply id="A3.1.p1.6.m2.2.2.2.cmml" xref="A3.1.p1.6.m2.2.2.2"><times id="A3.1.p1.6.m2.2.2.2.2.cmml" xref="A3.1.p1.6.m2.2.2.2.2"></times><ci id="A3.1.p1.6.m2.2.2.2.3.cmml" xref="A3.1.p1.6.m2.2.2.2.3">𝑓</ci><apply id="A3.1.p1.6.m2.2.2.2.1.1.1.cmml" xref="A3.1.p1.6.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="A3.1.p1.6.m2.2.2.2.1.1.1.1.cmml" xref="A3.1.p1.6.m2.2.2.2.1.1">superscript</csymbol><apply id="A3.1.p1.6.m2.2.2.2.1.1.1.2.cmml" xref="A3.1.p1.6.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="A3.1.p1.6.m2.2.2.2.1.1.1.2.1.cmml" xref="A3.1.p1.6.m2.2.2.2.1.1">subscript</csymbol><ci id="A3.1.p1.6.m2.2.2.2.1.1.1.2.2.cmml" xref="A3.1.p1.6.m2.2.2.2.1.1.1.2.2">𝑧</ci><cn id="A3.1.p1.6.m2.2.2.2.1.1.1.2.3.cmml" type="integer" xref="A3.1.p1.6.m2.2.2.2.1.1.1.2.3">1</cn></apply><ci id="A3.1.p1.6.m2.2.2.2.1.1.1.3.cmml" xref="A3.1.p1.6.m2.2.2.2.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.6.m2.2c">P(0|z_{1}^{k})-f(z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.6.m2.2d">italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) - italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>. <math alttext="P(0|z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.7.m3.1"><semantics id="A3.1.p1.7.m3.1a"><mrow id="A3.1.p1.7.m3.1.1" xref="A3.1.p1.7.m3.1.1.cmml"><mi id="A3.1.p1.7.m3.1.1.3" xref="A3.1.p1.7.m3.1.1.3.cmml">P</mi><mo id="A3.1.p1.7.m3.1.1.2" xref="A3.1.p1.7.m3.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.7.m3.1.1.1.1" xref="A3.1.p1.7.m3.1.1.1.1.1.cmml"><mo id="A3.1.p1.7.m3.1.1.1.1.2" stretchy="false" xref="A3.1.p1.7.m3.1.1.1.1.1.cmml">(</mo><mrow id="A3.1.p1.7.m3.1.1.1.1.1" xref="A3.1.p1.7.m3.1.1.1.1.1.cmml"><mn id="A3.1.p1.7.m3.1.1.1.1.1.2" xref="A3.1.p1.7.m3.1.1.1.1.1.2.cmml">0</mn><mo fence="false" id="A3.1.p1.7.m3.1.1.1.1.1.1" xref="A3.1.p1.7.m3.1.1.1.1.1.1.cmml">|</mo><msubsup id="A3.1.p1.7.m3.1.1.1.1.1.3" xref="A3.1.p1.7.m3.1.1.1.1.1.3.cmml"><mi id="A3.1.p1.7.m3.1.1.1.1.1.3.2.2" xref="A3.1.p1.7.m3.1.1.1.1.1.3.2.2.cmml">z</mi><mn id="A3.1.p1.7.m3.1.1.1.1.1.3.2.3" xref="A3.1.p1.7.m3.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="A3.1.p1.7.m3.1.1.1.1.1.3.3" xref="A3.1.p1.7.m3.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A3.1.p1.7.m3.1.1.1.1.3" stretchy="false" xref="A3.1.p1.7.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.7.m3.1b"><apply id="A3.1.p1.7.m3.1.1.cmml" xref="A3.1.p1.7.m3.1.1"><times id="A3.1.p1.7.m3.1.1.2.cmml" xref="A3.1.p1.7.m3.1.1.2"></times><ci id="A3.1.p1.7.m3.1.1.3.cmml" xref="A3.1.p1.7.m3.1.1.3">𝑃</ci><apply id="A3.1.p1.7.m3.1.1.1.1.1.cmml" xref="A3.1.p1.7.m3.1.1.1.1"><csymbol cd="latexml" id="A3.1.p1.7.m3.1.1.1.1.1.1.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.1">conditional</csymbol><cn id="A3.1.p1.7.m3.1.1.1.1.1.2.cmml" type="integer" xref="A3.1.p1.7.m3.1.1.1.1.1.2">0</cn><apply id="A3.1.p1.7.m3.1.1.1.1.1.3.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.7.m3.1.1.1.1.1.3.1.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.3">superscript</csymbol><apply id="A3.1.p1.7.m3.1.1.1.1.1.3.2.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.1.p1.7.m3.1.1.1.1.1.3.2.1.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.3">subscript</csymbol><ci id="A3.1.p1.7.m3.1.1.1.1.1.3.2.2.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.3.2.2">𝑧</ci><cn id="A3.1.p1.7.m3.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.1.p1.7.m3.1.1.1.1.1.3.2.3">1</cn></apply><ci id="A3.1.p1.7.m3.1.1.1.1.1.3.3.cmml" xref="A3.1.p1.7.m3.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.7.m3.1c">P(0|z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.7.m3.1d">italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> is a measurable function of <math alttext="f(z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.8.m4.1"><semantics id="A3.1.p1.8.m4.1a"><mrow id="A3.1.p1.8.m4.1.1" xref="A3.1.p1.8.m4.1.1.cmml"><mi id="A3.1.p1.8.m4.1.1.3" xref="A3.1.p1.8.m4.1.1.3.cmml">f</mi><mo id="A3.1.p1.8.m4.1.1.2" xref="A3.1.p1.8.m4.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.8.m4.1.1.1.1" xref="A3.1.p1.8.m4.1.1.1.1.1.cmml"><mo id="A3.1.p1.8.m4.1.1.1.1.2" stretchy="false" xref="A3.1.p1.8.m4.1.1.1.1.1.cmml">(</mo><msubsup id="A3.1.p1.8.m4.1.1.1.1.1" xref="A3.1.p1.8.m4.1.1.1.1.1.cmml"><mi id="A3.1.p1.8.m4.1.1.1.1.1.2.2" xref="A3.1.p1.8.m4.1.1.1.1.1.2.2.cmml">z</mi><mn id="A3.1.p1.8.m4.1.1.1.1.1.2.3" xref="A3.1.p1.8.m4.1.1.1.1.1.2.3.cmml">1</mn><mi id="A3.1.p1.8.m4.1.1.1.1.1.3" xref="A3.1.p1.8.m4.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="A3.1.p1.8.m4.1.1.1.1.3" stretchy="false" xref="A3.1.p1.8.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.8.m4.1b"><apply id="A3.1.p1.8.m4.1.1.cmml" xref="A3.1.p1.8.m4.1.1"><times id="A3.1.p1.8.m4.1.1.2.cmml" xref="A3.1.p1.8.m4.1.1.2"></times><ci id="A3.1.p1.8.m4.1.1.3.cmml" xref="A3.1.p1.8.m4.1.1.3">𝑓</ci><apply id="A3.1.p1.8.m4.1.1.1.1.1.cmml" xref="A3.1.p1.8.m4.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.8.m4.1.1.1.1.1.1.cmml" xref="A3.1.p1.8.m4.1.1.1.1">superscript</csymbol><apply id="A3.1.p1.8.m4.1.1.1.1.1.2.cmml" xref="A3.1.p1.8.m4.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.8.m4.1.1.1.1.1.2.1.cmml" xref="A3.1.p1.8.m4.1.1.1.1">subscript</csymbol><ci id="A3.1.p1.8.m4.1.1.1.1.1.2.2.cmml" xref="A3.1.p1.8.m4.1.1.1.1.1.2.2">𝑧</ci><cn id="A3.1.p1.8.m4.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.1.p1.8.m4.1.1.1.1.1.2.3">1</cn></apply><ci id="A3.1.p1.8.m4.1.1.1.1.1.3.cmml" xref="A3.1.p1.8.m4.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.8.m4.1c">f(z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.8.m4.1d">italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="e(z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.9.m5.1"><semantics id="A3.1.p1.9.m5.1a"><mrow id="A3.1.p1.9.m5.1.1" xref="A3.1.p1.9.m5.1.1.cmml"><mi id="A3.1.p1.9.m5.1.1.3" xref="A3.1.p1.9.m5.1.1.3.cmml">e</mi><mo id="A3.1.p1.9.m5.1.1.2" xref="A3.1.p1.9.m5.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.9.m5.1.1.1.1" xref="A3.1.p1.9.m5.1.1.1.1.1.cmml"><mo id="A3.1.p1.9.m5.1.1.1.1.2" stretchy="false" xref="A3.1.p1.9.m5.1.1.1.1.1.cmml">(</mo><msubsup id="A3.1.p1.9.m5.1.1.1.1.1" xref="A3.1.p1.9.m5.1.1.1.1.1.cmml"><mi id="A3.1.p1.9.m5.1.1.1.1.1.2.2" xref="A3.1.p1.9.m5.1.1.1.1.1.2.2.cmml">z</mi><mn id="A3.1.p1.9.m5.1.1.1.1.1.2.3" xref="A3.1.p1.9.m5.1.1.1.1.1.2.3.cmml">1</mn><mi id="A3.1.p1.9.m5.1.1.1.1.1.3" xref="A3.1.p1.9.m5.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="A3.1.p1.9.m5.1.1.1.1.3" stretchy="false" xref="A3.1.p1.9.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.9.m5.1b"><apply id="A3.1.p1.9.m5.1.1.cmml" xref="A3.1.p1.9.m5.1.1"><times id="A3.1.p1.9.m5.1.1.2.cmml" xref="A3.1.p1.9.m5.1.1.2"></times><ci id="A3.1.p1.9.m5.1.1.3.cmml" xref="A3.1.p1.9.m5.1.1.3">𝑒</ci><apply id="A3.1.p1.9.m5.1.1.1.1.1.cmml" xref="A3.1.p1.9.m5.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.9.m5.1.1.1.1.1.1.cmml" xref="A3.1.p1.9.m5.1.1.1.1">superscript</csymbol><apply id="A3.1.p1.9.m5.1.1.1.1.1.2.cmml" xref="A3.1.p1.9.m5.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.9.m5.1.1.1.1.1.2.1.cmml" xref="A3.1.p1.9.m5.1.1.1.1">subscript</csymbol><ci id="A3.1.p1.9.m5.1.1.1.1.1.2.2.cmml" xref="A3.1.p1.9.m5.1.1.1.1.1.2.2">𝑧</ci><cn id="A3.1.p1.9.m5.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.1.p1.9.m5.1.1.1.1.1.2.3">1</cn></apply><ci id="A3.1.p1.9.m5.1.1.1.1.1.3.cmml" xref="A3.1.p1.9.m5.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.9.m5.1c">e(z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.9.m5.1d">italic_e ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math>, and therefore,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx26"> <tbody id="A3.E47"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\text{Shannon}}\left(\left\{f(z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\cup\left\{e(z_{1}^{k}):z_{1}^{k}\in\mathcal{X}^{k}\right\}% \right)\geq H_{\text{Shannon}}\left(\left\{P(0|z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\right)" class="ltx_math_unparsed" display="inline" id="A3.E47.m1.1"><semantics id="A3.E47.m1.1a"><mrow id="A3.E47.m1.1b"><msub id="A3.E47.m1.1.1"><mi id="A3.E47.m1.1.1.2">H</mi><mtext id="A3.E47.m1.1.1.3">Shannon</mtext></msub><mrow id="A3.E47.m1.1.2"><mo id="A3.E47.m1.1.2.1">(</mo><mrow id="A3.E47.m1.1.2.2"><mo id="A3.E47.m1.1.2.2.1">{</mo><mi id="A3.E47.m1.1.2.2.2">f</mi><mrow id="A3.E47.m1.1.2.2.3"><mo id="A3.E47.m1.1.2.2.3.1" stretchy="false">(</mo><msubsup id="A3.E47.m1.1.2.2.3.2"><mi id="A3.E47.m1.1.2.2.3.2.2.2">z</mi><mn id="A3.E47.m1.1.2.2.3.2.2.3">1</mn><mi id="A3.E47.m1.1.2.2.3.2.3">k</mi></msubsup><mo id="A3.E47.m1.1.2.2.3.3" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="A3.E47.m1.1.2.2.4" rspace="0.278em">:</mo><msubsup id="A3.E47.m1.1.2.2.5"><mi id="A3.E47.m1.1.2.2.5.2.2">z</mi><mn id="A3.E47.m1.1.2.2.5.2.3">1</mn><mi id="A3.E47.m1.1.2.2.5.3">k</mi></msubsup><mo id="A3.E47.m1.1.2.2.6">∈</mo><msup id="A3.E47.m1.1.2.2.7"><mi class="ltx_font_mathcaligraphic" id="A3.E47.m1.1.2.2.7.2">𝒳</mi><mi id="A3.E47.m1.1.2.2.7.3">k</mi></msup><mo id="A3.E47.m1.1.2.2.8">}</mo></mrow><mo id="A3.E47.m1.1.2.3">∪</mo><mrow id="A3.E47.m1.1.2.4"><mo id="A3.E47.m1.1.2.4.1">{</mo><mi id="A3.E47.m1.1.2.4.2">e</mi><mrow id="A3.E47.m1.1.2.4.3"><mo id="A3.E47.m1.1.2.4.3.1" stretchy="false">(</mo><msubsup id="A3.E47.m1.1.2.4.3.2"><mi id="A3.E47.m1.1.2.4.3.2.2.2">z</mi><mn id="A3.E47.m1.1.2.4.3.2.2.3">1</mn><mi id="A3.E47.m1.1.2.4.3.2.3">k</mi></msubsup><mo id="A3.E47.m1.1.2.4.3.3" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="A3.E47.m1.1.2.4.4" rspace="0.278em">:</mo><msubsup id="A3.E47.m1.1.2.4.5"><mi id="A3.E47.m1.1.2.4.5.2.2">z</mi><mn id="A3.E47.m1.1.2.4.5.2.3">1</mn><mi id="A3.E47.m1.1.2.4.5.3">k</mi></msubsup><mo id="A3.E47.m1.1.2.4.6">∈</mo><msup id="A3.E47.m1.1.2.4.7"><mi class="ltx_font_mathcaligraphic" id="A3.E47.m1.1.2.4.7.2">𝒳</mi><mi id="A3.E47.m1.1.2.4.7.3">k</mi></msup><mo id="A3.E47.m1.1.2.4.8">}</mo></mrow><mo id="A3.E47.m1.1.2.5">)</mo></mrow><mo id="A3.E47.m1.1.3">≥</mo><msub id="A3.E47.m1.1.4"><mi id="A3.E47.m1.1.4.2">H</mi><mtext id="A3.E47.m1.1.4.3">Shannon</mtext></msub><mrow id="A3.E47.m1.1.5"><mo id="A3.E47.m1.1.5.1">(</mo><mrow id="A3.E47.m1.1.5.2"><mo id="A3.E47.m1.1.5.2.1">{</mo><mi id="A3.E47.m1.1.5.2.2">P</mi><mrow id="A3.E47.m1.1.5.2.3"><mo id="A3.E47.m1.1.5.2.3.1" stretchy="false">(</mo><mn id="A3.E47.m1.1.5.2.3.2">0</mn><mo fence="false" id="A3.E47.m1.1.5.2.3.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.E47.m1.1.5.2.3.4"><mi id="A3.E47.m1.1.5.2.3.4.2.2">z</mi><mn id="A3.E47.m1.1.5.2.3.4.2.3">1</mn><mi id="A3.E47.m1.1.5.2.3.4.3">k</mi></msubsup><mo id="A3.E47.m1.1.5.2.3.5" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="A3.E47.m1.1.5.2.4" rspace="0.278em">:</mo><msubsup id="A3.E47.m1.1.5.2.5"><mi id="A3.E47.m1.1.5.2.5.2.2">z</mi><mn id="A3.E47.m1.1.5.2.5.2.3">1</mn><mi id="A3.E47.m1.1.5.2.5.3">k</mi></msubsup><mo id="A3.E47.m1.1.5.2.6">∈</mo><msup id="A3.E47.m1.1.5.2.7"><mi class="ltx_font_mathcaligraphic" id="A3.E47.m1.1.5.2.7.2">𝒳</mi><mi id="A3.E47.m1.1.5.2.7.3">k</mi></msup><mo id="A3.E47.m1.1.5.2.8">}</mo></mrow><mo id="A3.E47.m1.1.5.3">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.E47.m1.1c">\displaystyle H_{\text{Shannon}}\left(\left\{f(z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\cup\left\{e(z_{1}^{k}):z_{1}^{k}\in\mathcal{X}^{k}\right\}% \right)\geq H_{\text{Shannon}}\left(\left\{P(0|z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\right)</annotation><annotation encoding="application/x-llamapun" id="A3.E47.m1.1d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ∪ { italic_e ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) ≥ italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(47)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.1.p1.14">Note that <math alttext="e(z_{1}^{k})" class="ltx_Math" display="inline" id="A3.1.p1.10.m1.1"><semantics id="A3.1.p1.10.m1.1a"><mrow id="A3.1.p1.10.m1.1.1" xref="A3.1.p1.10.m1.1.1.cmml"><mi id="A3.1.p1.10.m1.1.1.3" xref="A3.1.p1.10.m1.1.1.3.cmml">e</mi><mo id="A3.1.p1.10.m1.1.1.2" xref="A3.1.p1.10.m1.1.1.2.cmml">⁢</mo><mrow id="A3.1.p1.10.m1.1.1.1.1" xref="A3.1.p1.10.m1.1.1.1.1.1.cmml"><mo id="A3.1.p1.10.m1.1.1.1.1.2" stretchy="false" xref="A3.1.p1.10.m1.1.1.1.1.1.cmml">(</mo><msubsup id="A3.1.p1.10.m1.1.1.1.1.1" xref="A3.1.p1.10.m1.1.1.1.1.1.cmml"><mi id="A3.1.p1.10.m1.1.1.1.1.1.2.2" xref="A3.1.p1.10.m1.1.1.1.1.1.2.2.cmml">z</mi><mn id="A3.1.p1.10.m1.1.1.1.1.1.2.3" xref="A3.1.p1.10.m1.1.1.1.1.1.2.3.cmml">1</mn><mi id="A3.1.p1.10.m1.1.1.1.1.1.3" xref="A3.1.p1.10.m1.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="A3.1.p1.10.m1.1.1.1.1.3" stretchy="false" xref="A3.1.p1.10.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.10.m1.1b"><apply id="A3.1.p1.10.m1.1.1.cmml" xref="A3.1.p1.10.m1.1.1"><times id="A3.1.p1.10.m1.1.1.2.cmml" xref="A3.1.p1.10.m1.1.1.2"></times><ci id="A3.1.p1.10.m1.1.1.3.cmml" xref="A3.1.p1.10.m1.1.1.3">𝑒</ci><apply id="A3.1.p1.10.m1.1.1.1.1.1.cmml" xref="A3.1.p1.10.m1.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.10.m1.1.1.1.1.1.1.cmml" xref="A3.1.p1.10.m1.1.1.1.1">superscript</csymbol><apply id="A3.1.p1.10.m1.1.1.1.1.1.2.cmml" xref="A3.1.p1.10.m1.1.1.1.1"><csymbol cd="ambiguous" id="A3.1.p1.10.m1.1.1.1.1.1.2.1.cmml" xref="A3.1.p1.10.m1.1.1.1.1">subscript</csymbol><ci id="A3.1.p1.10.m1.1.1.1.1.1.2.2.cmml" xref="A3.1.p1.10.m1.1.1.1.1.1.2.2">𝑧</ci><cn id="A3.1.p1.10.m1.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.1.p1.10.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="A3.1.p1.10.m1.1.1.1.1.1.3.cmml" xref="A3.1.p1.10.m1.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.10.m1.1c">e(z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.10.m1.1d">italic_e ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> is bounded in the rate <math alttext="\{-r,\cdots,r\}" class="ltx_Math" display="inline" id="A3.1.p1.11.m2.3"><semantics id="A3.1.p1.11.m2.3a"><mrow id="A3.1.p1.11.m2.3.3.1" xref="A3.1.p1.11.m2.3.3.2.cmml"><mo id="A3.1.p1.11.m2.3.3.1.2" stretchy="false" xref="A3.1.p1.11.m2.3.3.2.cmml">{</mo><mrow id="A3.1.p1.11.m2.3.3.1.1" xref="A3.1.p1.11.m2.3.3.1.1.cmml"><mo id="A3.1.p1.11.m2.3.3.1.1a" xref="A3.1.p1.11.m2.3.3.1.1.cmml">−</mo><mi id="A3.1.p1.11.m2.3.3.1.1.2" xref="A3.1.p1.11.m2.3.3.1.1.2.cmml">r</mi></mrow><mo id="A3.1.p1.11.m2.3.3.1.3" xref="A3.1.p1.11.m2.3.3.2.cmml">,</mo><mi id="A3.1.p1.11.m2.1.1" mathvariant="normal" xref="A3.1.p1.11.m2.1.1.cmml">⋯</mi><mo id="A3.1.p1.11.m2.3.3.1.4" xref="A3.1.p1.11.m2.3.3.2.cmml">,</mo><mi id="A3.1.p1.11.m2.2.2" xref="A3.1.p1.11.m2.2.2.cmml">r</mi><mo id="A3.1.p1.11.m2.3.3.1.5" stretchy="false" xref="A3.1.p1.11.m2.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.11.m2.3b"><set id="A3.1.p1.11.m2.3.3.2.cmml" xref="A3.1.p1.11.m2.3.3.1"><apply id="A3.1.p1.11.m2.3.3.1.1.cmml" xref="A3.1.p1.11.m2.3.3.1.1"><minus id="A3.1.p1.11.m2.3.3.1.1.1.cmml" xref="A3.1.p1.11.m2.3.3.1.1"></minus><ci id="A3.1.p1.11.m2.3.3.1.1.2.cmml" xref="A3.1.p1.11.m2.3.3.1.1.2">𝑟</ci></apply><ci id="A3.1.p1.11.m2.1.1.cmml" xref="A3.1.p1.11.m2.1.1">⋯</ci><ci id="A3.1.p1.11.m2.2.2.cmml" xref="A3.1.p1.11.m2.2.2">𝑟</ci></set></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.11.m2.3c">\{-r,\cdots,r\}</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.11.m2.3d">{ - italic_r , ⋯ , italic_r }</annotation></semantics></math> and can take at most <math alttext="2r+1" class="ltx_Math" display="inline" id="A3.1.p1.12.m3.1"><semantics id="A3.1.p1.12.m3.1a"><mrow id="A3.1.p1.12.m3.1.1" xref="A3.1.p1.12.m3.1.1.cmml"><mrow id="A3.1.p1.12.m3.1.1.2" xref="A3.1.p1.12.m3.1.1.2.cmml"><mn id="A3.1.p1.12.m3.1.1.2.2" xref="A3.1.p1.12.m3.1.1.2.2.cmml">2</mn><mo id="A3.1.p1.12.m3.1.1.2.1" xref="A3.1.p1.12.m3.1.1.2.1.cmml">⁢</mo><mi id="A3.1.p1.12.m3.1.1.2.3" xref="A3.1.p1.12.m3.1.1.2.3.cmml">r</mi></mrow><mo id="A3.1.p1.12.m3.1.1.1" xref="A3.1.p1.12.m3.1.1.1.cmml">+</mo><mn id="A3.1.p1.12.m3.1.1.3" xref="A3.1.p1.12.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.12.m3.1b"><apply id="A3.1.p1.12.m3.1.1.cmml" xref="A3.1.p1.12.m3.1.1"><plus id="A3.1.p1.12.m3.1.1.1.cmml" xref="A3.1.p1.12.m3.1.1.1"></plus><apply id="A3.1.p1.12.m3.1.1.2.cmml" xref="A3.1.p1.12.m3.1.1.2"><times id="A3.1.p1.12.m3.1.1.2.1.cmml" xref="A3.1.p1.12.m3.1.1.2.1"></times><cn id="A3.1.p1.12.m3.1.1.2.2.cmml" type="integer" xref="A3.1.p1.12.m3.1.1.2.2">2</cn><ci id="A3.1.p1.12.m3.1.1.2.3.cmml" xref="A3.1.p1.12.m3.1.1.2.3">𝑟</ci></apply><cn id="A3.1.p1.12.m3.1.1.3.cmml" type="integer" xref="A3.1.p1.12.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.12.m3.1c">2r+1</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.12.m3.1d">2 italic_r + 1</annotation></semantics></math> values. 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id="A3.1.p1.13.m4.3.3.1.1.1.1.2.2.3.3.cmml" xref="A3.1.p1.13.m4.3.3.1.1.1.1.2.2.3.3">𝑘</ci></apply></apply></apply></apply><apply id="A3.1.p1.13.m4.4.4.2.cmml" xref="A3.1.p1.13.m4.4.4.2"><times id="A3.1.p1.13.m4.4.4.2.2.cmml" xref="A3.1.p1.13.m4.4.4.2.2"></times><apply id="A3.1.p1.13.m4.4.4.2.3.cmml" xref="A3.1.p1.13.m4.4.4.2.3"><csymbol cd="ambiguous" id="A3.1.p1.13.m4.4.4.2.3.1.cmml" xref="A3.1.p1.13.m4.4.4.2.3">superscript</csymbol><apply id="A3.1.p1.13.m4.4.4.2.3.2.1.cmml" xref="A3.1.p1.13.m4.4.4.2.3.2.2"><abs id="A3.1.p1.13.m4.4.4.2.3.2.1.1.cmml" xref="A3.1.p1.13.m4.4.4.2.3.2.2.1"></abs><ci id="A3.1.p1.13.m4.1.1.cmml" xref="A3.1.p1.13.m4.1.1">𝒳</ci></apply><ci id="A3.1.p1.13.m4.4.4.2.3.3.cmml" xref="A3.1.p1.13.m4.4.4.2.3.3">𝑘</ci></apply><apply id="A3.1.p1.13.m4.4.4.2.1.2.cmml" xref="A3.1.p1.13.m4.4.4.2.1.1"><log id="A3.1.p1.13.m4.2.2.cmml" xref="A3.1.p1.13.m4.2.2"></log><apply id="A3.1.p1.13.m4.4.4.2.1.1.1.1.cmml" xref="A3.1.p1.13.m4.4.4.2.1.1.1.1"><plus 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italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) ≤ | caligraphic_X | start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT roman_log ( 2 italic_r + 1 )</annotation></semantics></math>. Since <math alttext="H(A,B)\leq H(A)+H(B)" class="ltx_Math" display="inline" id="A3.1.p1.14.m5.4"><semantics id="A3.1.p1.14.m5.4a"><mrow id="A3.1.p1.14.m5.4.5" xref="A3.1.p1.14.m5.4.5.cmml"><mrow id="A3.1.p1.14.m5.4.5.2" xref="A3.1.p1.14.m5.4.5.2.cmml"><mi id="A3.1.p1.14.m5.4.5.2.2" xref="A3.1.p1.14.m5.4.5.2.2.cmml">H</mi><mo id="A3.1.p1.14.m5.4.5.2.1" xref="A3.1.p1.14.m5.4.5.2.1.cmml">⁢</mo><mrow id="A3.1.p1.14.m5.4.5.2.3.2" xref="A3.1.p1.14.m5.4.5.2.3.1.cmml"><mo id="A3.1.p1.14.m5.4.5.2.3.2.1" stretchy="false" xref="A3.1.p1.14.m5.4.5.2.3.1.cmml">(</mo><mi id="A3.1.p1.14.m5.1.1" xref="A3.1.p1.14.m5.1.1.cmml">A</mi><mo id="A3.1.p1.14.m5.4.5.2.3.2.2" xref="A3.1.p1.14.m5.4.5.2.3.1.cmml">,</mo><mi id="A3.1.p1.14.m5.2.2" xref="A3.1.p1.14.m5.2.2.cmml">B</mi><mo id="A3.1.p1.14.m5.4.5.2.3.2.3" stretchy="false" xref="A3.1.p1.14.m5.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.14.m5.4.5.1" xref="A3.1.p1.14.m5.4.5.1.cmml">≤</mo><mrow id="A3.1.p1.14.m5.4.5.3" xref="A3.1.p1.14.m5.4.5.3.cmml"><mrow id="A3.1.p1.14.m5.4.5.3.2" xref="A3.1.p1.14.m5.4.5.3.2.cmml"><mi id="A3.1.p1.14.m5.4.5.3.2.2" xref="A3.1.p1.14.m5.4.5.3.2.2.cmml">H</mi><mo id="A3.1.p1.14.m5.4.5.3.2.1" xref="A3.1.p1.14.m5.4.5.3.2.1.cmml">⁢</mo><mrow id="A3.1.p1.14.m5.4.5.3.2.3.2" xref="A3.1.p1.14.m5.4.5.3.2.cmml"><mo id="A3.1.p1.14.m5.4.5.3.2.3.2.1" stretchy="false" xref="A3.1.p1.14.m5.4.5.3.2.cmml">(</mo><mi id="A3.1.p1.14.m5.3.3" xref="A3.1.p1.14.m5.3.3.cmml">A</mi><mo id="A3.1.p1.14.m5.4.5.3.2.3.2.2" stretchy="false" xref="A3.1.p1.14.m5.4.5.3.2.cmml">)</mo></mrow></mrow><mo id="A3.1.p1.14.m5.4.5.3.1" xref="A3.1.p1.14.m5.4.5.3.1.cmml">+</mo><mrow id="A3.1.p1.14.m5.4.5.3.3" xref="A3.1.p1.14.m5.4.5.3.3.cmml"><mi id="A3.1.p1.14.m5.4.5.3.3.2" xref="A3.1.p1.14.m5.4.5.3.3.2.cmml">H</mi><mo id="A3.1.p1.14.m5.4.5.3.3.1" xref="A3.1.p1.14.m5.4.5.3.3.1.cmml">⁢</mo><mrow id="A3.1.p1.14.m5.4.5.3.3.3.2" xref="A3.1.p1.14.m5.4.5.3.3.cmml"><mo id="A3.1.p1.14.m5.4.5.3.3.3.2.1" stretchy="false" xref="A3.1.p1.14.m5.4.5.3.3.cmml">(</mo><mi id="A3.1.p1.14.m5.4.4" xref="A3.1.p1.14.m5.4.4.cmml">B</mi><mo id="A3.1.p1.14.m5.4.5.3.3.3.2.2" stretchy="false" xref="A3.1.p1.14.m5.4.5.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.1.p1.14.m5.4b"><apply id="A3.1.p1.14.m5.4.5.cmml" xref="A3.1.p1.14.m5.4.5"><leq id="A3.1.p1.14.m5.4.5.1.cmml" xref="A3.1.p1.14.m5.4.5.1"></leq><apply id="A3.1.p1.14.m5.4.5.2.cmml" xref="A3.1.p1.14.m5.4.5.2"><times id="A3.1.p1.14.m5.4.5.2.1.cmml" xref="A3.1.p1.14.m5.4.5.2.1"></times><ci id="A3.1.p1.14.m5.4.5.2.2.cmml" xref="A3.1.p1.14.m5.4.5.2.2">𝐻</ci><interval closure="open" id="A3.1.p1.14.m5.4.5.2.3.1.cmml" xref="A3.1.p1.14.m5.4.5.2.3.2"><ci id="A3.1.p1.14.m5.1.1.cmml" xref="A3.1.p1.14.m5.1.1">𝐴</ci><ci id="A3.1.p1.14.m5.2.2.cmml" xref="A3.1.p1.14.m5.2.2">𝐵</ci></interval></apply><apply id="A3.1.p1.14.m5.4.5.3.cmml" xref="A3.1.p1.14.m5.4.5.3"><plus id="A3.1.p1.14.m5.4.5.3.1.cmml" xref="A3.1.p1.14.m5.4.5.3.1"></plus><apply id="A3.1.p1.14.m5.4.5.3.2.cmml" xref="A3.1.p1.14.m5.4.5.3.2"><times id="A3.1.p1.14.m5.4.5.3.2.1.cmml" xref="A3.1.p1.14.m5.4.5.3.2.1"></times><ci id="A3.1.p1.14.m5.4.5.3.2.2.cmml" xref="A3.1.p1.14.m5.4.5.3.2.2">𝐻</ci><ci id="A3.1.p1.14.m5.3.3.cmml" xref="A3.1.p1.14.m5.3.3">𝐴</ci></apply><apply id="A3.1.p1.14.m5.4.5.3.3.cmml" xref="A3.1.p1.14.m5.4.5.3.3"><times id="A3.1.p1.14.m5.4.5.3.3.1.cmml" xref="A3.1.p1.14.m5.4.5.3.3.1"></times><ci id="A3.1.p1.14.m5.4.5.3.3.2.cmml" xref="A3.1.p1.14.m5.4.5.3.3.2">𝐻</ci><ci id="A3.1.p1.14.m5.4.4.cmml" xref="A3.1.p1.14.m5.4.4">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.1.p1.14.m5.4c">H(A,B)\leq H(A)+H(B)</annotation><annotation encoding="application/x-llamapun" id="A3.1.p1.14.m5.4d">italic_H ( italic_A , italic_B ) ≤ italic_H ( italic_A ) + italic_H ( italic_B )</annotation></semantics></math>, we have that,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx27"> <tbody id="A3.E48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\text{Shannon}}\left(\left\{f(z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\right)" class="ltx_Math" display="inline" id="A3.E48.m1.1"><semantics id="A3.E48.m1.1a"><mrow id="A3.E48.m1.1.1" xref="A3.E48.m1.1.1.cmml"><msub id="A3.E48.m1.1.1.3" xref="A3.E48.m1.1.1.3.cmml"><mi id="A3.E48.m1.1.1.3.2" xref="A3.E48.m1.1.1.3.2.cmml">H</mi><mtext id="A3.E48.m1.1.1.3.3" xref="A3.E48.m1.1.1.3.3a.cmml">Shannon</mtext></msub><mo id="A3.E48.m1.1.1.2" xref="A3.E48.m1.1.1.2.cmml">⁢</mo><mrow id="A3.E48.m1.1.1.1.1" xref="A3.E48.m1.1.1.cmml"><mo id="A3.E48.m1.1.1.1.1.2" xref="A3.E48.m1.1.1.cmml">(</mo><mrow id="A3.E48.m1.1.1.1.1.1.2" xref="A3.E48.m1.1.1.1.1.1.3.cmml"><mo id="A3.E48.m1.1.1.1.1.1.2.3" xref="A3.E48.m1.1.1.1.1.1.3.1.cmml">{</mo><mrow id="A3.E48.m1.1.1.1.1.1.1.1" xref="A3.E48.m1.1.1.1.1.1.1.1.cmml"><mi id="A3.E48.m1.1.1.1.1.1.1.1.3" xref="A3.E48.m1.1.1.1.1.1.1.1.3.cmml">f</mi><mo id="A3.E48.m1.1.1.1.1.1.1.1.2" xref="A3.E48.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.E48.m1.1.1.1.1.1.1.1.1.1" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.E48.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.2" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">z</mi><mn id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.3" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="A3.E48.m1.1.1.1.1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.E48.m1.1.1.1.1.1.2.4" rspace="0.278em" xref="A3.E48.m1.1.1.1.1.1.3.1.cmml">:</mo><mrow id="A3.E48.m1.1.1.1.1.1.2.2" xref="A3.E48.m1.1.1.1.1.1.2.2.cmml"><msubsup id="A3.E48.m1.1.1.1.1.1.2.2.2" xref="A3.E48.m1.1.1.1.1.1.2.2.2.cmml"><mi id="A3.E48.m1.1.1.1.1.1.2.2.2.2.2" xref="A3.E48.m1.1.1.1.1.1.2.2.2.2.2.cmml">z</mi><mn id="A3.E48.m1.1.1.1.1.1.2.2.2.2.3" xref="A3.E48.m1.1.1.1.1.1.2.2.2.2.3.cmml">1</mn><mi id="A3.E48.m1.1.1.1.1.1.2.2.2.3" xref="A3.E48.m1.1.1.1.1.1.2.2.2.3.cmml">k</mi></msubsup><mo id="A3.E48.m1.1.1.1.1.1.2.2.1" xref="A3.E48.m1.1.1.1.1.1.2.2.1.cmml">∈</mo><msup id="A3.E48.m1.1.1.1.1.1.2.2.3" xref="A3.E48.m1.1.1.1.1.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.E48.m1.1.1.1.1.1.2.2.3.2" xref="A3.E48.m1.1.1.1.1.1.2.2.3.2.cmml">𝒳</mi><mi id="A3.E48.m1.1.1.1.1.1.2.2.3.3" xref="A3.E48.m1.1.1.1.1.1.2.2.3.3.cmml">k</mi></msup></mrow><mo id="A3.E48.m1.1.1.1.1.1.2.5" xref="A3.E48.m1.1.1.1.1.1.3.1.cmml">}</mo></mrow><mo id="A3.E48.m1.1.1.1.1.3" xref="A3.E48.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E48.m1.1b"><apply id="A3.E48.m1.1.1.cmml" xref="A3.E48.m1.1.1"><times id="A3.E48.m1.1.1.2.cmml" xref="A3.E48.m1.1.1.2"></times><apply id="A3.E48.m1.1.1.3.cmml" xref="A3.E48.m1.1.1.3"><csymbol cd="ambiguous" id="A3.E48.m1.1.1.3.1.cmml" xref="A3.E48.m1.1.1.3">subscript</csymbol><ci id="A3.E48.m1.1.1.3.2.cmml" xref="A3.E48.m1.1.1.3.2">𝐻</ci><ci id="A3.E48.m1.1.1.3.3a.cmml" xref="A3.E48.m1.1.1.3.3"><mtext id="A3.E48.m1.1.1.3.3.cmml" mathsize="70%" xref="A3.E48.m1.1.1.3.3">Shannon</mtext></ci></apply><apply id="A3.E48.m1.1.1.1.1.1.3.cmml" xref="A3.E48.m1.1.1.1.1.1.2"><csymbol cd="latexml" id="A3.E48.m1.1.1.1.1.1.3.1.cmml" xref="A3.E48.m1.1.1.1.1.1.2.3">conditional-set</csymbol><apply id="A3.E48.m1.1.1.1.1.1.1.1.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1"><times id="A3.E48.m1.1.1.1.1.1.1.1.2.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.2"></times><ci id="A3.E48.m1.1.1.1.1.1.1.1.3.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.3">𝑓</ci><apply id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.2">𝑧</ci><cn id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A3.E48.m1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="A3.E48.m1.1.1.1.1.1.2.2.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2"><in id="A3.E48.m1.1.1.1.1.1.2.2.1.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.1"></in><apply id="A3.E48.m1.1.1.1.1.1.2.2.2.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A3.E48.m1.1.1.1.1.1.2.2.2.1.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A3.E48.m1.1.1.1.1.1.2.2.2.2.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A3.E48.m1.1.1.1.1.1.2.2.2.2.1.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A3.E48.m1.1.1.1.1.1.2.2.2.2.2.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.2.2.2">𝑧</ci><cn id="A3.E48.m1.1.1.1.1.1.2.2.2.2.3.cmml" type="integer" xref="A3.E48.m1.1.1.1.1.1.2.2.2.2.3">1</cn></apply><ci id="A3.E48.m1.1.1.1.1.1.2.2.2.3.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.2.3">𝑘</ci></apply><apply id="A3.E48.m1.1.1.1.1.1.2.2.3.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A3.E48.m1.1.1.1.1.1.2.2.3.1.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.3">superscript</csymbol><ci id="A3.E48.m1.1.1.1.1.1.2.2.3.2.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.3.2">𝒳</ci><ci id="A3.E48.m1.1.1.1.1.1.2.2.3.3.cmml" xref="A3.E48.m1.1.1.1.1.1.2.2.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E48.m1.1c">\displaystyle H_{\text{Shannon}}\left(\left\{f(z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\right)</annotation><annotation encoding="application/x-llamapun" id="A3.E48.m1.1d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq H_{\text{Shannon}}\left(\left\{P(0|z_{1}^{k}):z_{1}^{k}\in% \mathcal{X}^{k}\right\}\right)-H\left(\left\{e(z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\right)" class="ltx_math_unparsed" display="inline" id="A3.E48.m2.1"><semantics id="A3.E48.m2.1a"><mrow id="A3.E48.m2.1b"><mo id="A3.E48.m2.1.1">≥</mo><msub id="A3.E48.m2.1.2"><mi id="A3.E48.m2.1.2.2">H</mi><mtext id="A3.E48.m2.1.2.3">Shannon</mtext></msub><mrow id="A3.E48.m2.1.3"><mo id="A3.E48.m2.1.3.1">(</mo><mrow id="A3.E48.m2.1.3.2"><mo id="A3.E48.m2.1.3.2.1">{</mo><mi id="A3.E48.m2.1.3.2.2">P</mi><mrow id="A3.E48.m2.1.3.2.3"><mo id="A3.E48.m2.1.3.2.3.1" stretchy="false">(</mo><mn id="A3.E48.m2.1.3.2.3.2">0</mn><mo fence="false" id="A3.E48.m2.1.3.2.3.3" rspace="0.167em" stretchy="false">|</mo><msubsup id="A3.E48.m2.1.3.2.3.4"><mi id="A3.E48.m2.1.3.2.3.4.2.2">z</mi><mn id="A3.E48.m2.1.3.2.3.4.2.3">1</mn><mi id="A3.E48.m2.1.3.2.3.4.3">k</mi></msubsup><mo id="A3.E48.m2.1.3.2.3.5" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="A3.E48.m2.1.3.2.4" rspace="0.278em">:</mo><msubsup id="A3.E48.m2.1.3.2.5"><mi id="A3.E48.m2.1.3.2.5.2.2">z</mi><mn id="A3.E48.m2.1.3.2.5.2.3">1</mn><mi id="A3.E48.m2.1.3.2.5.3">k</mi></msubsup><mo id="A3.E48.m2.1.3.2.6">∈</mo><msup id="A3.E48.m2.1.3.2.7"><mi class="ltx_font_mathcaligraphic" id="A3.E48.m2.1.3.2.7.2">𝒳</mi><mi id="A3.E48.m2.1.3.2.7.3">k</mi></msup><mo id="A3.E48.m2.1.3.2.8">}</mo></mrow><mo id="A3.E48.m2.1.3.3">)</mo></mrow><mo id="A3.E48.m2.1.4">−</mo><mi id="A3.E48.m2.1.5">H</mi><mrow id="A3.E48.m2.1.6"><mo id="A3.E48.m2.1.6.1">(</mo><mrow id="A3.E48.m2.1.6.2"><mo id="A3.E48.m2.1.6.2.1">{</mo><mi id="A3.E48.m2.1.6.2.2">e</mi><mrow id="A3.E48.m2.1.6.2.3"><mo id="A3.E48.m2.1.6.2.3.1" stretchy="false">(</mo><msubsup id="A3.E48.m2.1.6.2.3.2"><mi id="A3.E48.m2.1.6.2.3.2.2.2">z</mi><mn id="A3.E48.m2.1.6.2.3.2.2.3">1</mn><mi id="A3.E48.m2.1.6.2.3.2.3">k</mi></msubsup><mo id="A3.E48.m2.1.6.2.3.3" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="A3.E48.m2.1.6.2.4" rspace="0.278em">:</mo><msubsup id="A3.E48.m2.1.6.2.5"><mi id="A3.E48.m2.1.6.2.5.2.2">z</mi><mn id="A3.E48.m2.1.6.2.5.2.3">1</mn><mi id="A3.E48.m2.1.6.2.5.3">k</mi></msubsup><mo id="A3.E48.m2.1.6.2.6">∈</mo><msup id="A3.E48.m2.1.6.2.7"><mi class="ltx_font_mathcaligraphic" id="A3.E48.m2.1.6.2.7.2">𝒳</mi><mi id="A3.E48.m2.1.6.2.7.3">k</mi></msup><mo id="A3.E48.m2.1.6.2.8">}</mo></mrow><mo id="A3.E48.m2.1.6.3">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.E48.m2.1c">\displaystyle\geq H_{\text{Shannon}}\left(\left\{P(0|z_{1}^{k}):z_{1}^{k}\in% \mathcal{X}^{k}\right\}\right)-H\left(\left\{e(z_{1}^{k}):z_{1}^{k}\in\mathcal% {X}^{k}\right\}\right)</annotation><annotation encoding="application/x-llamapun" id="A3.E48.m2.1d">≥ italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) - italic_H ( { italic_e ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(48)</span></td> </tr></tbody> <tbody id="A3.E49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq|\mathcal{X}|^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-% \log(2r+1)\right)." class="ltx_Math" display="inline" id="A3.E49.m1.4"><semantics id="A3.E49.m1.4a"><mrow id="A3.E49.m1.4.4.1" xref="A3.E49.m1.4.4.1.1.cmml"><mrow id="A3.E49.m1.4.4.1.1" xref="A3.E49.m1.4.4.1.1.cmml"><mi id="A3.E49.m1.4.4.1.1.3" xref="A3.E49.m1.4.4.1.1.3.cmml"></mi><mo id="A3.E49.m1.4.4.1.1.2" 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</table> <p class="ltx_p" id="A3.1.p1.15">∎</p> </div> </div> <div class="ltx_para" id="A3.p2"> <p class="ltx_p" id="A3.p2.10">Recall that we are guaranteed that <math alttext="g(x_{t+k+1}=0,H_{t+k})\in P(0|z_{1}^{k})+[-\varepsilon-\eta_{P}(t),\varepsilon% +\eta_{P}(t)]" class="ltx_Math" display="inline" id="A3.p2.1.m1.7"><semantics id="A3.p2.1.m1.7a"><mrow id="A3.p2.1.m1.7.7" xref="A3.p2.1.m1.7.7.cmml"><mrow id="A3.p2.1.m1.4.4.1" xref="A3.p2.1.m1.4.4.1.cmml"><mi id="A3.p2.1.m1.4.4.1.3" xref="A3.p2.1.m1.4.4.1.3.cmml">g</mi><mo id="A3.p2.1.m1.4.4.1.2" xref="A3.p2.1.m1.4.4.1.2.cmml">⁢</mo><mrow id="A3.p2.1.m1.4.4.1.1.1" xref="A3.p2.1.m1.4.4.1.1.1.1.cmml"><mo id="A3.p2.1.m1.4.4.1.1.1.2" stretchy="false" xref="A3.p2.1.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="A3.p2.1.m1.4.4.1.1.1.1" xref="A3.p2.1.m1.4.4.1.1.1.1.cmml"><msub id="A3.p2.1.m1.4.4.1.1.1.1.3" xref="A3.p2.1.m1.4.4.1.1.1.1.3.cmml"><mi id="A3.p2.1.m1.4.4.1.1.1.1.3.2" xref="A3.p2.1.m1.4.4.1.1.1.1.3.2.cmml">x</mi><mrow 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italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT ) ∈ italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) + [ - italic_ε - italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) , italic_ε + italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) ]</annotation></semantics></math>. This implies that the recurrent model is able to recover <math alttext="\mathrm{round}(p)" class="ltx_Math" display="inline" id="A3.p2.2.m2.1"><semantics id="A3.p2.2.m2.1a"><mrow id="A3.p2.2.m2.1.2" xref="A3.p2.2.m2.1.2.cmml"><mi id="A3.p2.2.m2.1.2.2" xref="A3.p2.2.m2.1.2.2.cmml">round</mi><mo id="A3.p2.2.m2.1.2.1" xref="A3.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="A3.p2.2.m2.1.2.3.2" xref="A3.p2.2.m2.1.2.cmml"><mo id="A3.p2.2.m2.1.2.3.2.1" stretchy="false" xref="A3.p2.2.m2.1.2.cmml">(</mo><mi id="A3.p2.2.m2.1.1" xref="A3.p2.2.m2.1.1.cmml">p</mi><mo id="A3.p2.2.m2.1.2.3.2.2" stretchy="false" xref="A3.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.2.m2.1b"><apply id="A3.p2.2.m2.1.2.cmml" xref="A3.p2.2.m2.1.2"><times id="A3.p2.2.m2.1.2.1.cmml" xref="A3.p2.2.m2.1.2.1"></times><ci id="A3.p2.2.m2.1.2.2.cmml" xref="A3.p2.2.m2.1.2.2">round</ci><ci id="A3.p2.2.m2.1.1.cmml" xref="A3.p2.2.m2.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.2.m2.1c">\mathrm{round}(p)</annotation><annotation encoding="application/x-llamapun" id="A3.p2.2.m2.1d">roman_round ( italic_p )</annotation></semantics></math> for <math alttext="p=P(0|z_{1}^{k})" class="ltx_Math" display="inline" id="A3.p2.3.m3.1"><semantics id="A3.p2.3.m3.1a"><mrow id="A3.p2.3.m3.1.1" xref="A3.p2.3.m3.1.1.cmml"><mi id="A3.p2.3.m3.1.1.3" xref="A3.p2.3.m3.1.1.3.cmml">p</mi><mo id="A3.p2.3.m3.1.1.2" xref="A3.p2.3.m3.1.1.2.cmml">=</mo><mrow id="A3.p2.3.m3.1.1.1" xref="A3.p2.3.m3.1.1.1.cmml"><mi id="A3.p2.3.m3.1.1.1.3" xref="A3.p2.3.m3.1.1.1.3.cmml">P</mi><mo id="A3.p2.3.m3.1.1.1.2" xref="A3.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="A3.p2.3.m3.1.1.1.1.1" xref="A3.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="A3.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="A3.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.p2.3.m3.1.1.1.1.1.1" xref="A3.p2.3.m3.1.1.1.1.1.1.cmml"><mn id="A3.p2.3.m3.1.1.1.1.1.1.2" xref="A3.p2.3.m3.1.1.1.1.1.1.2.cmml">0</mn><mo fence="false" id="A3.p2.3.m3.1.1.1.1.1.1.1" xref="A3.p2.3.m3.1.1.1.1.1.1.1.cmml">|</mo><msubsup id="A3.p2.3.m3.1.1.1.1.1.1.3" xref="A3.p2.3.m3.1.1.1.1.1.1.3.cmml"><mi id="A3.p2.3.m3.1.1.1.1.1.1.3.2.2" xref="A3.p2.3.m3.1.1.1.1.1.1.3.2.2.cmml">z</mi><mn id="A3.p2.3.m3.1.1.1.1.1.1.3.2.3" xref="A3.p2.3.m3.1.1.1.1.1.1.3.2.3.cmml">1</mn><mi id="A3.p2.3.m3.1.1.1.1.1.1.3.3" xref="A3.p2.3.m3.1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A3.p2.3.m3.1.1.1.1.1.3" stretchy="false" xref="A3.p2.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.3.m3.1b"><apply id="A3.p2.3.m3.1.1.cmml" xref="A3.p2.3.m3.1.1"><eq id="A3.p2.3.m3.1.1.2.cmml" xref="A3.p2.3.m3.1.1.2"></eq><ci id="A3.p2.3.m3.1.1.3.cmml" xref="A3.p2.3.m3.1.1.3">𝑝</ci><apply id="A3.p2.3.m3.1.1.1.cmml" xref="A3.p2.3.m3.1.1.1"><times id="A3.p2.3.m3.1.1.1.2.cmml" xref="A3.p2.3.m3.1.1.1.2"></times><ci id="A3.p2.3.m3.1.1.1.3.cmml" xref="A3.p2.3.m3.1.1.1.3">𝑃</ci><apply id="A3.p2.3.m3.1.1.1.1.1.1.cmml" xref="A3.p2.3.m3.1.1.1.1.1"><csymbol cd="latexml" id="A3.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.1">conditional</csymbol><cn id="A3.p2.3.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.p2.3.m3.1.1.1.1.1.1.2">0</cn><apply id="A3.p2.3.m3.1.1.1.1.1.1.3.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p2.3.m3.1.1.1.1.1.1.3.1.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.3">superscript</csymbol><apply id="A3.p2.3.m3.1.1.1.1.1.1.3.2.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p2.3.m3.1.1.1.1.1.1.3.2.1.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="A3.p2.3.m3.1.1.1.1.1.1.3.2.2.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.3.2.2">𝑧</ci><cn id="A3.p2.3.m3.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.p2.3.m3.1.1.1.1.1.1.3.2.3">1</cn></apply><ci id="A3.p2.3.m3.1.1.1.1.1.1.3.3.cmml" xref="A3.p2.3.m3.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.3.m3.1c">p=P(0|z_{1}^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.p2.3.m3.1d">italic_p = italic_P ( 0 | italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> up to an error of <math alttext="r=\lceil\eta(t)/\varepsilon\rceil" class="ltx_Math" display="inline" id="A3.p2.4.m4.2"><semantics id="A3.p2.4.m4.2a"><mrow id="A3.p2.4.m4.2.2" xref="A3.p2.4.m4.2.2.cmml"><mi id="A3.p2.4.m4.2.2.3" xref="A3.p2.4.m4.2.2.3.cmml">r</mi><mo id="A3.p2.4.m4.2.2.2" xref="A3.p2.4.m4.2.2.2.cmml">=</mo><mrow id="A3.p2.4.m4.2.2.1.1" xref="A3.p2.4.m4.2.2.1.2.cmml"><mo id="A3.p2.4.m4.2.2.1.1.2" stretchy="false" xref="A3.p2.4.m4.2.2.1.2.1.cmml">⌈</mo><mrow id="A3.p2.4.m4.2.2.1.1.1" xref="A3.p2.4.m4.2.2.1.1.1.cmml"><mrow id="A3.p2.4.m4.2.2.1.1.1.2" xref="A3.p2.4.m4.2.2.1.1.1.2.cmml"><mi id="A3.p2.4.m4.2.2.1.1.1.2.2" xref="A3.p2.4.m4.2.2.1.1.1.2.2.cmml">η</mi><mo id="A3.p2.4.m4.2.2.1.1.1.2.1" xref="A3.p2.4.m4.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.p2.4.m4.2.2.1.1.1.2.3.2" xref="A3.p2.4.m4.2.2.1.1.1.2.cmml"><mo id="A3.p2.4.m4.2.2.1.1.1.2.3.2.1" stretchy="false" xref="A3.p2.4.m4.2.2.1.1.1.2.cmml">(</mo><mi id="A3.p2.4.m4.1.1" xref="A3.p2.4.m4.1.1.cmml">t</mi><mo id="A3.p2.4.m4.2.2.1.1.1.2.3.2.2" stretchy="false" xref="A3.p2.4.m4.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.p2.4.m4.2.2.1.1.1.1" xref="A3.p2.4.m4.2.2.1.1.1.1.cmml">/</mo><mi id="A3.p2.4.m4.2.2.1.1.1.3" xref="A3.p2.4.m4.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A3.p2.4.m4.2.2.1.1.3" stretchy="false" xref="A3.p2.4.m4.2.2.1.2.1.cmml">⌉</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.4.m4.2b"><apply id="A3.p2.4.m4.2.2.cmml" xref="A3.p2.4.m4.2.2"><eq id="A3.p2.4.m4.2.2.2.cmml" xref="A3.p2.4.m4.2.2.2"></eq><ci id="A3.p2.4.m4.2.2.3.cmml" xref="A3.p2.4.m4.2.2.3">𝑟</ci><apply id="A3.p2.4.m4.2.2.1.2.cmml" xref="A3.p2.4.m4.2.2.1.1"><ceiling id="A3.p2.4.m4.2.2.1.2.1.cmml" xref="A3.p2.4.m4.2.2.1.1.2"></ceiling><apply id="A3.p2.4.m4.2.2.1.1.1.cmml" xref="A3.p2.4.m4.2.2.1.1.1"><divide id="A3.p2.4.m4.2.2.1.1.1.1.cmml" xref="A3.p2.4.m4.2.2.1.1.1.1"></divide><apply id="A3.p2.4.m4.2.2.1.1.1.2.cmml" xref="A3.p2.4.m4.2.2.1.1.1.2"><times id="A3.p2.4.m4.2.2.1.1.1.2.1.cmml" xref="A3.p2.4.m4.2.2.1.1.1.2.1"></times><ci id="A3.p2.4.m4.2.2.1.1.1.2.2.cmml" xref="A3.p2.4.m4.2.2.1.1.1.2.2">𝜂</ci><ci id="A3.p2.4.m4.1.1.cmml" xref="A3.p2.4.m4.1.1">𝑡</ci></apply><ci id="A3.p2.4.m4.2.2.1.1.1.3.cmml" xref="A3.p2.4.m4.2.2.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.4.m4.2c">r=\lceil\eta(t)/\varepsilon\rceil</annotation><annotation encoding="application/x-llamapun" id="A3.p2.4.m4.2d">italic_r = ⌈ italic_η ( italic_t ) / italic_ε ⌉</annotation></semantics></math> for each <math alttext="z_{1}^{k}\in\mathcal{X}^{k}" class="ltx_Math" display="inline" id="A3.p2.5.m5.1"><semantics id="A3.p2.5.m5.1a"><mrow id="A3.p2.5.m5.1.1" xref="A3.p2.5.m5.1.1.cmml"><msubsup id="A3.p2.5.m5.1.1.2" xref="A3.p2.5.m5.1.1.2.cmml"><mi id="A3.p2.5.m5.1.1.2.2.2" xref="A3.p2.5.m5.1.1.2.2.2.cmml">z</mi><mn id="A3.p2.5.m5.1.1.2.2.3" xref="A3.p2.5.m5.1.1.2.2.3.cmml">1</mn><mi id="A3.p2.5.m5.1.1.2.3" xref="A3.p2.5.m5.1.1.2.3.cmml">k</mi></msubsup><mo id="A3.p2.5.m5.1.1.1" xref="A3.p2.5.m5.1.1.1.cmml">∈</mo><msup id="A3.p2.5.m5.1.1.3" xref="A3.p2.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.p2.5.m5.1.1.3.2" xref="A3.p2.5.m5.1.1.3.2.cmml">𝒳</mi><mi id="A3.p2.5.m5.1.1.3.3" xref="A3.p2.5.m5.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.5.m5.1b"><apply id="A3.p2.5.m5.1.1.cmml" xref="A3.p2.5.m5.1.1"><in id="A3.p2.5.m5.1.1.1.cmml" xref="A3.p2.5.m5.1.1.1"></in><apply id="A3.p2.5.m5.1.1.2.cmml" xref="A3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="A3.p2.5.m5.1.1.2.1.cmml" xref="A3.p2.5.m5.1.1.2">superscript</csymbol><apply id="A3.p2.5.m5.1.1.2.2.cmml" xref="A3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="A3.p2.5.m5.1.1.2.2.1.cmml" xref="A3.p2.5.m5.1.1.2">subscript</csymbol><ci id="A3.p2.5.m5.1.1.2.2.2.cmml" xref="A3.p2.5.m5.1.1.2.2.2">𝑧</ci><cn id="A3.p2.5.m5.1.1.2.2.3.cmml" type="integer" xref="A3.p2.5.m5.1.1.2.2.3">1</cn></apply><ci id="A3.p2.5.m5.1.1.2.3.cmml" xref="A3.p2.5.m5.1.1.2.3">𝑘</ci></apply><apply id="A3.p2.5.m5.1.1.3.cmml" xref="A3.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="A3.p2.5.m5.1.1.3.1.cmml" xref="A3.p2.5.m5.1.1.3">superscript</csymbol><ci id="A3.p2.5.m5.1.1.3.2.cmml" xref="A3.p2.5.m5.1.1.3.2">𝒳</ci><ci id="A3.p2.5.m5.1.1.3.3.cmml" xref="A3.p2.5.m5.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.5.m5.1c">z_{1}^{k}\in\mathcal{X}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.5.m5.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> by computing <math alttext="\mathrm{round}(\hat{p})" class="ltx_Math" display="inline" id="A3.p2.6.m6.1"><semantics id="A3.p2.6.m6.1a"><mrow id="A3.p2.6.m6.1.2" xref="A3.p2.6.m6.1.2.cmml"><mi id="A3.p2.6.m6.1.2.2" xref="A3.p2.6.m6.1.2.2.cmml">round</mi><mo id="A3.p2.6.m6.1.2.1" xref="A3.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="A3.p2.6.m6.1.2.3.2" xref="A3.p2.6.m6.1.1.cmml"><mo id="A3.p2.6.m6.1.2.3.2.1" stretchy="false" xref="A3.p2.6.m6.1.1.cmml">(</mo><mover accent="true" id="A3.p2.6.m6.1.1" xref="A3.p2.6.m6.1.1.cmml"><mi id="A3.p2.6.m6.1.1.2" xref="A3.p2.6.m6.1.1.2.cmml">p</mi><mo id="A3.p2.6.m6.1.1.1" xref="A3.p2.6.m6.1.1.1.cmml">^</mo></mover><mo id="A3.p2.6.m6.1.2.3.2.2" stretchy="false" xref="A3.p2.6.m6.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.6.m6.1b"><apply id="A3.p2.6.m6.1.2.cmml" xref="A3.p2.6.m6.1.2"><times id="A3.p2.6.m6.1.2.1.cmml" xref="A3.p2.6.m6.1.2.1"></times><ci id="A3.p2.6.m6.1.2.2.cmml" xref="A3.p2.6.m6.1.2.2">round</ci><apply id="A3.p2.6.m6.1.1.cmml" xref="A3.p2.6.m6.1.2.3.2"><ci id="A3.p2.6.m6.1.1.1.cmml" xref="A3.p2.6.m6.1.1.1">^</ci><ci id="A3.p2.6.m6.1.1.2.cmml" xref="A3.p2.6.m6.1.1.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.6.m6.1c">\mathrm{round}(\hat{p})</annotation><annotation encoding="application/x-llamapun" id="A3.p2.6.m6.1d">roman_round ( over^ start_ARG italic_p end_ARG )</annotation></semantics></math> where <math alttext="\hat{p}=g(x_{t+k+1}=0,H_{t+k})" class="ltx_Math" display="inline" id="A3.p2.7.m7.2"><semantics id="A3.p2.7.m7.2a"><mrow id="A3.p2.7.m7.2.2" xref="A3.p2.7.m7.2.2.cmml"><mover accent="true" id="A3.p2.7.m7.2.2.3" xref="A3.p2.7.m7.2.2.3.cmml"><mi id="A3.p2.7.m7.2.2.3.2" xref="A3.p2.7.m7.2.2.3.2.cmml">p</mi><mo id="A3.p2.7.m7.2.2.3.1" xref="A3.p2.7.m7.2.2.3.1.cmml">^</mo></mover><mo id="A3.p2.7.m7.2.2.2" xref="A3.p2.7.m7.2.2.2.cmml">=</mo><mrow id="A3.p2.7.m7.2.2.1" xref="A3.p2.7.m7.2.2.1.cmml"><mi id="A3.p2.7.m7.2.2.1.3" xref="A3.p2.7.m7.2.2.1.3.cmml">g</mi><mo id="A3.p2.7.m7.2.2.1.2" xref="A3.p2.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="A3.p2.7.m7.2.2.1.1.1" xref="A3.p2.7.m7.2.2.1.1.1.1.cmml"><mo id="A3.p2.7.m7.2.2.1.1.1.2" stretchy="false" xref="A3.p2.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="A3.p2.7.m7.2.2.1.1.1.1" xref="A3.p2.7.m7.2.2.1.1.1.1.cmml"><msub id="A3.p2.7.m7.2.2.1.1.1.1.3" xref="A3.p2.7.m7.2.2.1.1.1.1.3.cmml"><mi id="A3.p2.7.m7.2.2.1.1.1.1.3.2" xref="A3.p2.7.m7.2.2.1.1.1.1.3.2.cmml">x</mi><mrow id="A3.p2.7.m7.2.2.1.1.1.1.3.3" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.cmml"><mi id="A3.p2.7.m7.2.2.1.1.1.1.3.3.2" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.2.cmml">t</mi><mo id="A3.p2.7.m7.2.2.1.1.1.1.3.3.1" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.1.cmml">+</mo><mi id="A3.p2.7.m7.2.2.1.1.1.1.3.3.3" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.3.cmml">k</mi><mo id="A3.p2.7.m7.2.2.1.1.1.1.3.3.1a" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.1.cmml">+</mo><mn id="A3.p2.7.m7.2.2.1.1.1.1.3.3.4" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.4.cmml">1</mn></mrow></msub><mo id="A3.p2.7.m7.2.2.1.1.1.1.2" xref="A3.p2.7.m7.2.2.1.1.1.1.2.cmml">=</mo><mrow id="A3.p2.7.m7.2.2.1.1.1.1.1.1" xref="A3.p2.7.m7.2.2.1.1.1.1.1.2.cmml"><mn id="A3.p2.7.m7.1.1" xref="A3.p2.7.m7.1.1.cmml">0</mn><mo id="A3.p2.7.m7.2.2.1.1.1.1.1.1.2" xref="A3.p2.7.m7.2.2.1.1.1.1.1.2.cmml">,</mo><msub id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.cmml"><mi id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.2" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.2.cmml">H</mi><mrow id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.cmml"><mi id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.2" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.1" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.3" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.3.cmml">k</mi></mrow></msub></mrow></mrow><mo id="A3.p2.7.m7.2.2.1.1.1.3" stretchy="false" xref="A3.p2.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.7.m7.2b"><apply id="A3.p2.7.m7.2.2.cmml" xref="A3.p2.7.m7.2.2"><eq id="A3.p2.7.m7.2.2.2.cmml" xref="A3.p2.7.m7.2.2.2"></eq><apply id="A3.p2.7.m7.2.2.3.cmml" xref="A3.p2.7.m7.2.2.3"><ci id="A3.p2.7.m7.2.2.3.1.cmml" xref="A3.p2.7.m7.2.2.3.1">^</ci><ci id="A3.p2.7.m7.2.2.3.2.cmml" xref="A3.p2.7.m7.2.2.3.2">𝑝</ci></apply><apply id="A3.p2.7.m7.2.2.1.cmml" xref="A3.p2.7.m7.2.2.1"><times id="A3.p2.7.m7.2.2.1.2.cmml" xref="A3.p2.7.m7.2.2.1.2"></times><ci id="A3.p2.7.m7.2.2.1.3.cmml" xref="A3.p2.7.m7.2.2.1.3">𝑔</ci><apply id="A3.p2.7.m7.2.2.1.1.1.1.cmml" xref="A3.p2.7.m7.2.2.1.1.1"><eq id="A3.p2.7.m7.2.2.1.1.1.1.2.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.2"></eq><apply id="A3.p2.7.m7.2.2.1.1.1.1.3.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.p2.7.m7.2.2.1.1.1.1.3.1.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3">subscript</csymbol><ci id="A3.p2.7.m7.2.2.1.1.1.1.3.2.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3.2">𝑥</ci><apply id="A3.p2.7.m7.2.2.1.1.1.1.3.3.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3"><plus id="A3.p2.7.m7.2.2.1.1.1.1.3.3.1.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.1"></plus><ci id="A3.p2.7.m7.2.2.1.1.1.1.3.3.2.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.2">𝑡</ci><ci id="A3.p2.7.m7.2.2.1.1.1.1.3.3.3.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.3">𝑘</ci><cn id="A3.p2.7.m7.2.2.1.1.1.1.3.3.4.cmml" type="integer" xref="A3.p2.7.m7.2.2.1.1.1.1.3.3.4">1</cn></apply></apply><list id="A3.p2.7.m7.2.2.1.1.1.1.1.2.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1"><cn id="A3.p2.7.m7.1.1.cmml" type="integer" xref="A3.p2.7.m7.1.1">0</cn><apply id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.1.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.2.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.2">𝐻</ci><apply id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3"><plus id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.1.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.1"></plus><ci id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.2.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.2">𝑡</ci><ci id="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.3.cmml" xref="A3.p2.7.m7.2.2.1.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.7.m7.2c">\hat{p}=g(x_{t+k+1}=0,H_{t+k})</annotation><annotation encoding="application/x-llamapun" id="A3.p2.7.m7.2d">over^ start_ARG italic_p end_ARG = italic_g ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. Informally, this just means that <math alttext="\hat{p}" class="ltx_Math" display="inline" id="A3.p2.8.m8.1"><semantics id="A3.p2.8.m8.1a"><mover accent="true" id="A3.p2.8.m8.1.1" xref="A3.p2.8.m8.1.1.cmml"><mi id="A3.p2.8.m8.1.1.2" xref="A3.p2.8.m8.1.1.2.cmml">p</mi><mo id="A3.p2.8.m8.1.1.1" xref="A3.p2.8.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A3.p2.8.m8.1b"><apply id="A3.p2.8.m8.1.1.cmml" xref="A3.p2.8.m8.1.1"><ci id="A3.p2.8.m8.1.1.1.cmml" xref="A3.p2.8.m8.1.1.1">^</ci><ci id="A3.p2.8.m8.1.1.2.cmml" xref="A3.p2.8.m8.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.8.m8.1c">\hat{p}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.8.m8.1d">over^ start_ARG italic_p end_ARG</annotation></semantics></math> is likely to fall in a bucket close to <math alttext="p" class="ltx_Math" display="inline" id="A3.p2.9.m9.1"><semantics id="A3.p2.9.m9.1a"><mi id="A3.p2.9.m9.1.1" xref="A3.p2.9.m9.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.p2.9.m9.1b"><ci id="A3.p2.9.m9.1.1.cmml" xref="A3.p2.9.m9.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.9.m9.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.p2.9.m9.1d">italic_p</annotation></semantics></math>. In combination with <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#Thmlemma1" title="Lemma 1. ‣ Appendix C Proof of Thm. 2 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">1</span></a>, for <math alttext="f(z_{1}^{k})=g(x_{t+k+1}=0,H_{t+k})" class="ltx_Math" display="inline" id="A3.p2.10.m10.3"><semantics id="A3.p2.10.m10.3a"><mrow id="A3.p2.10.m10.3.3" xref="A3.p2.10.m10.3.3.cmml"><mrow id="A3.p2.10.m10.2.2.1" xref="A3.p2.10.m10.2.2.1.cmml"><mi id="A3.p2.10.m10.2.2.1.3" xref="A3.p2.10.m10.2.2.1.3.cmml">f</mi><mo id="A3.p2.10.m10.2.2.1.2" xref="A3.p2.10.m10.2.2.1.2.cmml">⁢</mo><mrow id="A3.p2.10.m10.2.2.1.1.1" xref="A3.p2.10.m10.2.2.1.1.1.1.cmml"><mo id="A3.p2.10.m10.2.2.1.1.1.2" stretchy="false" xref="A3.p2.10.m10.2.2.1.1.1.1.cmml">(</mo><msubsup id="A3.p2.10.m10.2.2.1.1.1.1" xref="A3.p2.10.m10.2.2.1.1.1.1.cmml"><mi id="A3.p2.10.m10.2.2.1.1.1.1.2.2" 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xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.1.cmml" xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1">subscript</csymbol><ci id="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.2.cmml" xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.2">𝐻</ci><apply id="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.cmml" xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3"><plus id="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.1.cmml" xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.1"></plus><ci id="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.2.cmml" xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.2">𝑡</ci><ci id="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.3.cmml" xref="A3.p2.10.m10.3.3.2.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.10.m10.3c">f(z_{1}^{k})=g(x_{t+k+1}=0,H_{t+k})</annotation><annotation encoding="application/x-llamapun" id="A3.p2.10.m10.3d">italic_f ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = italic_g ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> we have that,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx28"> <tbody id="A3.E50"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\text{Shannon}}(\{g(x_{t+k+1}=0,H_{t+k}):z_{1}^{k}\in\{0,1\}^% {k}\})\geq 2^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-\log(2\lceil\eta_{P}% (t)/\varepsilon\rceil+1)\right)" class="ltx_Math" display="inline" id="A3.E50.m1.8"><semantics id="A3.E50.m1.8a"><mrow id="A3.E50.m1.8.8" xref="A3.E50.m1.8.8.cmml"><mrow id="A3.E50.m1.7.7.1" xref="A3.E50.m1.7.7.1.cmml"><msub id="A3.E50.m1.7.7.1.3" xref="A3.E50.m1.7.7.1.3.cmml"><mi id="A3.E50.m1.7.7.1.3.2" xref="A3.E50.m1.7.7.1.3.2.cmml">H</mi><mtext 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xref="A3.E50.m1.8.8.2.1.1.1.3.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E50.m1.8c">\displaystyle H_{\text{Shannon}}(\{g(x_{t+k+1}=0,H_{t+k}):z_{1}^{k}\in\{0,1\}^% {k}\})\geq 2^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-\log(2\lceil\eta_{P}% (t)/\varepsilon\rceil+1)\right)</annotation><annotation encoding="application/x-llamapun" id="A3.E50.m1.8d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_g ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) ≥ 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( ( 1 - 3 italic_ε ) roman_log ( 1 / italic_ε ) - roman_log ( 2 ⌈ italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) / italic_ε ⌉ + 1 ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(50)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p2.17">Note however, that <math alttext="g(x_{t+k+1}=0,H_{t+k})" class="ltx_Math" display="inline" id="A3.p2.11.m1.2"><semantics id="A3.p2.11.m1.2a"><mrow id="A3.p2.11.m1.2.2" xref="A3.p2.11.m1.2.2.cmml"><mi id="A3.p2.11.m1.2.2.3" xref="A3.p2.11.m1.2.2.3.cmml">g</mi><mo id="A3.p2.11.m1.2.2.2" xref="A3.p2.11.m1.2.2.2.cmml">⁢</mo><mrow id="A3.p2.11.m1.2.2.1.1" xref="A3.p2.11.m1.2.2.1.1.1.cmml"><mo id="A3.p2.11.m1.2.2.1.1.2" stretchy="false" xref="A3.p2.11.m1.2.2.1.1.1.cmml">(</mo><mrow id="A3.p2.11.m1.2.2.1.1.1" xref="A3.p2.11.m1.2.2.1.1.1.cmml"><msub id="A3.p2.11.m1.2.2.1.1.1.3" xref="A3.p2.11.m1.2.2.1.1.1.3.cmml"><mi id="A3.p2.11.m1.2.2.1.1.1.3.2" xref="A3.p2.11.m1.2.2.1.1.1.3.2.cmml">x</mi><mrow id="A3.p2.11.m1.2.2.1.1.1.3.3" xref="A3.p2.11.m1.2.2.1.1.1.3.3.cmml"><mi id="A3.p2.11.m1.2.2.1.1.1.3.3.2" xref="A3.p2.11.m1.2.2.1.1.1.3.3.2.cmml">t</mi><mo id="A3.p2.11.m1.2.2.1.1.1.3.3.1" xref="A3.p2.11.m1.2.2.1.1.1.3.3.1.cmml">+</mo><mi id="A3.p2.11.m1.2.2.1.1.1.3.3.3" xref="A3.p2.11.m1.2.2.1.1.1.3.3.3.cmml">k</mi><mo id="A3.p2.11.m1.2.2.1.1.1.3.3.1a" xref="A3.p2.11.m1.2.2.1.1.1.3.3.1.cmml">+</mo><mn id="A3.p2.11.m1.2.2.1.1.1.3.3.4" xref="A3.p2.11.m1.2.2.1.1.1.3.3.4.cmml">1</mn></mrow></msub><mo id="A3.p2.11.m1.2.2.1.1.1.2" xref="A3.p2.11.m1.2.2.1.1.1.2.cmml">=</mo><mrow id="A3.p2.11.m1.2.2.1.1.1.1.1" xref="A3.p2.11.m1.2.2.1.1.1.1.2.cmml"><mn id="A3.p2.11.m1.1.1" xref="A3.p2.11.m1.1.1.cmml">0</mn><mo id="A3.p2.11.m1.2.2.1.1.1.1.1.2" xref="A3.p2.11.m1.2.2.1.1.1.1.2.cmml">,</mo><msub id="A3.p2.11.m1.2.2.1.1.1.1.1.1" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.cmml"><mi id="A3.p2.11.m1.2.2.1.1.1.1.1.1.2" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.2.cmml">H</mi><mrow id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.cmml"><mi id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.2" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.1" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.3" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.3.cmml">k</mi></mrow></msub></mrow></mrow><mo id="A3.p2.11.m1.2.2.1.1.3" stretchy="false" xref="A3.p2.11.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.11.m1.2b"><apply id="A3.p2.11.m1.2.2.cmml" xref="A3.p2.11.m1.2.2"><times id="A3.p2.11.m1.2.2.2.cmml" xref="A3.p2.11.m1.2.2.2"></times><ci id="A3.p2.11.m1.2.2.3.cmml" xref="A3.p2.11.m1.2.2.3">𝑔</ci><apply id="A3.p2.11.m1.2.2.1.1.1.cmml" xref="A3.p2.11.m1.2.2.1.1"><eq id="A3.p2.11.m1.2.2.1.1.1.2.cmml" xref="A3.p2.11.m1.2.2.1.1.1.2"></eq><apply id="A3.p2.11.m1.2.2.1.1.1.3.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.p2.11.m1.2.2.1.1.1.3.1.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3">subscript</csymbol><ci id="A3.p2.11.m1.2.2.1.1.1.3.2.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3.2">𝑥</ci><apply id="A3.p2.11.m1.2.2.1.1.1.3.3.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3.3"><plus id="A3.p2.11.m1.2.2.1.1.1.3.3.1.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3.3.1"></plus><ci id="A3.p2.11.m1.2.2.1.1.1.3.3.2.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3.3.2">𝑡</ci><ci id="A3.p2.11.m1.2.2.1.1.1.3.3.3.cmml" xref="A3.p2.11.m1.2.2.1.1.1.3.3.3">𝑘</ci><cn id="A3.p2.11.m1.2.2.1.1.1.3.3.4.cmml" type="integer" xref="A3.p2.11.m1.2.2.1.1.1.3.3.4">1</cn></apply></apply><list id="A3.p2.11.m1.2.2.1.1.1.1.2.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1"><cn id="A3.p2.11.m1.1.1.cmml" type="integer" xref="A3.p2.11.m1.1.1">0</cn><apply id="A3.p2.11.m1.2.2.1.1.1.1.1.1.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.p2.11.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="A3.p2.11.m1.2.2.1.1.1.1.1.1.2.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.2">𝐻</ci><apply id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3"><plus id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.1"></plus><ci id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.2">𝑡</ci><ci id="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="A3.p2.11.m1.2.2.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.11.m1.2c">g(x_{t+k+1}=0,H_{t+k})</annotation><annotation encoding="application/x-llamapun" id="A3.p2.11.m1.2d">italic_g ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> is a function of <math alttext="z_{1}^{k}" class="ltx_Math" display="inline" id="A3.p2.12.m2.1"><semantics id="A3.p2.12.m2.1a"><msubsup id="A3.p2.12.m2.1.1" xref="A3.p2.12.m2.1.1.cmml"><mi id="A3.p2.12.m2.1.1.2.2" xref="A3.p2.12.m2.1.1.2.2.cmml">z</mi><mn id="A3.p2.12.m2.1.1.2.3" xref="A3.p2.12.m2.1.1.2.3.cmml">1</mn><mi id="A3.p2.12.m2.1.1.3" xref="A3.p2.12.m2.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p2.12.m2.1b"><apply id="A3.p2.12.m2.1.1.cmml" xref="A3.p2.12.m2.1.1"><csymbol cd="ambiguous" id="A3.p2.12.m2.1.1.1.cmml" xref="A3.p2.12.m2.1.1">superscript</csymbol><apply id="A3.p2.12.m2.1.1.2.cmml" xref="A3.p2.12.m2.1.1"><csymbol cd="ambiguous" id="A3.p2.12.m2.1.1.2.1.cmml" xref="A3.p2.12.m2.1.1">subscript</csymbol><ci id="A3.p2.12.m2.1.1.2.2.cmml" xref="A3.p2.12.m2.1.1.2.2">𝑧</ci><cn id="A3.p2.12.m2.1.1.2.3.cmml" type="integer" xref="A3.p2.12.m2.1.1.2.3">1</cn></apply><ci id="A3.p2.12.m2.1.1.3.cmml" xref="A3.p2.12.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.12.m2.1c">z_{1}^{k}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.12.m2.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> implicitly, through <math alttext="H_{t+k}" class="ltx_Math" display="inline" id="A3.p2.13.m3.1"><semantics id="A3.p2.13.m3.1a"><msub id="A3.p2.13.m3.1.1" xref="A3.p2.13.m3.1.1.cmml"><mi id="A3.p2.13.m3.1.1.2" xref="A3.p2.13.m3.1.1.2.cmml">H</mi><mrow id="A3.p2.13.m3.1.1.3" xref="A3.p2.13.m3.1.1.3.cmml"><mi id="A3.p2.13.m3.1.1.3.2" xref="A3.p2.13.m3.1.1.3.2.cmml">t</mi><mo id="A3.p2.13.m3.1.1.3.1" xref="A3.p2.13.m3.1.1.3.1.cmml">+</mo><mi id="A3.p2.13.m3.1.1.3.3" xref="A3.p2.13.m3.1.1.3.3.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.p2.13.m3.1b"><apply id="A3.p2.13.m3.1.1.cmml" xref="A3.p2.13.m3.1.1"><csymbol cd="ambiguous" id="A3.p2.13.m3.1.1.1.cmml" xref="A3.p2.13.m3.1.1">subscript</csymbol><ci id="A3.p2.13.m3.1.1.2.cmml" xref="A3.p2.13.m3.1.1.2">𝐻</ci><apply id="A3.p2.13.m3.1.1.3.cmml" xref="A3.p2.13.m3.1.1.3"><plus id="A3.p2.13.m3.1.1.3.1.cmml" xref="A3.p2.13.m3.1.1.3.1"></plus><ci id="A3.p2.13.m3.1.1.3.2.cmml" xref="A3.p2.13.m3.1.1.3.2">𝑡</ci><ci id="A3.p2.13.m3.1.1.3.3.cmml" xref="A3.p2.13.m3.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.13.m3.1c">H_{t+k}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.13.m3.1d">italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT</annotation></semantics></math> (which is also a function of <math alttext="x_{1}^{t}" class="ltx_Math" display="inline" id="A3.p2.14.m4.1"><semantics id="A3.p2.14.m4.1a"><msubsup id="A3.p2.14.m4.1.1" xref="A3.p2.14.m4.1.1.cmml"><mi id="A3.p2.14.m4.1.1.2.2" xref="A3.p2.14.m4.1.1.2.2.cmml">x</mi><mn id="A3.p2.14.m4.1.1.2.3" xref="A3.p2.14.m4.1.1.2.3.cmml">1</mn><mi id="A3.p2.14.m4.1.1.3" xref="A3.p2.14.m4.1.1.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A3.p2.14.m4.1b"><apply id="A3.p2.14.m4.1.1.cmml" xref="A3.p2.14.m4.1.1"><csymbol cd="ambiguous" id="A3.p2.14.m4.1.1.1.cmml" xref="A3.p2.14.m4.1.1">superscript</csymbol><apply id="A3.p2.14.m4.1.1.2.cmml" xref="A3.p2.14.m4.1.1"><csymbol cd="ambiguous" id="A3.p2.14.m4.1.1.2.1.cmml" xref="A3.p2.14.m4.1.1">subscript</csymbol><ci id="A3.p2.14.m4.1.1.2.2.cmml" xref="A3.p2.14.m4.1.1.2.2">𝑥</ci><cn id="A3.p2.14.m4.1.1.2.3.cmml" type="integer" xref="A3.p2.14.m4.1.1.2.3">1</cn></apply><ci id="A3.p2.14.m4.1.1.3.cmml" xref="A3.p2.14.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.14.m4.1c">x_{1}^{t}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.14.m4.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>). Since the dimensionality of <math alttext="H_{t+k}" class="ltx_Math" display="inline" id="A3.p2.15.m5.1"><semantics id="A3.p2.15.m5.1a"><msub id="A3.p2.15.m5.1.1" xref="A3.p2.15.m5.1.1.cmml"><mi id="A3.p2.15.m5.1.1.2" xref="A3.p2.15.m5.1.1.2.cmml">H</mi><mrow id="A3.p2.15.m5.1.1.3" xref="A3.p2.15.m5.1.1.3.cmml"><mi id="A3.p2.15.m5.1.1.3.2" xref="A3.p2.15.m5.1.1.3.2.cmml">t</mi><mo id="A3.p2.15.m5.1.1.3.1" xref="A3.p2.15.m5.1.1.3.1.cmml">+</mo><mi id="A3.p2.15.m5.1.1.3.3" xref="A3.p2.15.m5.1.1.3.3.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A3.p2.15.m5.1b"><apply id="A3.p2.15.m5.1.1.cmml" xref="A3.p2.15.m5.1.1"><csymbol cd="ambiguous" id="A3.p2.15.m5.1.1.1.cmml" xref="A3.p2.15.m5.1.1">subscript</csymbol><ci id="A3.p2.15.m5.1.1.2.cmml" xref="A3.p2.15.m5.1.1.2">𝐻</ci><apply id="A3.p2.15.m5.1.1.3.cmml" xref="A3.p2.15.m5.1.1.3"><plus id="A3.p2.15.m5.1.1.3.1.cmml" xref="A3.p2.15.m5.1.1.3.1"></plus><ci id="A3.p2.15.m5.1.1.3.2.cmml" xref="A3.p2.15.m5.1.1.3.2">𝑡</ci><ci id="A3.p2.15.m5.1.1.3.3.cmml" xref="A3.p2.15.m5.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.15.m5.1c">H_{t+k}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.15.m5.1d">italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="d" class="ltx_Math" display="inline" id="A3.p2.16.m6.1"><semantics id="A3.p2.16.m6.1a"><mi id="A3.p2.16.m6.1.1" xref="A3.p2.16.m6.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="A3.p2.16.m6.1b"><ci id="A3.p2.16.m6.1.1.cmml" xref="A3.p2.16.m6.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.16.m6.1c">d</annotation><annotation encoding="application/x-llamapun" id="A3.p2.16.m6.1d">italic_d</annotation></semantics></math> and the model is implemented to <math alttext="\mathtt{p}" class="ltx_Math" display="inline" id="A3.p2.17.m7.1"><semantics id="A3.p2.17.m7.1a"><mi id="A3.p2.17.m7.1.1" xref="A3.p2.17.m7.1.1.cmml">𝚙</mi><annotation-xml encoding="MathML-Content" id="A3.p2.17.m7.1b"><ci id="A3.p2.17.m7.1.1.cmml" xref="A3.p2.17.m7.1.1">𝚙</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.17.m7.1c">\mathtt{p}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.17.m7.1d">typewriter_p</annotation></semantics></math> bits of precision,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx29"> <tbody id="A3.E51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\text{Shannon}}(\{g(x_{t+k+1}=0,H_{t+k}):z_{1}^{k}\in\{0,1\}^% {k}\})\leq H_{\text{Shannon}}(H_{t+k})\leq d\mathtt{p}" class="ltx_Math" display="inline" id="A3.E51.m1.5"><semantics id="A3.E51.m1.5a"><mrow id="A3.E51.m1.5.5" xref="A3.E51.m1.5.5.cmml"><mrow id="A3.E51.m1.4.4.1" 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xref="A3.E51.m1.4.4.1.1.1.1.2.2.3.2.1.cmml">{</mo><mn id="A3.E51.m1.2.2" xref="A3.E51.m1.2.2.cmml">0</mn><mo id="A3.E51.m1.4.4.1.1.1.1.2.2.3.2.2.2" xref="A3.E51.m1.4.4.1.1.1.1.2.2.3.2.1.cmml">,</mo><mn id="A3.E51.m1.3.3" xref="A3.E51.m1.3.3.cmml">1</mn><mo id="A3.E51.m1.4.4.1.1.1.1.2.2.3.2.2.3" stretchy="false" xref="A3.E51.m1.4.4.1.1.1.1.2.2.3.2.1.cmml">}</mo></mrow><mi id="A3.E51.m1.4.4.1.1.1.1.2.2.3.3" xref="A3.E51.m1.4.4.1.1.1.1.2.2.3.3.cmml">k</mi></msup></mrow><mo id="A3.E51.m1.4.4.1.1.1.1.2.5" stretchy="false" xref="A3.E51.m1.4.4.1.1.1.1.3.1.cmml">}</mo></mrow><mo id="A3.E51.m1.4.4.1.1.1.3" stretchy="false" xref="A3.E51.m1.4.4.1.cmml">)</mo></mrow></mrow><mo id="A3.E51.m1.5.5.4" xref="A3.E51.m1.5.5.4.cmml">≤</mo><mrow id="A3.E51.m1.5.5.2" xref="A3.E51.m1.5.5.2.cmml"><msub id="A3.E51.m1.5.5.2.3" xref="A3.E51.m1.5.5.2.3.cmml"><mi id="A3.E51.m1.5.5.2.3.2" xref="A3.E51.m1.5.5.2.3.2.cmml">H</mi><mtext id="A3.E51.m1.5.5.2.3.3" xref="A3.E51.m1.5.5.2.3.3a.cmml">Shannon</mtext></msub><mo 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xref="A3.E51.m1.5.5.6.1.cmml">⁢</mo><mi id="A3.E51.m1.5.5.6.3" xref="A3.E51.m1.5.5.6.3.cmml">𝚙</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E51.m1.5b"><apply id="A3.E51.m1.5.5.cmml" xref="A3.E51.m1.5.5"><and id="A3.E51.m1.5.5a.cmml" xref="A3.E51.m1.5.5"></and><apply id="A3.E51.m1.5.5b.cmml" xref="A3.E51.m1.5.5"><leq id="A3.E51.m1.5.5.4.cmml" xref="A3.E51.m1.5.5.4"></leq><apply id="A3.E51.m1.4.4.1.cmml" xref="A3.E51.m1.4.4.1"><times id="A3.E51.m1.4.4.1.2.cmml" xref="A3.E51.m1.4.4.1.2"></times><apply id="A3.E51.m1.4.4.1.3.cmml" xref="A3.E51.m1.4.4.1.3"><csymbol cd="ambiguous" id="A3.E51.m1.4.4.1.3.1.cmml" xref="A3.E51.m1.4.4.1.3">subscript</csymbol><ci id="A3.E51.m1.4.4.1.3.2.cmml" xref="A3.E51.m1.4.4.1.3.2">𝐻</ci><ci id="A3.E51.m1.4.4.1.3.3a.cmml" xref="A3.E51.m1.4.4.1.3.3"><mtext id="A3.E51.m1.4.4.1.3.3.cmml" mathsize="70%" xref="A3.E51.m1.4.4.1.3.3">Shannon</mtext></ci></apply><apply id="A3.E51.m1.4.4.1.1.1.1.3.cmml" xref="A3.E51.m1.4.4.1.1.1.1.2"><csymbol 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xref="A3.E51.m1.5.5.2.2"></times><apply id="A3.E51.m1.5.5.2.3.cmml" xref="A3.E51.m1.5.5.2.3"><csymbol cd="ambiguous" id="A3.E51.m1.5.5.2.3.1.cmml" xref="A3.E51.m1.5.5.2.3">subscript</csymbol><ci id="A3.E51.m1.5.5.2.3.2.cmml" xref="A3.E51.m1.5.5.2.3.2">𝐻</ci><ci id="A3.E51.m1.5.5.2.3.3a.cmml" xref="A3.E51.m1.5.5.2.3.3"><mtext id="A3.E51.m1.5.5.2.3.3.cmml" mathsize="70%" xref="A3.E51.m1.5.5.2.3.3">Shannon</mtext></ci></apply><apply id="A3.E51.m1.5.5.2.1.1.1.cmml" xref="A3.E51.m1.5.5.2.1.1"><csymbol cd="ambiguous" id="A3.E51.m1.5.5.2.1.1.1.1.cmml" xref="A3.E51.m1.5.5.2.1.1">subscript</csymbol><ci id="A3.E51.m1.5.5.2.1.1.1.2.cmml" xref="A3.E51.m1.5.5.2.1.1.1.2">𝐻</ci><apply id="A3.E51.m1.5.5.2.1.1.1.3.cmml" xref="A3.E51.m1.5.5.2.1.1.1.3"><plus id="A3.E51.m1.5.5.2.1.1.1.3.1.cmml" xref="A3.E51.m1.5.5.2.1.1.1.3.1"></plus><ci id="A3.E51.m1.5.5.2.1.1.1.3.2.cmml" xref="A3.E51.m1.5.5.2.1.1.1.3.2">𝑡</ci><ci id="A3.E51.m1.5.5.2.1.1.1.3.3.cmml" xref="A3.E51.m1.5.5.2.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply><apply id="A3.E51.m1.5.5c.cmml" xref="A3.E51.m1.5.5"><leq id="A3.E51.m1.5.5.5.cmml" xref="A3.E51.m1.5.5.5"></leq><share href="https://arxiv.org/html/2502.10178v1#A3.E51.m1.5.5.2.cmml" id="A3.E51.m1.5.5d.cmml" xref="A3.E51.m1.5.5"></share><apply id="A3.E51.m1.5.5.6.cmml" xref="A3.E51.m1.5.5.6"><times id="A3.E51.m1.5.5.6.1.cmml" xref="A3.E51.m1.5.5.6.1"></times><ci id="A3.E51.m1.5.5.6.2.cmml" xref="A3.E51.m1.5.5.6.2">𝑑</ci><ci id="A3.E51.m1.5.5.6.3.cmml" xref="A3.E51.m1.5.5.6.3">𝚙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E51.m1.5c">\displaystyle H_{\text{Shannon}}(\{g(x_{t+k+1}=0,H_{t+k}):z_{1}^{k}\in\{0,1\}^% {k}\})\leq H_{\text{Shannon}}(H_{t+k})\leq d\mathtt{p}</annotation><annotation encoding="application/x-llamapun" id="A3.E51.m1.5d">italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( { italic_g ( italic_x start_POSTSUBSCRIPT italic_t + italic_k + 1 end_POSTSUBSCRIPT = 0 , italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT ) : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } ) ≤ italic_H start_POSTSUBSCRIPT Shannon end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_t + italic_k end_POSTSUBSCRIPT ) ≤ italic_d typewriter_p</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(51)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p2.19">where all randomness here is induced by the random draw of the <math alttext="k^{\text{th}}" class="ltx_Math" display="inline" id="A3.p2.18.m1.1"><semantics id="A3.p2.18.m1.1a"><msup id="A3.p2.18.m1.1.1" xref="A3.p2.18.m1.1.1.cmml"><mi id="A3.p2.18.m1.1.1.2" xref="A3.p2.18.m1.1.1.2.cmml">k</mi><mtext id="A3.p2.18.m1.1.1.3" xref="A3.p2.18.m1.1.1.3a.cmml">th</mtext></msup><annotation-xml encoding="MathML-Content" id="A3.p2.18.m1.1b"><apply id="A3.p2.18.m1.1.1.cmml" xref="A3.p2.18.m1.1.1"><csymbol cd="ambiguous" id="A3.p2.18.m1.1.1.1.cmml" xref="A3.p2.18.m1.1.1">superscript</csymbol><ci id="A3.p2.18.m1.1.1.2.cmml" xref="A3.p2.18.m1.1.1.2">𝑘</ci><ci id="A3.p2.18.m1.1.1.3a.cmml" xref="A3.p2.18.m1.1.1.3"><mtext id="A3.p2.18.m1.1.1.3.cmml" mathsize="70%" xref="A3.p2.18.m1.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.18.m1.1c">k^{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.18.m1.1d">italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT</annotation></semantics></math>-order Markov kernel <math alttext="P" class="ltx_Math" display="inline" id="A3.p2.19.m2.1"><semantics id="A3.p2.19.m2.1a"><mi id="A3.p2.19.m2.1.1" xref="A3.p2.19.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p2.19.m2.1b"><ci id="A3.p2.19.m2.1.1.cmml" xref="A3.p2.19.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.19.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p2.19.m2.1d">italic_P</annotation></semantics></math>. Therefore, for the correctness guarantee <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#A3.E38" title="In Appendix C Proof of Thm. 2 ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">38</span></a> to hold, we need,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx30"> <tbody id="A3.E52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d\mathtt{p}\geq 2^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-% \log(2\lceil\eta_{P}(t)/\varepsilon\rceil+1)\right)" class="ltx_Math" display="inline" id="A3.E52.m1.4"><semantics id="A3.E52.m1.4a"><mrow id="A3.E52.m1.4.4" xref="A3.E52.m1.4.4.cmml"><mrow id="A3.E52.m1.4.4.3" xref="A3.E52.m1.4.4.3.cmml"><mi id="A3.E52.m1.4.4.3.2" xref="A3.E52.m1.4.4.3.2.cmml">d</mi><mo 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id="A3.E52.m1.4.4.1.1.1.1.3.1.1.1.1.1.1.1.3.cmml" xref="A3.E52.m1.4.4.1.1.1.1.3.1.1.1.1.1.1.1.3">𝜀</ci></apply></apply></apply><cn id="A3.E52.m1.4.4.1.1.1.1.3.1.1.1.3.cmml" type="integer" xref="A3.E52.m1.4.4.1.1.1.1.3.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E52.m1.4c">\displaystyle d\mathtt{p}\geq 2^{k}\left((1-3\varepsilon)\log(1/\varepsilon)-% \log(2\lceil\eta_{P}(t)/\varepsilon\rceil+1)\right)</annotation><annotation encoding="application/x-llamapun" id="A3.E52.m1.4d">italic_d typewriter_p ≥ 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( ( 1 - 3 italic_ε ) roman_log ( 1 / italic_ε ) - roman_log ( 2 ⌈ italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) / italic_ε ⌉ + 1 ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(52)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p2.22">in the limit <math alttext="t\to\infty" class="ltx_Math" display="inline" id="A3.p2.20.m1.1"><semantics id="A3.p2.20.m1.1a"><mrow id="A3.p2.20.m1.1.1" xref="A3.p2.20.m1.1.1.cmml"><mi id="A3.p2.20.m1.1.1.2" xref="A3.p2.20.m1.1.1.2.cmml">t</mi><mo id="A3.p2.20.m1.1.1.1" stretchy="false" xref="A3.p2.20.m1.1.1.1.cmml">→</mo><mi id="A3.p2.20.m1.1.1.3" mathvariant="normal" xref="A3.p2.20.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.20.m1.1b"><apply id="A3.p2.20.m1.1.1.cmml" xref="A3.p2.20.m1.1.1"><ci id="A3.p2.20.m1.1.1.1.cmml" xref="A3.p2.20.m1.1.1.1">→</ci><ci id="A3.p2.20.m1.1.1.2.cmml" xref="A3.p2.20.m1.1.1.2">𝑡</ci><infinity id="A3.p2.20.m1.1.1.3.cmml" xref="A3.p2.20.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.20.m1.1c">t\to\infty</annotation><annotation encoding="application/x-llamapun" id="A3.p2.20.m1.1d">italic_t → ∞</annotation></semantics></math>, and noting that <math alttext="\limsup_{t\to\infty}\eta_{P}(t)=0" class="ltx_Math" display="inline" id="A3.p2.21.m2.1"><semantics id="A3.p2.21.m2.1a"><mrow id="A3.p2.21.m2.1.2" xref="A3.p2.21.m2.1.2.cmml"><mrow id="A3.p2.21.m2.1.2.2" xref="A3.p2.21.m2.1.2.2.cmml"><msub id="A3.p2.21.m2.1.2.2.1" xref="A3.p2.21.m2.1.2.2.1.cmml"><mo id="A3.p2.21.m2.1.2.2.1.2" xref="A3.p2.21.m2.1.2.2.1.2.cmml">lim sup</mo><mrow id="A3.p2.21.m2.1.2.2.1.3" xref="A3.p2.21.m2.1.2.2.1.3.cmml"><mi id="A3.p2.21.m2.1.2.2.1.3.2" xref="A3.p2.21.m2.1.2.2.1.3.2.cmml">t</mi><mo id="A3.p2.21.m2.1.2.2.1.3.1" stretchy="false" xref="A3.p2.21.m2.1.2.2.1.3.1.cmml">→</mo><mi id="A3.p2.21.m2.1.2.2.1.3.3" mathvariant="normal" xref="A3.p2.21.m2.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="A3.p2.21.m2.1.2.2.2" xref="A3.p2.21.m2.1.2.2.2.cmml"><msub id="A3.p2.21.m2.1.2.2.2.2" xref="A3.p2.21.m2.1.2.2.2.2.cmml"><mi id="A3.p2.21.m2.1.2.2.2.2.2" xref="A3.p2.21.m2.1.2.2.2.2.2.cmml">η</mi><mi id="A3.p2.21.m2.1.2.2.2.2.3" xref="A3.p2.21.m2.1.2.2.2.2.3.cmml">P</mi></msub><mo id="A3.p2.21.m2.1.2.2.2.1" xref="A3.p2.21.m2.1.2.2.2.1.cmml">⁢</mo><mrow id="A3.p2.21.m2.1.2.2.2.3.2" xref="A3.p2.21.m2.1.2.2.2.cmml"><mo id="A3.p2.21.m2.1.2.2.2.3.2.1" stretchy="false" xref="A3.p2.21.m2.1.2.2.2.cmml">(</mo><mi id="A3.p2.21.m2.1.1" xref="A3.p2.21.m2.1.1.cmml">t</mi><mo id="A3.p2.21.m2.1.2.2.2.3.2.2" stretchy="false" xref="A3.p2.21.m2.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A3.p2.21.m2.1.2.1" xref="A3.p2.21.m2.1.2.1.cmml">=</mo><mn id="A3.p2.21.m2.1.2.3" xref="A3.p2.21.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.21.m2.1b"><apply id="A3.p2.21.m2.1.2.cmml" xref="A3.p2.21.m2.1.2"><eq id="A3.p2.21.m2.1.2.1.cmml" xref="A3.p2.21.m2.1.2.1"></eq><apply id="A3.p2.21.m2.1.2.2.cmml" xref="A3.p2.21.m2.1.2.2"><apply id="A3.p2.21.m2.1.2.2.1.cmml" xref="A3.p2.21.m2.1.2.2.1"><csymbol cd="ambiguous" id="A3.p2.21.m2.1.2.2.1.1.cmml" xref="A3.p2.21.m2.1.2.2.1">subscript</csymbol><csymbol cd="latexml" id="A3.p2.21.m2.1.2.2.1.2.cmml" xref="A3.p2.21.m2.1.2.2.1.2">limit-supremum</csymbol><apply id="A3.p2.21.m2.1.2.2.1.3.cmml" xref="A3.p2.21.m2.1.2.2.1.3"><ci id="A3.p2.21.m2.1.2.2.1.3.1.cmml" xref="A3.p2.21.m2.1.2.2.1.3.1">→</ci><ci id="A3.p2.21.m2.1.2.2.1.3.2.cmml" xref="A3.p2.21.m2.1.2.2.1.3.2">𝑡</ci><infinity id="A3.p2.21.m2.1.2.2.1.3.3.cmml" xref="A3.p2.21.m2.1.2.2.1.3.3"></infinity></apply></apply><apply id="A3.p2.21.m2.1.2.2.2.cmml" xref="A3.p2.21.m2.1.2.2.2"><times id="A3.p2.21.m2.1.2.2.2.1.cmml" xref="A3.p2.21.m2.1.2.2.2.1"></times><apply id="A3.p2.21.m2.1.2.2.2.2.cmml" xref="A3.p2.21.m2.1.2.2.2.2"><csymbol cd="ambiguous" id="A3.p2.21.m2.1.2.2.2.2.1.cmml" xref="A3.p2.21.m2.1.2.2.2.2">subscript</csymbol><ci id="A3.p2.21.m2.1.2.2.2.2.2.cmml" xref="A3.p2.21.m2.1.2.2.2.2.2">𝜂</ci><ci id="A3.p2.21.m2.1.2.2.2.2.3.cmml" xref="A3.p2.21.m2.1.2.2.2.2.3">𝑃</ci></apply><ci id="A3.p2.21.m2.1.1.cmml" xref="A3.p2.21.m2.1.1">𝑡</ci></apply></apply><cn id="A3.p2.21.m2.1.2.3.cmml" type="integer" xref="A3.p2.21.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.21.m2.1c">\limsup_{t\to\infty}\eta_{P}(t)=0</annotation><annotation encoding="application/x-llamapun" id="A3.p2.21.m2.1d">lim sup start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) = 0</annotation></semantics></math> almost surely over <math alttext="P" class="ltx_Math" display="inline" id="A3.p2.22.m3.1"><semantics id="A3.p2.22.m3.1a"><mi id="A3.p2.22.m3.1.1" xref="A3.p2.22.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A3.p2.22.m3.1b"><ci id="A3.p2.22.m3.1.1.cmml" xref="A3.p2.22.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.22.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="A3.p2.22.m3.1d">italic_P</annotation></semantics></math> drawn from the prior, it is necessary that,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx31"> <tbody id="A3.E53"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d\mathtt{p}\geq 2^{k}(1-3\varepsilon)\log(1/\varepsilon)." class="ltx_Math" display="inline" id="A3.E53.m1.2"><semantics id="A3.E53.m1.2a"><mrow id="A3.E53.m1.2.2.1" xref="A3.E53.m1.2.2.1.1.cmml"><mrow id="A3.E53.m1.2.2.1.1" xref="A3.E53.m1.2.2.1.1.cmml"><mrow id="A3.E53.m1.2.2.1.1.4" xref="A3.E53.m1.2.2.1.1.4.cmml"><mi id="A3.E53.m1.2.2.1.1.4.2" xref="A3.E53.m1.2.2.1.1.4.2.cmml">d</mi><mo id="A3.E53.m1.2.2.1.1.4.1" xref="A3.E53.m1.2.2.1.1.4.1.cmml">⁢</mo><mi id="A3.E53.m1.2.2.1.1.4.3" xref="A3.E53.m1.2.2.1.1.4.3.cmml">𝚙</mi></mrow><mo id="A3.E53.m1.2.2.1.1.3" xref="A3.E53.m1.2.2.1.1.3.cmml">≥</mo><mrow id="A3.E53.m1.2.2.1.1.2" xref="A3.E53.m1.2.2.1.1.2.cmml"><msup id="A3.E53.m1.2.2.1.1.2.4" xref="A3.E53.m1.2.2.1.1.2.4.cmml"><mn id="A3.E53.m1.2.2.1.1.2.4.2" xref="A3.E53.m1.2.2.1.1.2.4.2.cmml">2</mn><mi id="A3.E53.m1.2.2.1.1.2.4.3" xref="A3.E53.m1.2.2.1.1.2.4.3.cmml">k</mi></msup><mo id="A3.E53.m1.2.2.1.1.2.3" xref="A3.E53.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A3.E53.m1.2.2.1.1.1.1.1" xref="A3.E53.m1.2.2.1.1.1.1.1.1.cmml"><mo id="A3.E53.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.E53.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.E53.m1.2.2.1.1.1.1.1.1" xref="A3.E53.m1.2.2.1.1.1.1.1.1.cmml"><mn id="A3.E53.m1.2.2.1.1.1.1.1.1.2" xref="A3.E53.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="A3.E53.m1.2.2.1.1.1.1.1.1.1" xref="A3.E53.m1.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.E53.m1.2.2.1.1.1.1.1.1.3" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.cmml"><mn id="A3.E53.m1.2.2.1.1.1.1.1.1.3.2" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.2.cmml">3</mn><mo id="A3.E53.m1.2.2.1.1.1.1.1.1.3.1" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A3.E53.m1.2.2.1.1.1.1.1.1.3.3" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.3.cmml">ε</mi></mrow></mrow><mo id="A3.E53.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="A3.E53.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.E53.m1.2.2.1.1.2.3a" lspace="0.167em" xref="A3.E53.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A3.E53.m1.2.2.1.1.2.2.1" xref="A3.E53.m1.2.2.1.1.2.2.2.cmml"><mi id="A3.E53.m1.1.1" xref="A3.E53.m1.1.1.cmml">log</mi><mo id="A3.E53.m1.2.2.1.1.2.2.1a" xref="A3.E53.m1.2.2.1.1.2.2.2.cmml">⁡</mo><mrow id="A3.E53.m1.2.2.1.1.2.2.1.1" xref="A3.E53.m1.2.2.1.1.2.2.2.cmml"><mo id="A3.E53.m1.2.2.1.1.2.2.1.1.2" stretchy="false" xref="A3.E53.m1.2.2.1.1.2.2.2.cmml">(</mo><mrow id="A3.E53.m1.2.2.1.1.2.2.1.1.1" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.cmml"><mn id="A3.E53.m1.2.2.1.1.2.2.1.1.1.2" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.2.cmml">1</mn><mo id="A3.E53.m1.2.2.1.1.2.2.1.1.1.1" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.1.cmml">/</mo><mi id="A3.E53.m1.2.2.1.1.2.2.1.1.1.3" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A3.E53.m1.2.2.1.1.2.2.1.1.3" stretchy="false" xref="A3.E53.m1.2.2.1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A3.E53.m1.2.2.1.2" lspace="0em" xref="A3.E53.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E53.m1.2b"><apply id="A3.E53.m1.2.2.1.1.cmml" xref="A3.E53.m1.2.2.1"><geq id="A3.E53.m1.2.2.1.1.3.cmml" xref="A3.E53.m1.2.2.1.1.3"></geq><apply id="A3.E53.m1.2.2.1.1.4.cmml" xref="A3.E53.m1.2.2.1.1.4"><times id="A3.E53.m1.2.2.1.1.4.1.cmml" xref="A3.E53.m1.2.2.1.1.4.1"></times><ci id="A3.E53.m1.2.2.1.1.4.2.cmml" xref="A3.E53.m1.2.2.1.1.4.2">𝑑</ci><ci id="A3.E53.m1.2.2.1.1.4.3.cmml" xref="A3.E53.m1.2.2.1.1.4.3">𝚙</ci></apply><apply id="A3.E53.m1.2.2.1.1.2.cmml" xref="A3.E53.m1.2.2.1.1.2"><times id="A3.E53.m1.2.2.1.1.2.3.cmml" xref="A3.E53.m1.2.2.1.1.2.3"></times><apply id="A3.E53.m1.2.2.1.1.2.4.cmml" xref="A3.E53.m1.2.2.1.1.2.4"><csymbol cd="ambiguous" id="A3.E53.m1.2.2.1.1.2.4.1.cmml" xref="A3.E53.m1.2.2.1.1.2.4">superscript</csymbol><cn id="A3.E53.m1.2.2.1.1.2.4.2.cmml" type="integer" xref="A3.E53.m1.2.2.1.1.2.4.2">2</cn><ci id="A3.E53.m1.2.2.1.1.2.4.3.cmml" xref="A3.E53.m1.2.2.1.1.2.4.3">𝑘</ci></apply><apply id="A3.E53.m1.2.2.1.1.1.1.1.1.cmml" xref="A3.E53.m1.2.2.1.1.1.1.1"><minus id="A3.E53.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.E53.m1.2.2.1.1.1.1.1.1.1"></minus><cn id="A3.E53.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="A3.E53.m1.2.2.1.1.1.1.1.1.2">1</cn><apply id="A3.E53.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3"><times id="A3.E53.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.1"></times><cn id="A3.E53.m1.2.2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.2">3</cn><ci id="A3.E53.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="A3.E53.m1.2.2.1.1.1.1.1.1.3.3">𝜀</ci></apply></apply><apply id="A3.E53.m1.2.2.1.1.2.2.2.cmml" xref="A3.E53.m1.2.2.1.1.2.2.1"><log id="A3.E53.m1.1.1.cmml" xref="A3.E53.m1.1.1"></log><apply id="A3.E53.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1"><divide id="A3.E53.m1.2.2.1.1.2.2.1.1.1.1.cmml" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.1"></divide><cn id="A3.E53.m1.2.2.1.1.2.2.1.1.1.2.cmml" type="integer" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.2">1</cn><ci id="A3.E53.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="A3.E53.m1.2.2.1.1.2.2.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E53.m1.2c">\displaystyle d\mathtt{p}\geq 2^{k}(1-3\varepsilon)\log(1/\varepsilon).</annotation><annotation encoding="application/x-llamapun" id="A3.E53.m1.2d">italic_d typewriter_p ≥ 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( 1 - 3 italic_ε ) roman_log ( 1 / italic_ε ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(53)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p2.23">∎</p> </div> <div class="ltx_para" id="A3.p3"> <p class="ltx_p" id="A3.p3.6"><span class="ltx_text ltx_font_bold" id="A3.p3.6.1">Remark.</span> The proof above assumes that the <math alttext="k^{\text{th}}" class="ltx_Math" display="inline" id="A3.p3.1.m1.1"><semantics id="A3.p3.1.m1.1a"><msup id="A3.p3.1.m1.1.1" xref="A3.p3.1.m1.1.1.cmml"><mi id="A3.p3.1.m1.1.1.2" xref="A3.p3.1.m1.1.1.2.cmml">k</mi><mtext id="A3.p3.1.m1.1.1.3" xref="A3.p3.1.m1.1.1.3a.cmml">th</mtext></msup><annotation-xml encoding="MathML-Content" id="A3.p3.1.m1.1b"><apply id="A3.p3.1.m1.1.1.cmml" xref="A3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A3.p3.1.m1.1.1.1.cmml" xref="A3.p3.1.m1.1.1">superscript</csymbol><ci id="A3.p3.1.m1.1.1.2.cmml" xref="A3.p3.1.m1.1.1.2">𝑘</ci><ci id="A3.p3.1.m1.1.1.3a.cmml" xref="A3.p3.1.m1.1.1.3"><mtext id="A3.p3.1.m1.1.1.3.cmml" mathsize="70%" xref="A3.p3.1.m1.1.1.3">th</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.1.m1.1c">k^{\text{th}}</annotation><annotation encoding="application/x-llamapun" id="A3.p3.1.m1.1d">italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT</annotation></semantics></math>-order Markov chain is on a binary state space. However, the result can easily be extended to give the lower bound <math alttext="d\cdot\mathtt{p}\geq\Omega(|\mathcal{X}|^{k})" class="ltx_Math" display="inline" id="A3.p3.2.m2.2"><semantics id="A3.p3.2.m2.2a"><mrow id="A3.p3.2.m2.2.2" xref="A3.p3.2.m2.2.2.cmml"><mrow id="A3.p3.2.m2.2.2.3" xref="A3.p3.2.m2.2.2.3.cmml"><mi id="A3.p3.2.m2.2.2.3.2" xref="A3.p3.2.m2.2.2.3.2.cmml">d</mi><mo id="A3.p3.2.m2.2.2.3.1" lspace="0.222em" rspace="0.222em" xref="A3.p3.2.m2.2.2.3.1.cmml">⋅</mo><mi id="A3.p3.2.m2.2.2.3.3" xref="A3.p3.2.m2.2.2.3.3.cmml">𝚙</mi></mrow><mo id="A3.p3.2.m2.2.2.2" xref="A3.p3.2.m2.2.2.2.cmml">≥</mo><mrow id="A3.p3.2.m2.2.2.1" xref="A3.p3.2.m2.2.2.1.cmml"><mi id="A3.p3.2.m2.2.2.1.3" mathvariant="normal" xref="A3.p3.2.m2.2.2.1.3.cmml">Ω</mi><mo id="A3.p3.2.m2.2.2.1.2" xref="A3.p3.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="A3.p3.2.m2.2.2.1.1.1" xref="A3.p3.2.m2.2.2.1.1.1.1.cmml"><mo id="A3.p3.2.m2.2.2.1.1.1.2" stretchy="false" xref="A3.p3.2.m2.2.2.1.1.1.1.cmml">(</mo><msup id="A3.p3.2.m2.2.2.1.1.1.1" xref="A3.p3.2.m2.2.2.1.1.1.1.cmml"><mrow id="A3.p3.2.m2.2.2.1.1.1.1.2.2" xref="A3.p3.2.m2.2.2.1.1.1.1.2.1.cmml"><mo id="A3.p3.2.m2.2.2.1.1.1.1.2.2.1" stretchy="false" xref="A3.p3.2.m2.2.2.1.1.1.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="A3.p3.2.m2.1.1" xref="A3.p3.2.m2.1.1.cmml">𝒳</mi><mo id="A3.p3.2.m2.2.2.1.1.1.1.2.2.2" stretchy="false" xref="A3.p3.2.m2.2.2.1.1.1.1.2.1.1.cmml">|</mo></mrow><mi id="A3.p3.2.m2.2.2.1.1.1.1.3" xref="A3.p3.2.m2.2.2.1.1.1.1.3.cmml">k</mi></msup><mo id="A3.p3.2.m2.2.2.1.1.1.3" stretchy="false" xref="A3.p3.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.2.m2.2b"><apply id="A3.p3.2.m2.2.2.cmml" xref="A3.p3.2.m2.2.2"><geq id="A3.p3.2.m2.2.2.2.cmml" xref="A3.p3.2.m2.2.2.2"></geq><apply id="A3.p3.2.m2.2.2.3.cmml" xref="A3.p3.2.m2.2.2.3"><ci id="A3.p3.2.m2.2.2.3.1.cmml" xref="A3.p3.2.m2.2.2.3.1">⋅</ci><ci id="A3.p3.2.m2.2.2.3.2.cmml" xref="A3.p3.2.m2.2.2.3.2">𝑑</ci><ci id="A3.p3.2.m2.2.2.3.3.cmml" xref="A3.p3.2.m2.2.2.3.3">𝚙</ci></apply><apply id="A3.p3.2.m2.2.2.1.cmml" xref="A3.p3.2.m2.2.2.1"><times id="A3.p3.2.m2.2.2.1.2.cmml" xref="A3.p3.2.m2.2.2.1.2"></times><ci id="A3.p3.2.m2.2.2.1.3.cmml" xref="A3.p3.2.m2.2.2.1.3">Ω</ci><apply id="A3.p3.2.m2.2.2.1.1.1.1.cmml" xref="A3.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.p3.2.m2.2.2.1.1.1.1.1.cmml" xref="A3.p3.2.m2.2.2.1.1.1">superscript</csymbol><apply id="A3.p3.2.m2.2.2.1.1.1.1.2.1.cmml" xref="A3.p3.2.m2.2.2.1.1.1.1.2.2"><abs id="A3.p3.2.m2.2.2.1.1.1.1.2.1.1.cmml" xref="A3.p3.2.m2.2.2.1.1.1.1.2.2.1"></abs><ci id="A3.p3.2.m2.1.1.cmml" xref="A3.p3.2.m2.1.1">𝒳</ci></apply><ci id="A3.p3.2.m2.2.2.1.1.1.1.3.cmml" xref="A3.p3.2.m2.2.2.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.2.m2.2c">d\cdot\mathtt{p}\geq\Omega(|\mathcal{X}|^{k})</annotation><annotation encoding="application/x-llamapun" id="A3.p3.2.m2.2d">italic_d ⋅ typewriter_p ≥ roman_Ω ( | caligraphic_X | start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT )</annotation></semantics></math> for larger state spaces, as well as similar scaling results for priors <math alttext="\mathrm{Dir}(\beta\cdot\bm{1})" class="ltx_Math" display="inline" id="A3.p3.3.m3.1"><semantics id="A3.p3.3.m3.1a"><mrow id="A3.p3.3.m3.1.1" xref="A3.p3.3.m3.1.1.cmml"><mi id="A3.p3.3.m3.1.1.3" xref="A3.p3.3.m3.1.1.3.cmml">Dir</mi><mo id="A3.p3.3.m3.1.1.2" xref="A3.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="A3.p3.3.m3.1.1.1.1" xref="A3.p3.3.m3.1.1.1.1.1.cmml"><mo id="A3.p3.3.m3.1.1.1.1.2" stretchy="false" xref="A3.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="A3.p3.3.m3.1.1.1.1.1" xref="A3.p3.3.m3.1.1.1.1.1.cmml"><mi id="A3.p3.3.m3.1.1.1.1.1.2" xref="A3.p3.3.m3.1.1.1.1.1.2.cmml">β</mi><mo id="A3.p3.3.m3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="A3.p3.3.m3.1.1.1.1.1.1.cmml">⋅</mo><mn id="A3.p3.3.m3.1.1.1.1.1.3" xref="A3.p3.3.m3.1.1.1.1.1.3.cmml">𝟏</mn></mrow><mo id="A3.p3.3.m3.1.1.1.1.3" stretchy="false" xref="A3.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.3.m3.1b"><apply id="A3.p3.3.m3.1.1.cmml" xref="A3.p3.3.m3.1.1"><times id="A3.p3.3.m3.1.1.2.cmml" xref="A3.p3.3.m3.1.1.2"></times><ci id="A3.p3.3.m3.1.1.3.cmml" xref="A3.p3.3.m3.1.1.3">Dir</ci><apply id="A3.p3.3.m3.1.1.1.1.1.cmml" xref="A3.p3.3.m3.1.1.1.1"><ci id="A3.p3.3.m3.1.1.1.1.1.1.cmml" xref="A3.p3.3.m3.1.1.1.1.1.1">⋅</ci><ci id="A3.p3.3.m3.1.1.1.1.1.2.cmml" xref="A3.p3.3.m3.1.1.1.1.1.2">𝛽</ci><cn id="A3.p3.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="A3.p3.3.m3.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.3.m3.1c">\mathrm{Dir}(\beta\cdot\bm{1})</annotation><annotation encoding="application/x-llamapun" id="A3.p3.3.m3.1d">roman_Dir ( italic_β ⋅ bold_1 )</annotation></semantics></math> for any <math alttext="\beta>0" class="ltx_Math" display="inline" id="A3.p3.4.m4.1"><semantics id="A3.p3.4.m4.1a"><mrow id="A3.p3.4.m4.1.1" xref="A3.p3.4.m4.1.1.cmml"><mi id="A3.p3.4.m4.1.1.2" xref="A3.p3.4.m4.1.1.2.cmml">β</mi><mo id="A3.p3.4.m4.1.1.1" xref="A3.p3.4.m4.1.1.1.cmml">></mo><mn id="A3.p3.4.m4.1.1.3" xref="A3.p3.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.4.m4.1b"><apply id="A3.p3.4.m4.1.1.cmml" xref="A3.p3.4.m4.1.1"><gt id="A3.p3.4.m4.1.1.1.cmml" xref="A3.p3.4.m4.1.1.1"></gt><ci id="A3.p3.4.m4.1.1.2.cmml" xref="A3.p3.4.m4.1.1.2">𝛽</ci><cn id="A3.p3.4.m4.1.1.3.cmml" type="integer" xref="A3.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.4.m4.1c">\beta>0</annotation><annotation encoding="application/x-llamapun" id="A3.p3.4.m4.1d">italic_β > 0</annotation></semantics></math>. Furthermore, we believe it should be possible to replace the <math alttext="L_{\infty}" class="ltx_Math" display="inline" id="A3.p3.5.m5.1"><semantics id="A3.p3.5.m5.1a"><msub id="A3.p3.5.m5.1.1" xref="A3.p3.5.m5.1.1.cmml"><mi id="A3.p3.5.m5.1.1.2" xref="A3.p3.5.m5.1.1.2.cmml">L</mi><mi id="A3.p3.5.m5.1.1.3" mathvariant="normal" xref="A3.p3.5.m5.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="A3.p3.5.m5.1b"><apply id="A3.p3.5.m5.1.1.cmml" xref="A3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="A3.p3.5.m5.1.1.1.cmml" xref="A3.p3.5.m5.1.1">subscript</csymbol><ci id="A3.p3.5.m5.1.1.2.cmml" xref="A3.p3.5.m5.1.1.2">𝐿</ci><infinity id="A3.p3.5.m5.1.1.3.cmml" xref="A3.p3.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.5.m5.1c">L_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A3.p3.5.m5.1d">italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> error guarantee in <a class="ltx_ref" href="https://arxiv.org/html/2502.10178v1#S4.E4" title="In Theorem 2. ‣ 4.3 Higher order processes (𝑘>1) ‣ 4 Theoretical results ‣ From Markov to Laplace: How Mamba In-Context Learns Markov Chains"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">4</span></a> by the KL-divergence between the two distributions without significantly changing the conclusion (<math alttext="d\cdot\mathtt{p}=2^{\Omega(k)}" class="ltx_Math" display="inline" id="A3.p3.6.m6.1"><semantics id="A3.p3.6.m6.1a"><mrow id="A3.p3.6.m6.1.2" xref="A3.p3.6.m6.1.2.cmml"><mrow id="A3.p3.6.m6.1.2.2" xref="A3.p3.6.m6.1.2.2.cmml"><mi id="A3.p3.6.m6.1.2.2.2" xref="A3.p3.6.m6.1.2.2.2.cmml">d</mi><mo id="A3.p3.6.m6.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="A3.p3.6.m6.1.2.2.1.cmml">⋅</mo><mi id="A3.p3.6.m6.1.2.2.3" xref="A3.p3.6.m6.1.2.2.3.cmml">𝚙</mi></mrow><mo id="A3.p3.6.m6.1.2.1" xref="A3.p3.6.m6.1.2.1.cmml">=</mo><msup id="A3.p3.6.m6.1.2.3" 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id="A3.p3.6.m6.1.2.2.3.cmml" xref="A3.p3.6.m6.1.2.2.3">𝚙</ci></apply><apply id="A3.p3.6.m6.1.2.3.cmml" xref="A3.p3.6.m6.1.2.3"><csymbol cd="ambiguous" id="A3.p3.6.m6.1.2.3.1.cmml" xref="A3.p3.6.m6.1.2.3">superscript</csymbol><cn id="A3.p3.6.m6.1.2.3.2.cmml" type="integer" xref="A3.p3.6.m6.1.2.3.2">2</cn><apply id="A3.p3.6.m6.1.1.1.cmml" xref="A3.p3.6.m6.1.1.1"><times id="A3.p3.6.m6.1.1.1.2.cmml" xref="A3.p3.6.m6.1.1.1.2"></times><ci id="A3.p3.6.m6.1.1.1.3.cmml" xref="A3.p3.6.m6.1.1.1.3">Ω</ci><ci id="A3.p3.6.m6.1.1.1.1.cmml" xref="A3.p3.6.m6.1.1.1.1">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.6.m6.1c">d\cdot\mathtt{p}=2^{\Omega(k)}</annotation><annotation encoding="application/x-llamapun" id="A3.p3.6.m6.1d">italic_d ⋅ typewriter_p = 2 start_POSTSUPERSCRIPT roman_Ω ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math>).</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_appendix" id="A4"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix D </span>Model architectures and hyper-parameters</h2> <figure class="ltx_table" id="A4.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A4.T2.11.1.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="A4.T2.12.2" style="font-size:90%;">Parameters in the Mamba architecture with their shape.</span></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A4.T2.9"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A4.T2.9.10.1"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="A4.T2.9.10.1.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.9.10.1.1.1"> <span class="ltx_p" id="A4.T2.9.10.1.1.1.1"><span class="ltx_text" id="A4.T2.9.10.1.1.1.1.1" style="font-size:90%;">Parameter</span></span> </span> </th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="A4.T2.9.10.1.2"><span class="ltx_text" id="A4.T2.9.10.1.2.1" style="font-size:90%;">Matrix shape</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A4.T2.1.1"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row ltx_border_t" id="A4.T2.1.1.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.1.1.2.1"> <span class="ltx_p" id="A4.T2.1.1.2.1.1"><span class="ltx_text" id="A4.T2.1.1.2.1.1.1" style="font-size:90%;">embedding</span></span> </span> </th> <td class="ltx_td ltx_align_left ltx_border_t" id="A4.T2.1.1.1"><math alttext="2\times d" class="ltx_Math" display="inline" id="A4.T2.1.1.1.m1.1"><semantics id="A4.T2.1.1.1.m1.1a"><mrow id="A4.T2.1.1.1.m1.1.1" xref="A4.T2.1.1.1.m1.1.1.cmml"><mn id="A4.T2.1.1.1.m1.1.1.2" mathsize="90%" xref="A4.T2.1.1.1.m1.1.1.2.cmml">2</mn><mo id="A4.T2.1.1.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T2.1.1.1.m1.1.1.1.cmml">×</mo><mi id="A4.T2.1.1.1.m1.1.1.3" mathsize="90%" xref="A4.T2.1.1.1.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T2.1.1.1.m1.1b"><apply id="A4.T2.1.1.1.m1.1.1.cmml" xref="A4.T2.1.1.1.m1.1.1"><times id="A4.T2.1.1.1.m1.1.1.1.cmml" xref="A4.T2.1.1.1.m1.1.1.1"></times><cn id="A4.T2.1.1.1.m1.1.1.2.cmml" type="integer" xref="A4.T2.1.1.1.m1.1.1.2">2</cn><ci id="A4.T2.1.1.1.m1.1.1.3.cmml" xref="A4.T2.1.1.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.1.1.1.m1.1c">2\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T2.1.1.1.m1.1d">2 × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.2.2"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.2.2.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.2.2.2.1"> <span class="ltx_p" id="A4.T2.2.2.2.1.1"><span class="ltx_text" id="A4.T2.2.2.2.1.1.1" style="font-size:90%;">mamba.A</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.2.2.1"><math alttext="1" class="ltx_Math" display="inline" id="A4.T2.2.2.1.m1.1"><semantics id="A4.T2.2.2.1.m1.1a"><mn id="A4.T2.2.2.1.m1.1.1" mathsize="90%" xref="A4.T2.2.2.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.T2.2.2.1.m1.1b"><cn id="A4.T2.2.2.1.m1.1.1.cmml" type="integer" xref="A4.T2.2.2.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.2.2.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A4.T2.2.2.1.m1.1d">1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.3.3"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.3.3.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.3.3.2.1"> <span class="ltx_p" id="A4.T2.3.3.2.1.1"><span class="ltx_text" id="A4.T2.3.3.2.1.1.1" style="font-size:90%;">mamba.dt</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.3.3.1"><math alttext="1" class="ltx_Math" display="inline" id="A4.T2.3.3.1.m1.1"><semantics id="A4.T2.3.3.1.m1.1a"><mn id="A4.T2.3.3.1.m1.1.1" mathsize="90%" xref="A4.T2.3.3.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.T2.3.3.1.m1.1b"><cn id="A4.T2.3.3.1.m1.1.1.cmml" type="integer" xref="A4.T2.3.3.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.3.3.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A4.T2.3.3.1.m1.1d">1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.4.4"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.4.4.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.4.4.2.1"> <span class="ltx_p" id="A4.T2.4.4.2.1.1"><span class="ltx_text" id="A4.T2.4.4.2.1.1.1" style="font-size:90%;">mamba.in_proj</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.4.4.1"><math alttext="(2ed+2N+1)\times d" class="ltx_Math" display="inline" id="A4.T2.4.4.1.m1.1"><semantics id="A4.T2.4.4.1.m1.1a"><mrow 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xref="A4.T2.4.4.1.m1.1.1.1.1.1.3.3">𝑁</ci></apply><cn id="A4.T2.4.4.1.m1.1.1.1.1.1.4.cmml" type="integer" xref="A4.T2.4.4.1.m1.1.1.1.1.1.4">1</cn></apply><ci id="A4.T2.4.4.1.m1.1.1.3.cmml" xref="A4.T2.4.4.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.4.4.1.m1.1c">(2ed+2N+1)\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T2.4.4.1.m1.1d">( 2 italic_e italic_d + 2 italic_N + 1 ) × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.5.5"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.5.5.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.5.5.2.1"> <span class="ltx_p" id="A4.T2.5.5.2.1.1"><span class="ltx_text" id="A4.T2.5.5.2.1.1.1" style="font-size:90%;">mamba.conv1d</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.5.5.1"><math alttext="(ed+2N)\times w" class="ltx_Math" display="inline" id="A4.T2.5.5.1.m1.1"><semantics 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xref="A4.T2.5.5.1.m1.1.1.1.1.1.2.1"></times><ci id="A4.T2.5.5.1.m1.1.1.1.1.1.2.2.cmml" xref="A4.T2.5.5.1.m1.1.1.1.1.1.2.2">𝑒</ci><ci id="A4.T2.5.5.1.m1.1.1.1.1.1.2.3.cmml" xref="A4.T2.5.5.1.m1.1.1.1.1.1.2.3">𝑑</ci></apply><apply id="A4.T2.5.5.1.m1.1.1.1.1.1.3.cmml" xref="A4.T2.5.5.1.m1.1.1.1.1.1.3"><times id="A4.T2.5.5.1.m1.1.1.1.1.1.3.1.cmml" xref="A4.T2.5.5.1.m1.1.1.1.1.1.3.1"></times><cn id="A4.T2.5.5.1.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="A4.T2.5.5.1.m1.1.1.1.1.1.3.2">2</cn><ci id="A4.T2.5.5.1.m1.1.1.1.1.1.3.3.cmml" xref="A4.T2.5.5.1.m1.1.1.1.1.1.3.3">𝑁</ci></apply></apply><ci id="A4.T2.5.5.1.m1.1.1.3.cmml" xref="A4.T2.5.5.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.5.5.1.m1.1c">(ed+2N)\times w</annotation><annotation encoding="application/x-llamapun" id="A4.T2.5.5.1.m1.1d">( italic_e italic_d + 2 italic_N ) × italic_w</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.6.6"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.6.6.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.6.6.2.1"> <span class="ltx_p" id="A4.T2.6.6.2.1.1"><span class="ltx_text" id="A4.T2.6.6.2.1.1.1" style="font-size:90%;">mamba.out_proj</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.6.6.1"><math alttext="d\times(2ed+2N+1)" class="ltx_Math" display="inline" id="A4.T2.6.6.1.m1.1"><semantics id="A4.T2.6.6.1.m1.1a"><mrow id="A4.T2.6.6.1.m1.1.1" xref="A4.T2.6.6.1.m1.1.1.cmml"><mi id="A4.T2.6.6.1.m1.1.1.3" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.3.cmml">d</mi><mo id="A4.T2.6.6.1.m1.1.1.2" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T2.6.6.1.m1.1.1.2.cmml">×</mo><mrow id="A4.T2.6.6.1.m1.1.1.1.1" xref="A4.T2.6.6.1.m1.1.1.1.1.1.cmml"><mo id="A4.T2.6.6.1.m1.1.1.1.1.2" maxsize="90%" minsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="A4.T2.6.6.1.m1.1.1.1.1.1" xref="A4.T2.6.6.1.m1.1.1.1.1.1.cmml"><mrow id="A4.T2.6.6.1.m1.1.1.1.1.1.2" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.cmml"><mn id="A4.T2.6.6.1.m1.1.1.1.1.1.2.2" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.2.cmml">2</mn><mo id="A4.T2.6.6.1.m1.1.1.1.1.1.2.1" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="A4.T2.6.6.1.m1.1.1.1.1.1.2.3" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.3.cmml">e</mi><mo id="A4.T2.6.6.1.m1.1.1.1.1.1.2.1a" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="A4.T2.6.6.1.m1.1.1.1.1.1.2.4" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.4.cmml">d</mi></mrow><mo id="A4.T2.6.6.1.m1.1.1.1.1.1.1" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.1.cmml">+</mo><mrow id="A4.T2.6.6.1.m1.1.1.1.1.1.3" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.cmml"><mn id="A4.T2.6.6.1.m1.1.1.1.1.1.3.2" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.2.cmml">2</mn><mo id="A4.T2.6.6.1.m1.1.1.1.1.1.3.1" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A4.T2.6.6.1.m1.1.1.1.1.1.3.3" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.3.cmml">N</mi></mrow><mo id="A4.T2.6.6.1.m1.1.1.1.1.1.1a" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.1.cmml">+</mo><mn id="A4.T2.6.6.1.m1.1.1.1.1.1.4" mathsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.4.cmml">1</mn></mrow><mo id="A4.T2.6.6.1.m1.1.1.1.1.3" maxsize="90%" minsize="90%" xref="A4.T2.6.6.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.T2.6.6.1.m1.1b"><apply id="A4.T2.6.6.1.m1.1.1.cmml" xref="A4.T2.6.6.1.m1.1.1"><times id="A4.T2.6.6.1.m1.1.1.2.cmml" xref="A4.T2.6.6.1.m1.1.1.2"></times><ci id="A4.T2.6.6.1.m1.1.1.3.cmml" xref="A4.T2.6.6.1.m1.1.1.3">𝑑</ci><apply id="A4.T2.6.6.1.m1.1.1.1.1.1.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1"><plus id="A4.T2.6.6.1.m1.1.1.1.1.1.1.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.1"></plus><apply id="A4.T2.6.6.1.m1.1.1.1.1.1.2.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2"><times id="A4.T2.6.6.1.m1.1.1.1.1.1.2.1.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.1"></times><cn id="A4.T2.6.6.1.m1.1.1.1.1.1.2.2.cmml" type="integer" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.2">2</cn><ci id="A4.T2.6.6.1.m1.1.1.1.1.1.2.3.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.3">𝑒</ci><ci id="A4.T2.6.6.1.m1.1.1.1.1.1.2.4.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.2.4">𝑑</ci></apply><apply id="A4.T2.6.6.1.m1.1.1.1.1.1.3.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3"><times id="A4.T2.6.6.1.m1.1.1.1.1.1.3.1.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.1"></times><cn id="A4.T2.6.6.1.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.2">2</cn><ci id="A4.T2.6.6.1.m1.1.1.1.1.1.3.3.cmml" xref="A4.T2.6.6.1.m1.1.1.1.1.1.3.3">𝑁</ci></apply><cn id="A4.T2.6.6.1.m1.1.1.1.1.1.4.cmml" type="integer" xref="A4.T2.6.6.1.m1.1.1.1.1.1.4">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.6.6.1.m1.1c">d\times(2ed+2N+1)</annotation><annotation encoding="application/x-llamapun" id="A4.T2.6.6.1.m1.1d">italic_d × ( 2 italic_e italic_d + 2 italic_N + 1 )</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.7.7"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.7.7.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.7.7.2.1"> <span class="ltx_p" id="A4.T2.7.7.2.1.1"><span class="ltx_text" id="A4.T2.7.7.2.1.1.1" style="font-size:90%;">mlp.fc1</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.7.7.1"><math alttext="4d\times d" class="ltx_Math" display="inline" id="A4.T2.7.7.1.m1.1"><semantics id="A4.T2.7.7.1.m1.1a"><mrow id="A4.T2.7.7.1.m1.1.1" xref="A4.T2.7.7.1.m1.1.1.cmml"><mrow id="A4.T2.7.7.1.m1.1.1.2" xref="A4.T2.7.7.1.m1.1.1.2.cmml"><mn id="A4.T2.7.7.1.m1.1.1.2.2" mathsize="90%" xref="A4.T2.7.7.1.m1.1.1.2.2.cmml">4</mn><mo id="A4.T2.7.7.1.m1.1.1.2.1" xref="A4.T2.7.7.1.m1.1.1.2.1.cmml">⁢</mo><mi id="A4.T2.7.7.1.m1.1.1.2.3" mathsize="90%" xref="A4.T2.7.7.1.m1.1.1.2.3.cmml">d</mi></mrow><mo id="A4.T2.7.7.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T2.7.7.1.m1.1.1.1.cmml">×</mo><mi id="A4.T2.7.7.1.m1.1.1.3" mathsize="90%" xref="A4.T2.7.7.1.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T2.7.7.1.m1.1b"><apply id="A4.T2.7.7.1.m1.1.1.cmml" xref="A4.T2.7.7.1.m1.1.1"><times id="A4.T2.7.7.1.m1.1.1.1.cmml" xref="A4.T2.7.7.1.m1.1.1.1"></times><apply id="A4.T2.7.7.1.m1.1.1.2.cmml" xref="A4.T2.7.7.1.m1.1.1.2"><times id="A4.T2.7.7.1.m1.1.1.2.1.cmml" xref="A4.T2.7.7.1.m1.1.1.2.1"></times><cn id="A4.T2.7.7.1.m1.1.1.2.2.cmml" type="integer" xref="A4.T2.7.7.1.m1.1.1.2.2">4</cn><ci id="A4.T2.7.7.1.m1.1.1.2.3.cmml" xref="A4.T2.7.7.1.m1.1.1.2.3">𝑑</ci></apply><ci id="A4.T2.7.7.1.m1.1.1.3.cmml" xref="A4.T2.7.7.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.7.7.1.m1.1c">4d\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T2.7.7.1.m1.1d">4 italic_d × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.8.8"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row" id="A4.T2.8.8.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.8.8.2.1"> <span class="ltx_p" id="A4.T2.8.8.2.1.1"><span class="ltx_text" id="A4.T2.8.8.2.1.1.1" style="font-size:90%;">mlp.fc2</span></span> </span> </th> <td class="ltx_td ltx_align_left" id="A4.T2.8.8.1"><math alttext="d\times 4d" class="ltx_Math" display="inline" id="A4.T2.8.8.1.m1.1"><semantics id="A4.T2.8.8.1.m1.1a"><mrow id="A4.T2.8.8.1.m1.1.1" xref="A4.T2.8.8.1.m1.1.1.cmml"><mrow id="A4.T2.8.8.1.m1.1.1.2" xref="A4.T2.8.8.1.m1.1.1.2.cmml"><mi id="A4.T2.8.8.1.m1.1.1.2.2" mathsize="90%" xref="A4.T2.8.8.1.m1.1.1.2.2.cmml">d</mi><mo id="A4.T2.8.8.1.m1.1.1.2.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T2.8.8.1.m1.1.1.2.1.cmml">×</mo><mn id="A4.T2.8.8.1.m1.1.1.2.3" mathsize="90%" xref="A4.T2.8.8.1.m1.1.1.2.3.cmml">4</mn></mrow><mo id="A4.T2.8.8.1.m1.1.1.1" xref="A4.T2.8.8.1.m1.1.1.1.cmml">⁢</mo><mi id="A4.T2.8.8.1.m1.1.1.3" mathsize="90%" xref="A4.T2.8.8.1.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T2.8.8.1.m1.1b"><apply id="A4.T2.8.8.1.m1.1.1.cmml" xref="A4.T2.8.8.1.m1.1.1"><times id="A4.T2.8.8.1.m1.1.1.1.cmml" xref="A4.T2.8.8.1.m1.1.1.1"></times><apply id="A4.T2.8.8.1.m1.1.1.2.cmml" xref="A4.T2.8.8.1.m1.1.1.2"><times id="A4.T2.8.8.1.m1.1.1.2.1.cmml" xref="A4.T2.8.8.1.m1.1.1.2.1"></times><ci id="A4.T2.8.8.1.m1.1.1.2.2.cmml" xref="A4.T2.8.8.1.m1.1.1.2.2">𝑑</ci><cn id="A4.T2.8.8.1.m1.1.1.2.3.cmml" type="integer" xref="A4.T2.8.8.1.m1.1.1.2.3">4</cn></apply><ci id="A4.T2.8.8.1.m1.1.1.3.cmml" xref="A4.T2.8.8.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.8.8.1.m1.1c">d\times 4d</annotation><annotation encoding="application/x-llamapun" id="A4.T2.8.8.1.m1.1d">italic_d × 4 italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T2.9.9"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_row ltx_border_bb" id="A4.T2.9.9.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T2.9.9.2.1"> <span class="ltx_p" id="A4.T2.9.9.2.1.1"><span class="ltx_text" id="A4.T2.9.9.2.1.1.1" style="font-size:90%;">lm_head</span></span> </span> </th> <td class="ltx_td ltx_align_left ltx_border_bb" id="A4.T2.9.9.1"><math alttext="d\times 2" class="ltx_Math" display="inline" id="A4.T2.9.9.1.m1.1"><semantics id="A4.T2.9.9.1.m1.1a"><mrow id="A4.T2.9.9.1.m1.1.1" xref="A4.T2.9.9.1.m1.1.1.cmml"><mi id="A4.T2.9.9.1.m1.1.1.2" mathsize="90%" xref="A4.T2.9.9.1.m1.1.1.2.cmml">d</mi><mo id="A4.T2.9.9.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T2.9.9.1.m1.1.1.1.cmml">×</mo><mn id="A4.T2.9.9.1.m1.1.1.3" mathsize="90%" xref="A4.T2.9.9.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.T2.9.9.1.m1.1b"><apply id="A4.T2.9.9.1.m1.1.1.cmml" xref="A4.T2.9.9.1.m1.1.1"><times id="A4.T2.9.9.1.m1.1.1.1.cmml" xref="A4.T2.9.9.1.m1.1.1.1"></times><ci id="A4.T2.9.9.1.m1.1.1.2.cmml" xref="A4.T2.9.9.1.m1.1.1.2">𝑑</ci><cn id="A4.T2.9.9.1.m1.1.1.3.cmml" type="integer" xref="A4.T2.9.9.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T2.9.9.1.m1.1c">d\times 2</annotation><annotation encoding="application/x-llamapun" id="A4.T2.9.9.1.m1.1d">italic_d × 2</annotation></semantics></math></td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="A4.T3"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A4.T3.16.1.1" style="font-size:90%;">Table 3</span>: </span><span class="ltx_text" id="A4.T3.17.2" style="font-size:90%;">Settings and parameters for the Mamba model used in the experiments.</span></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A4.T3.14"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A4.T3.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="A4.T3.1.1.2"><span class="ltx_text" id="A4.T3.1.1.2.1" style="font-size:90%;">Dataset</span></th> <td class="ltx_td ltx_align_justify ltx_border_tt" id="A4.T3.1.1.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.1.1.1.1"> <span class="ltx_p" id="A4.T3.1.1.1.1.1"><math alttext="k" class="ltx_Math" display="inline" id="A4.T3.1.1.1.1.1.m1.1"><semantics id="A4.T3.1.1.1.1.1.m1.1a"><mi id="A4.T3.1.1.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.1.1.1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A4.T3.1.1.1.1.1.m1.1b"><ci id="A4.T3.1.1.1.1.1.m1.1.1.cmml" xref="A4.T3.1.1.1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.1.1.1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="A4.T3.1.1.1.1.1.m1.1d">italic_k</annotation></semantics></math><span class="ltx_text" id="A4.T3.1.1.1.1.1.1" style="font-size:90%;">-th order binary Markov source</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.14.15.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.14.15.1.1"><span class="ltx_text" id="A4.T3.14.15.1.1.1" style="font-size:90%;">Architecture</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.14.15.1.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.14.15.1.2.1"> <span class="ltx_p" id="A4.T3.14.15.1.2.1.1"><span class="ltx_text" id="A4.T3.14.15.1.2.1.1.1" style="font-size:90%;">Based on the Mamba-2 architecture as implemented in </span><cite class="ltx_cite ltx_citemacro_cite"><span class="ltx_text" id="A4.T3.14.15.1.2.1.1.2.1" style="font-size:90%;">[</span><span class="ltx_ref ltx_missing_citation ltx_ref_self">dao2024transformers</span><span class="ltx_text" id="A4.T3.14.15.1.2.1.1.3.2" style="font-size:90%;">]</span></cite></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.2.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A4.T3.2.2.2"><span class="ltx_text" id="A4.T3.2.2.2.1" style="font-size:90%;">Batch size</span></th> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T3.2.2.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.2.2.1.1"> <span class="ltx_p" id="A4.T3.2.2.1.1.1"><span class="ltx_text" id="A4.T3.2.2.1.1.1.1" style="font-size:90%;">Grid-searched in </span><math alttext="\{16,32,64,128,256\}" class="ltx_Math" display="inline" id="A4.T3.2.2.1.1.1.m1.5"><semantics id="A4.T3.2.2.1.1.1.m1.5a"><mrow id="A4.T3.2.2.1.1.1.m1.5.6.2" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml"><mo id="A4.T3.2.2.1.1.1.m1.5.6.2.1" maxsize="90%" minsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml">{</mo><mn id="A4.T3.2.2.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.1.1.cmml">16</mn><mo id="A4.T3.2.2.1.1.1.m1.5.6.2.2" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.2.2.1.1.1.m1.2.2" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.2.2.cmml">32</mn><mo id="A4.T3.2.2.1.1.1.m1.5.6.2.3" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.2.2.1.1.1.m1.3.3" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.3.3.cmml">64</mn><mo id="A4.T3.2.2.1.1.1.m1.5.6.2.4" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.2.2.1.1.1.m1.4.4" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.4.4.cmml">128</mn><mo id="A4.T3.2.2.1.1.1.m1.5.6.2.5" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.2.2.1.1.1.m1.5.5" mathsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.5.cmml">256</mn><mo id="A4.T3.2.2.1.1.1.m1.5.6.2.6" maxsize="90%" minsize="90%" xref="A4.T3.2.2.1.1.1.m1.5.6.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.T3.2.2.1.1.1.m1.5b"><set id="A4.T3.2.2.1.1.1.m1.5.6.1.cmml" xref="A4.T3.2.2.1.1.1.m1.5.6.2"><cn id="A4.T3.2.2.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.2.2.1.1.1.m1.1.1">16</cn><cn id="A4.T3.2.2.1.1.1.m1.2.2.cmml" type="integer" xref="A4.T3.2.2.1.1.1.m1.2.2">32</cn><cn id="A4.T3.2.2.1.1.1.m1.3.3.cmml" type="integer" xref="A4.T3.2.2.1.1.1.m1.3.3">64</cn><cn id="A4.T3.2.2.1.1.1.m1.4.4.cmml" type="integer" xref="A4.T3.2.2.1.1.1.m1.4.4">128</cn><cn id="A4.T3.2.2.1.1.1.m1.5.5.cmml" type="integer" 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style="font-size:90%;">Optimizer</span></th> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T3.3.3.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.3.3.1.1"> <span class="ltx_p" id="A4.T3.3.3.1.1.1"><span class="ltx_text" id="A4.T3.3.3.1.1.1.1" style="font-size:90%;">AdamW (</span><math alttext="\beta_{1}=0.9,\beta_{2}=0.95" class="ltx_Math" display="inline" id="A4.T3.3.3.1.1.1.m1.2"><semantics id="A4.T3.3.3.1.1.1.m1.2a"><mrow id="A4.T3.3.3.1.1.1.m1.2.2.2" xref="A4.T3.3.3.1.1.1.m1.2.2.3.cmml"><mrow id="A4.T3.3.3.1.1.1.m1.1.1.1.1" xref="A4.T3.3.3.1.1.1.m1.1.1.1.1.cmml"><msub id="A4.T3.3.3.1.1.1.m1.1.1.1.1.2" xref="A4.T3.3.3.1.1.1.m1.1.1.1.1.2.cmml"><mi id="A4.T3.3.3.1.1.1.m1.1.1.1.1.2.2" mathsize="90%" xref="A4.T3.3.3.1.1.1.m1.1.1.1.1.2.2.cmml">β</mi><mn id="A4.T3.3.3.1.1.1.m1.1.1.1.1.2.3" mathsize="90%" xref="A4.T3.3.3.1.1.1.m1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="A4.T3.3.3.1.1.1.m1.1.1.1.1.1" mathsize="90%" xref="A4.T3.3.3.1.1.1.m1.1.1.1.1.1.cmml">=</mo><mn 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cd="ambiguous" id="A4.T3.3.3.1.1.1.m1.2.2.2.2.2.1.cmml" xref="A4.T3.3.3.1.1.1.m1.2.2.2.2.2">subscript</csymbol><ci id="A4.T3.3.3.1.1.1.m1.2.2.2.2.2.2.cmml" xref="A4.T3.3.3.1.1.1.m1.2.2.2.2.2.2">𝛽</ci><cn id="A4.T3.3.3.1.1.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="A4.T3.3.3.1.1.1.m1.2.2.2.2.2.3">2</cn></apply><cn id="A4.T3.3.3.1.1.1.m1.2.2.2.2.3.cmml" type="float" xref="A4.T3.3.3.1.1.1.m1.2.2.2.2.3">0.95</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.3.3.1.1.1.m1.2c">\beta_{1}=0.9,\beta_{2}=0.95</annotation><annotation encoding="application/x-llamapun" id="A4.T3.3.3.1.1.1.m1.2d">italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.9 , italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.95</annotation></semantics></math><span class="ltx_text" id="A4.T3.3.3.1.1.1.2" style="font-size:90%;">)</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.4.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.4.4.2"><span class="ltx_text" id="A4.T3.4.4.2.1" style="font-size:90%;">Learning rate</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.4.4.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.4.4.1.1"> <span class="ltx_p" id="A4.T3.4.4.1.1.1"><math alttext="0.001" class="ltx_Math" display="inline" id="A4.T3.4.4.1.1.1.m1.1"><semantics id="A4.T3.4.4.1.1.1.m1.1a"><mn id="A4.T3.4.4.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.4.4.1.1.1.m1.1.1.cmml">0.001</mn><annotation-xml encoding="MathML-Content" id="A4.T3.4.4.1.1.1.m1.1b"><cn id="A4.T3.4.4.1.1.1.m1.1.1.cmml" type="float" xref="A4.T3.4.4.1.1.1.m1.1.1">0.001</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.4.4.1.1.1.m1.1c">0.001</annotation><annotation encoding="application/x-llamapun" id="A4.T3.4.4.1.1.1.m1.1d">0.001</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.14.17.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.14.17.3.1"><span class="ltx_text" id="A4.T3.14.17.3.1.1" style="font-size:90%;">Scheduler</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.14.17.3.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.14.17.3.2.1"> <span class="ltx_p" id="A4.T3.14.17.3.2.1.1"><span class="ltx_text" id="A4.T3.14.17.3.2.1.1.1" style="font-size:90%;">Cosine</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.5.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.5.5.2"><span class="ltx_text" id="A4.T3.5.5.2.1" style="font-size:90%;"># Iterations</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.5.5.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.5.5.1.1"> <span class="ltx_p" id="A4.T3.5.5.1.1.1"><math alttext="10000" class="ltx_Math" display="inline" id="A4.T3.5.5.1.1.1.m1.1"><semantics id="A4.T3.5.5.1.1.1.m1.1a"><mn id="A4.T3.5.5.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.5.5.1.1.1.m1.1.1.cmml">10000</mn><annotation-xml encoding="MathML-Content" id="A4.T3.5.5.1.1.1.m1.1b"><cn id="A4.T3.5.5.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.5.5.1.1.1.m1.1.1">10000</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.5.5.1.1.1.m1.1c">10000</annotation><annotation encoding="application/x-llamapun" id="A4.T3.5.5.1.1.1.m1.1d">10000</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.6.6.2"><span class="ltx_text" id="A4.T3.6.6.2.1" style="font-size:90%;">Weight decay</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.6.6.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.6.6.1.1"> <span class="ltx_p" id="A4.T3.6.6.1.1.1"><math alttext="1\text{\times}{10}^{-3}" class="ltx_markedasmath" display="inline" id="A4.T3.6.6.1.1.1.m1.1.1.m1.3"><semantics id="A4.T3.6.6.1.1.1.m1.1.1.m1.3a"><mrow id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml"><mn id="A4.T3.6.6.1.1.1.m1.1.1.m1.1.1.1.1.1.1.1" mathsize="90%" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">1</mn><mtext id="A4.T3.6.6.1.1.1.m1.1.1.m1.2.2.2.2.2.2.2" mathsize="90%" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">×</mtext><msup id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml"><mn id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.2" mathsize="90%" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">10</mn><mrow id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.3.2" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml"><mo id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.3.2a" mathsize="90%" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">−</mo><mn id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.3.2.2" mathsize="90%" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.T3.6.6.1.1.1.m1.1.1.m1.3b"><csymbol cd="latexml" id="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml" xref="A4.T3.6.6.1.1.1.m1.1.1.m1.3.3.3">1E-3</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.6.6.1.1.1.m1.1.1.m1.3c">1\text{\times}{10}^{-3}</annotation><annotation encoding="application/x-llamapun" id="A4.T3.6.6.1.1.1.m1.1.1.m1.3d">start_ARG 1 end_ARG start_ARG times end_ARG start_ARG power start_ARG 10 end_ARG start_ARG - 3 end_ARG end_ARG</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.7.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A4.T3.7.7.2"><span class="ltx_text" id="A4.T3.7.7.2.1" style="font-size:90%;">Dropout</span></th> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T3.7.7.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.7.7.1.1"> <span class="ltx_p" id="A4.T3.7.7.1.1.1"><math alttext="0" class="ltx_Math" display="inline" id="A4.T3.7.7.1.1.1.m1.1"><semantics id="A4.T3.7.7.1.1.1.m1.1a"><mn id="A4.T3.7.7.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.7.7.1.1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A4.T3.7.7.1.1.1.m1.1b"><cn id="A4.T3.7.7.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.7.7.1.1.1.m1.1.1">0</cn></annotation-xml></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.8.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.8.8.2"><span class="ltx_text" id="A4.T3.8.8.2.1" style="font-size:90%;">Sequence length</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.8.8.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.8.8.1.1"> <span class="ltx_p" id="A4.T3.8.8.1.1.1"><span class="ltx_text" id="A4.T3.8.8.1.1.1.1" style="font-size:90%;">Grid-searched in </span><math alttext="\{128,256,512\}" class="ltx_Math" display="inline" id="A4.T3.8.8.1.1.1.m1.3"><semantics id="A4.T3.8.8.1.1.1.m1.3a"><mrow id="A4.T3.8.8.1.1.1.m1.3.4.2" xref="A4.T3.8.8.1.1.1.m1.3.4.1.cmml"><mo id="A4.T3.8.8.1.1.1.m1.3.4.2.1" maxsize="90%" minsize="90%" xref="A4.T3.8.8.1.1.1.m1.3.4.1.cmml">{</mo><mn id="A4.T3.8.8.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.8.8.1.1.1.m1.1.1.cmml">128</mn><mo id="A4.T3.8.8.1.1.1.m1.3.4.2.2" mathsize="90%" xref="A4.T3.8.8.1.1.1.m1.3.4.1.cmml">,</mo><mn id="A4.T3.8.8.1.1.1.m1.2.2" mathsize="90%" xref="A4.T3.8.8.1.1.1.m1.2.2.cmml">256</mn><mo id="A4.T3.8.8.1.1.1.m1.3.4.2.3" mathsize="90%" xref="A4.T3.8.8.1.1.1.m1.3.4.1.cmml">,</mo><mn id="A4.T3.8.8.1.1.1.m1.3.3" mathsize="90%" xref="A4.T3.8.8.1.1.1.m1.3.3.cmml">512</mn><mo id="A4.T3.8.8.1.1.1.m1.3.4.2.4" maxsize="90%" minsize="90%" xref="A4.T3.8.8.1.1.1.m1.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.T3.8.8.1.1.1.m1.3b"><set id="A4.T3.8.8.1.1.1.m1.3.4.1.cmml" xref="A4.T3.8.8.1.1.1.m1.3.4.2"><cn id="A4.T3.8.8.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.8.8.1.1.1.m1.1.1">128</cn><cn id="A4.T3.8.8.1.1.1.m1.2.2.cmml" type="integer" xref="A4.T3.8.8.1.1.1.m1.2.2">256</cn><cn id="A4.T3.8.8.1.1.1.m1.3.3.cmml" type="integer" xref="A4.T3.8.8.1.1.1.m1.3.3">512</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.8.8.1.1.1.m1.3c">\{128,256,512\}</annotation><annotation encoding="application/x-llamapun" id="A4.T3.8.8.1.1.1.m1.3d">{ 128 , 256 , 512 }</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.9.9"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.9.9.2"><span class="ltx_text" id="A4.T3.9.9.2.1" style="font-size:90%;">Embedding dimension</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.9.9.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.9.9.1.1"> <span class="ltx_p" id="A4.T3.9.9.1.1.1"><span class="ltx_text" id="A4.T3.9.9.1.1.1.1" style="font-size:90%;">Grid-searched in </span><math alttext="\{2,4,8,16,32\}" class="ltx_Math" display="inline" id="A4.T3.9.9.1.1.1.m1.5"><semantics id="A4.T3.9.9.1.1.1.m1.5a"><mrow id="A4.T3.9.9.1.1.1.m1.5.6.2" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml"><mo id="A4.T3.9.9.1.1.1.m1.5.6.2.1" maxsize="90%" minsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml">{</mo><mn id="A4.T3.9.9.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.1.1.cmml">2</mn><mo id="A4.T3.9.9.1.1.1.m1.5.6.2.2" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.9.9.1.1.1.m1.2.2" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.2.2.cmml">4</mn><mo id="A4.T3.9.9.1.1.1.m1.5.6.2.3" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.9.9.1.1.1.m1.3.3" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.3.3.cmml">8</mn><mo id="A4.T3.9.9.1.1.1.m1.5.6.2.4" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.9.9.1.1.1.m1.4.4" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.4.4.cmml">16</mn><mo id="A4.T3.9.9.1.1.1.m1.5.6.2.5" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T3.9.9.1.1.1.m1.5.5" mathsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.5.cmml">32</mn><mo id="A4.T3.9.9.1.1.1.m1.5.6.2.6" maxsize="90%" minsize="90%" xref="A4.T3.9.9.1.1.1.m1.5.6.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.T3.9.9.1.1.1.m1.5b"><set id="A4.T3.9.9.1.1.1.m1.5.6.1.cmml" xref="A4.T3.9.9.1.1.1.m1.5.6.2"><cn id="A4.T3.9.9.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.9.9.1.1.1.m1.1.1">2</cn><cn id="A4.T3.9.9.1.1.1.m1.2.2.cmml" type="integer" xref="A4.T3.9.9.1.1.1.m1.2.2">4</cn><cn id="A4.T3.9.9.1.1.1.m1.3.3.cmml" type="integer" xref="A4.T3.9.9.1.1.1.m1.3.3">8</cn><cn id="A4.T3.9.9.1.1.1.m1.4.4.cmml" type="integer" xref="A4.T3.9.9.1.1.1.m1.4.4">16</cn><cn id="A4.T3.9.9.1.1.1.m1.5.5.cmml" type="integer" xref="A4.T3.9.9.1.1.1.m1.5.5">32</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.9.9.1.1.1.m1.5c">\{2,4,8,16,32\}</annotation><annotation encoding="application/x-llamapun" id="A4.T3.9.9.1.1.1.m1.5d">{ 2 , 4 , 8 , 16 , 32 }</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.10.10"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.10.10.2"><span class="ltx_text" id="A4.T3.10.10.2.1" style="font-size:90%;">Mamba layers</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.10.10.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.10.10.1.1"> <span class="ltx_p" id="A4.T3.10.10.1.1.1"><math alttext="1" class="ltx_Math" display="inline" id="A4.T3.10.10.1.1.1.m1.1"><semantics id="A4.T3.10.10.1.1.1.m1.1a"><mn id="A4.T3.10.10.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.10.10.1.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.T3.10.10.1.1.1.m1.1b"><cn id="A4.T3.10.10.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.10.10.1.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.10.10.1.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A4.T3.10.10.1.1.1.m1.1d">1</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.11.11"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.11.11.2"><span class="ltx_text" id="A4.T3.11.11.2.1" style="font-size:90%;">Heads</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.11.11.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.11.11.1.1"> <span class="ltx_p" id="A4.T3.11.11.1.1.1"><math alttext="1" class="ltx_Math" display="inline" id="A4.T3.11.11.1.1.1.m1.1"><semantics id="A4.T3.11.11.1.1.1.m1.1a"><mn id="A4.T3.11.11.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.11.11.1.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.T3.11.11.1.1.1.m1.1b"><cn id="A4.T3.11.11.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.11.11.1.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.11.11.1.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A4.T3.11.11.1.1.1.m1.1d">1</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.13.13"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T3.13.13.3"><span class="ltx_text" id="A4.T3.13.13.3.1" style="font-size:90%;">Convolution window</span></th> <td class="ltx_td ltx_align_justify" id="A4.T3.13.13.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.13.13.2.2"> <span class="ltx_p" id="A4.T3.13.13.2.2.2"><span class="ltx_text" id="A4.T3.13.13.2.2.2.1" style="font-size:90%;">Between </span><math alttext="2" class="ltx_Math" display="inline" id="A4.T3.12.12.1.1.1.m1.1"><semantics id="A4.T3.12.12.1.1.1.m1.1a"><mn id="A4.T3.12.12.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.12.12.1.1.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="A4.T3.12.12.1.1.1.m1.1b"><cn id="A4.T3.12.12.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.12.12.1.1.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.12.12.1.1.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="A4.T3.12.12.1.1.1.m1.1d">2</annotation></semantics></math><span class="ltx_text" id="A4.T3.13.13.2.2.2.2" style="font-size:90%;"> and </span><math alttext="6" class="ltx_Math" display="inline" id="A4.T3.13.13.2.2.2.m2.1"><semantics id="A4.T3.13.13.2.2.2.m2.1a"><mn id="A4.T3.13.13.2.2.2.m2.1.1" mathsize="90%" xref="A4.T3.13.13.2.2.2.m2.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="A4.T3.13.13.2.2.2.m2.1b"><cn id="A4.T3.13.13.2.2.2.m2.1.1.cmml" type="integer" xref="A4.T3.13.13.2.2.2.m2.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.13.13.2.2.2.m2.1c">6</annotation><annotation encoding="application/x-llamapun" id="A4.T3.13.13.2.2.2.m2.1d">6</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T3.14.14"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="A4.T3.14.14.2"><span class="ltx_text" id="A4.T3.14.14.2.1" style="font-size:90%;">Repetitions</span></th> <td class="ltx_td ltx_align_justify ltx_border_bb ltx_border_t" id="A4.T3.14.14.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T3.14.14.1.1"> <span class="ltx_p" id="A4.T3.14.14.1.1.1"><math alttext="5" class="ltx_Math" display="inline" id="A4.T3.14.14.1.1.1.m1.1"><semantics id="A4.T3.14.14.1.1.1.m1.1a"><mn id="A4.T3.14.14.1.1.1.m1.1.1" mathsize="90%" xref="A4.T3.14.14.1.1.1.m1.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="A4.T3.14.14.1.1.1.m1.1b"><cn id="A4.T3.14.14.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T3.14.14.1.1.1.m1.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T3.14.14.1.1.1.m1.1c">5</annotation><annotation encoding="application/x-llamapun" id="A4.T3.14.14.1.1.1.m1.1d">5</annotation></semantics></math></span> </span> </td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="A4.T4"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A4.T4.17.1.1" style="font-size:90%;">Table 4</span>: </span><span class="ltx_text" id="A4.T4.18.2" style="font-size:90%;">Parameters in the transformer architecture with their shape.</span></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A4.T4.15"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A4.T4.15.16.1"> <th class="ltx_td ltx_align_justify ltx_th ltx_th_column ltx_border_tt" id="A4.T4.15.16.1.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.15.16.1.1.1"> <span class="ltx_p" id="A4.T4.15.16.1.1.1.1"><span class="ltx_text" id="A4.T4.15.16.1.1.1.1.1" style="font-size:90%;">Parameter</span></span> </span> </th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="A4.T4.15.16.1.2"><span class="ltx_text" id="A4.T4.15.16.1.2.1" style="font-size:90%;">Matrix shape</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A4.T4.1.1"> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T4.1.1.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.1.1.2.1"> <span class="ltx_p" id="A4.T4.1.1.2.1.1"><span class="ltx_text" id="A4.T4.1.1.2.1.1.1" style="font-size:90%;">transformer.wte</span></span> </span> </td> <td class="ltx_td ltx_align_left ltx_border_t" id="A4.T4.1.1.1"><math alttext="2\times d" class="ltx_Math" display="inline" id="A4.T4.1.1.1.m1.1"><semantics id="A4.T4.1.1.1.m1.1a"><mrow id="A4.T4.1.1.1.m1.1.1" xref="A4.T4.1.1.1.m1.1.1.cmml"><mn id="A4.T4.1.1.1.m1.1.1.2" mathsize="90%" xref="A4.T4.1.1.1.m1.1.1.2.cmml">2</mn><mo id="A4.T4.1.1.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.1.1.1.m1.1.1.1.cmml">×</mo><mi id="A4.T4.1.1.1.m1.1.1.3" mathsize="90%" xref="A4.T4.1.1.1.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.1.1.1.m1.1b"><apply id="A4.T4.1.1.1.m1.1.1.cmml" xref="A4.T4.1.1.1.m1.1.1"><times id="A4.T4.1.1.1.m1.1.1.1.cmml" xref="A4.T4.1.1.1.m1.1.1.1"></times><cn id="A4.T4.1.1.1.m1.1.1.2.cmml" type="integer" xref="A4.T4.1.1.1.m1.1.1.2">2</cn><ci id="A4.T4.1.1.1.m1.1.1.3.cmml" xref="A4.T4.1.1.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.1.1.1.m1.1c">2\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T4.1.1.1.m1.1d">2 × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.2.2"> <td class="ltx_td ltx_align_justify" id="A4.T4.2.2.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.2.2.2.1"> <span class="ltx_p" id="A4.T4.2.2.2.1.1"><span class="ltx_text" id="A4.T4.2.2.2.1.1.1" style="font-size:90%;">transformer.wpe</span></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.2.2.1"><math alttext="N\times d" class="ltx_Math" display="inline" id="A4.T4.2.2.1.m1.1"><semantics id="A4.T4.2.2.1.m1.1a"><mrow id="A4.T4.2.2.1.m1.1.1" xref="A4.T4.2.2.1.m1.1.1.cmml"><mi id="A4.T4.2.2.1.m1.1.1.2" mathsize="90%" xref="A4.T4.2.2.1.m1.1.1.2.cmml">N</mi><mo id="A4.T4.2.2.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.2.2.1.m1.1.1.1.cmml">×</mo><mi id="A4.T4.2.2.1.m1.1.1.3" mathsize="90%" xref="A4.T4.2.2.1.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.2.2.1.m1.1b"><apply id="A4.T4.2.2.1.m1.1.1.cmml" xref="A4.T4.2.2.1.m1.1.1"><times id="A4.T4.2.2.1.m1.1.1.1.cmml" xref="A4.T4.2.2.1.m1.1.1.1"></times><ci id="A4.T4.2.2.1.m1.1.1.2.cmml" xref="A4.T4.2.2.1.m1.1.1.2">𝑁</ci><ci id="A4.T4.2.2.1.m1.1.1.3.cmml" xref="A4.T4.2.2.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.2.2.1.m1.1c">N\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T4.2.2.1.m1.1d">italic_N × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.4.4"> <td class="ltx_td ltx_align_justify" id="A4.T4.3.3.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.3.3.1.1"> <span class="ltx_p" id="A4.T4.3.3.1.1.1"><span class="ltx_text" id="A4.T4.3.3.1.1.1.1" style="font-size:90%;">transformer.h.ln_1 </span><math alttext="(\times\ell)" class="ltx_math_unparsed" display="inline" id="A4.T4.3.3.1.1.1.m1.1"><semantics id="A4.T4.3.3.1.1.1.m1.1a"><mrow id="A4.T4.3.3.1.1.1.m1.1b"><mo id="A4.T4.3.3.1.1.1.m1.1.1" maxsize="90%" minsize="90%">(</mo><mo id="A4.T4.3.3.1.1.1.m1.1.2" lspace="0em" mathsize="90%" rspace="0.222em">×</mo><mi id="A4.T4.3.3.1.1.1.m1.1.3" mathsize="90%" mathvariant="normal">ℓ</mi><mo id="A4.T4.3.3.1.1.1.m1.1.4" maxsize="90%" minsize="90%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.T4.3.3.1.1.1.m1.1c">(\times\ell)</annotation><annotation encoding="application/x-llamapun" id="A4.T4.3.3.1.1.1.m1.1d">( × roman_ℓ )</annotation></semantics></math></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.4.4.2"><math alttext="d\times 1" class="ltx_Math" display="inline" id="A4.T4.4.4.2.m1.1"><semantics id="A4.T4.4.4.2.m1.1a"><mrow id="A4.T4.4.4.2.m1.1.1" xref="A4.T4.4.4.2.m1.1.1.cmml"><mi id="A4.T4.4.4.2.m1.1.1.2" mathsize="90%" xref="A4.T4.4.4.2.m1.1.1.2.cmml">d</mi><mo id="A4.T4.4.4.2.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.4.4.2.m1.1.1.1.cmml">×</mo><mn id="A4.T4.4.4.2.m1.1.1.3" mathsize="90%" xref="A4.T4.4.4.2.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.4.4.2.m1.1b"><apply id="A4.T4.4.4.2.m1.1.1.cmml" xref="A4.T4.4.4.2.m1.1.1"><times id="A4.T4.4.4.2.m1.1.1.1.cmml" xref="A4.T4.4.4.2.m1.1.1.1"></times><ci id="A4.T4.4.4.2.m1.1.1.2.cmml" xref="A4.T4.4.4.2.m1.1.1.2">𝑑</ci><cn id="A4.T4.4.4.2.m1.1.1.3.cmml" type="integer" xref="A4.T4.4.4.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.4.4.2.m1.1c">d\times 1</annotation><annotation encoding="application/x-llamapun" id="A4.T4.4.4.2.m1.1d">italic_d × 1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.6.6"> <td class="ltx_td ltx_align_justify" id="A4.T4.5.5.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.5.5.1.1"> <span class="ltx_p" id="A4.T4.5.5.1.1.1"><span class="ltx_text" id="A4.T4.5.5.1.1.1.1" style="font-size:90%;">transformer.h.attn.c_attn </span><math alttext="(\times\ell)" class="ltx_math_unparsed" display="inline" id="A4.T4.5.5.1.1.1.m1.1"><semantics id="A4.T4.5.5.1.1.1.m1.1a"><mrow id="A4.T4.5.5.1.1.1.m1.1b"><mo id="A4.T4.5.5.1.1.1.m1.1.1" maxsize="90%" minsize="90%">(</mo><mo id="A4.T4.5.5.1.1.1.m1.1.2" lspace="0em" mathsize="90%" rspace="0.222em">×</mo><mi id="A4.T4.5.5.1.1.1.m1.1.3" mathsize="90%" mathvariant="normal">ℓ</mi><mo id="A4.T4.5.5.1.1.1.m1.1.4" maxsize="90%" minsize="90%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.T4.5.5.1.1.1.m1.1c">(\times\ell)</annotation><annotation encoding="application/x-llamapun" id="A4.T4.5.5.1.1.1.m1.1d">( × roman_ℓ )</annotation></semantics></math></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.6.6.2"><math alttext="3d\times d" class="ltx_Math" display="inline" id="A4.T4.6.6.2.m1.1"><semantics id="A4.T4.6.6.2.m1.1a"><mrow id="A4.T4.6.6.2.m1.1.1" xref="A4.T4.6.6.2.m1.1.1.cmml"><mrow id="A4.T4.6.6.2.m1.1.1.2" xref="A4.T4.6.6.2.m1.1.1.2.cmml"><mn id="A4.T4.6.6.2.m1.1.1.2.2" mathsize="90%" xref="A4.T4.6.6.2.m1.1.1.2.2.cmml">3</mn><mo id="A4.T4.6.6.2.m1.1.1.2.1" xref="A4.T4.6.6.2.m1.1.1.2.1.cmml">⁢</mo><mi id="A4.T4.6.6.2.m1.1.1.2.3" mathsize="90%" xref="A4.T4.6.6.2.m1.1.1.2.3.cmml">d</mi></mrow><mo id="A4.T4.6.6.2.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.6.6.2.m1.1.1.1.cmml">×</mo><mi id="A4.T4.6.6.2.m1.1.1.3" mathsize="90%" xref="A4.T4.6.6.2.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.6.6.2.m1.1b"><apply id="A4.T4.6.6.2.m1.1.1.cmml" xref="A4.T4.6.6.2.m1.1.1"><times id="A4.T4.6.6.2.m1.1.1.1.cmml" xref="A4.T4.6.6.2.m1.1.1.1"></times><apply id="A4.T4.6.6.2.m1.1.1.2.cmml" xref="A4.T4.6.6.2.m1.1.1.2"><times id="A4.T4.6.6.2.m1.1.1.2.1.cmml" xref="A4.T4.6.6.2.m1.1.1.2.1"></times><cn id="A4.T4.6.6.2.m1.1.1.2.2.cmml" type="integer" xref="A4.T4.6.6.2.m1.1.1.2.2">3</cn><ci id="A4.T4.6.6.2.m1.1.1.2.3.cmml" xref="A4.T4.6.6.2.m1.1.1.2.3">𝑑</ci></apply><ci id="A4.T4.6.6.2.m1.1.1.3.cmml" xref="A4.T4.6.6.2.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.6.6.2.m1.1c">3d\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T4.6.6.2.m1.1d">3 italic_d × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.8.8"> <td class="ltx_td ltx_align_justify" id="A4.T4.7.7.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.7.7.1.1"> <span class="ltx_p" id="A4.T4.7.7.1.1.1"><span class="ltx_text" id="A4.T4.7.7.1.1.1.1" style="font-size:90%;">transformer.h.attn.c_proj </span><math alttext="(\times\ell)" class="ltx_math_unparsed" display="inline" id="A4.T4.7.7.1.1.1.m1.1"><semantics id="A4.T4.7.7.1.1.1.m1.1a"><mrow id="A4.T4.7.7.1.1.1.m1.1b"><mo id="A4.T4.7.7.1.1.1.m1.1.1" maxsize="90%" minsize="90%">(</mo><mo id="A4.T4.7.7.1.1.1.m1.1.2" lspace="0em" mathsize="90%" rspace="0.222em">×</mo><mi id="A4.T4.7.7.1.1.1.m1.1.3" mathsize="90%" mathvariant="normal">ℓ</mi><mo id="A4.T4.7.7.1.1.1.m1.1.4" maxsize="90%" minsize="90%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.T4.7.7.1.1.1.m1.1c">(\times\ell)</annotation><annotation encoding="application/x-llamapun" id="A4.T4.7.7.1.1.1.m1.1d">( × roman_ℓ )</annotation></semantics></math></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.8.8.2"><math alttext="d\times d" class="ltx_Math" display="inline" id="A4.T4.8.8.2.m1.1"><semantics id="A4.T4.8.8.2.m1.1a"><mrow id="A4.T4.8.8.2.m1.1.1" xref="A4.T4.8.8.2.m1.1.1.cmml"><mi id="A4.T4.8.8.2.m1.1.1.2" mathsize="90%" xref="A4.T4.8.8.2.m1.1.1.2.cmml">d</mi><mo id="A4.T4.8.8.2.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.8.8.2.m1.1.1.1.cmml">×</mo><mi id="A4.T4.8.8.2.m1.1.1.3" mathsize="90%" xref="A4.T4.8.8.2.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.8.8.2.m1.1b"><apply id="A4.T4.8.8.2.m1.1.1.cmml" xref="A4.T4.8.8.2.m1.1.1"><times id="A4.T4.8.8.2.m1.1.1.1.cmml" xref="A4.T4.8.8.2.m1.1.1.1"></times><ci id="A4.T4.8.8.2.m1.1.1.2.cmml" xref="A4.T4.8.8.2.m1.1.1.2">𝑑</ci><ci id="A4.T4.8.8.2.m1.1.1.3.cmml" xref="A4.T4.8.8.2.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.8.8.2.m1.1c">d\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T4.8.8.2.m1.1d">italic_d × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.10.10"> <td class="ltx_td ltx_align_justify" id="A4.T4.9.9.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.9.9.1.1"> <span class="ltx_p" id="A4.T4.9.9.1.1.1"><span class="ltx_text" id="A4.T4.9.9.1.1.1.1" style="font-size:90%;">transformer.h.ln_2 </span><math alttext="(\times\ell)" class="ltx_math_unparsed" display="inline" id="A4.T4.9.9.1.1.1.m1.1"><semantics id="A4.T4.9.9.1.1.1.m1.1a"><mrow id="A4.T4.9.9.1.1.1.m1.1b"><mo id="A4.T4.9.9.1.1.1.m1.1.1" maxsize="90%" minsize="90%">(</mo><mo id="A4.T4.9.9.1.1.1.m1.1.2" lspace="0em" mathsize="90%" rspace="0.222em">×</mo><mi id="A4.T4.9.9.1.1.1.m1.1.3" mathsize="90%" mathvariant="normal">ℓ</mi><mo id="A4.T4.9.9.1.1.1.m1.1.4" maxsize="90%" minsize="90%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.T4.9.9.1.1.1.m1.1c">(\times\ell)</annotation><annotation encoding="application/x-llamapun" id="A4.T4.9.9.1.1.1.m1.1d">( × roman_ℓ )</annotation></semantics></math></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.10.10.2"><math alttext="d\times 1" class="ltx_Math" display="inline" id="A4.T4.10.10.2.m1.1"><semantics id="A4.T4.10.10.2.m1.1a"><mrow id="A4.T4.10.10.2.m1.1.1" xref="A4.T4.10.10.2.m1.1.1.cmml"><mi id="A4.T4.10.10.2.m1.1.1.2" mathsize="90%" xref="A4.T4.10.10.2.m1.1.1.2.cmml">d</mi><mo id="A4.T4.10.10.2.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.10.10.2.m1.1.1.1.cmml">×</mo><mn id="A4.T4.10.10.2.m1.1.1.3" mathsize="90%" xref="A4.T4.10.10.2.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.10.10.2.m1.1b"><apply id="A4.T4.10.10.2.m1.1.1.cmml" xref="A4.T4.10.10.2.m1.1.1"><times id="A4.T4.10.10.2.m1.1.1.1.cmml" xref="A4.T4.10.10.2.m1.1.1.1"></times><ci id="A4.T4.10.10.2.m1.1.1.2.cmml" xref="A4.T4.10.10.2.m1.1.1.2">𝑑</ci><cn id="A4.T4.10.10.2.m1.1.1.3.cmml" type="integer" xref="A4.T4.10.10.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.10.10.2.m1.1c">d\times 1</annotation><annotation encoding="application/x-llamapun" id="A4.T4.10.10.2.m1.1d">italic_d × 1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.12.12"> <td class="ltx_td ltx_align_justify" id="A4.T4.11.11.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.11.11.1.1"> <span class="ltx_p" id="A4.T4.11.11.1.1.1"><span class="ltx_text" id="A4.T4.11.11.1.1.1.1" style="font-size:90%;">transformer.h.mlp.c_fc </span><math alttext="(\times\ell)" class="ltx_math_unparsed" display="inline" id="A4.T4.11.11.1.1.1.m1.1"><semantics id="A4.T4.11.11.1.1.1.m1.1a"><mrow id="A4.T4.11.11.1.1.1.m1.1b"><mo id="A4.T4.11.11.1.1.1.m1.1.1" maxsize="90%" minsize="90%">(</mo><mo id="A4.T4.11.11.1.1.1.m1.1.2" lspace="0em" mathsize="90%" rspace="0.222em">×</mo><mi id="A4.T4.11.11.1.1.1.m1.1.3" mathsize="90%" mathvariant="normal">ℓ</mi><mo id="A4.T4.11.11.1.1.1.m1.1.4" maxsize="90%" minsize="90%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.T4.11.11.1.1.1.m1.1c">(\times\ell)</annotation><annotation encoding="application/x-llamapun" id="A4.T4.11.11.1.1.1.m1.1d">( × roman_ℓ )</annotation></semantics></math></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.12.12.2"><math alttext="4d\times d" class="ltx_Math" display="inline" id="A4.T4.12.12.2.m1.1"><semantics id="A4.T4.12.12.2.m1.1a"><mrow id="A4.T4.12.12.2.m1.1.1" xref="A4.T4.12.12.2.m1.1.1.cmml"><mrow id="A4.T4.12.12.2.m1.1.1.2" xref="A4.T4.12.12.2.m1.1.1.2.cmml"><mn id="A4.T4.12.12.2.m1.1.1.2.2" mathsize="90%" xref="A4.T4.12.12.2.m1.1.1.2.2.cmml">4</mn><mo id="A4.T4.12.12.2.m1.1.1.2.1" xref="A4.T4.12.12.2.m1.1.1.2.1.cmml">⁢</mo><mi id="A4.T4.12.12.2.m1.1.1.2.3" mathsize="90%" xref="A4.T4.12.12.2.m1.1.1.2.3.cmml">d</mi></mrow><mo id="A4.T4.12.12.2.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.12.12.2.m1.1.1.1.cmml">×</mo><mi id="A4.T4.12.12.2.m1.1.1.3" mathsize="90%" xref="A4.T4.12.12.2.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.12.12.2.m1.1b"><apply id="A4.T4.12.12.2.m1.1.1.cmml" xref="A4.T4.12.12.2.m1.1.1"><times id="A4.T4.12.12.2.m1.1.1.1.cmml" xref="A4.T4.12.12.2.m1.1.1.1"></times><apply id="A4.T4.12.12.2.m1.1.1.2.cmml" xref="A4.T4.12.12.2.m1.1.1.2"><times id="A4.T4.12.12.2.m1.1.1.2.1.cmml" xref="A4.T4.12.12.2.m1.1.1.2.1"></times><cn id="A4.T4.12.12.2.m1.1.1.2.2.cmml" type="integer" xref="A4.T4.12.12.2.m1.1.1.2.2">4</cn><ci id="A4.T4.12.12.2.m1.1.1.2.3.cmml" xref="A4.T4.12.12.2.m1.1.1.2.3">𝑑</ci></apply><ci id="A4.T4.12.12.2.m1.1.1.3.cmml" xref="A4.T4.12.12.2.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.12.12.2.m1.1c">4d\times d</annotation><annotation encoding="application/x-llamapun" id="A4.T4.12.12.2.m1.1d">4 italic_d × italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.14.14"> <td class="ltx_td ltx_align_justify" id="A4.T4.13.13.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.13.13.1.1"> <span class="ltx_p" id="A4.T4.13.13.1.1.1"><span class="ltx_text" id="A4.T4.13.13.1.1.1.1" style="font-size:90%;">transformer.h.mlp.c_proj </span><math alttext="(\times\ell)" class="ltx_math_unparsed" display="inline" id="A4.T4.13.13.1.1.1.m1.1"><semantics id="A4.T4.13.13.1.1.1.m1.1a"><mrow id="A4.T4.13.13.1.1.1.m1.1b"><mo id="A4.T4.13.13.1.1.1.m1.1.1" maxsize="90%" minsize="90%">(</mo><mo id="A4.T4.13.13.1.1.1.m1.1.2" lspace="0em" mathsize="90%" rspace="0.222em">×</mo><mi id="A4.T4.13.13.1.1.1.m1.1.3" mathsize="90%" mathvariant="normal">ℓ</mi><mo id="A4.T4.13.13.1.1.1.m1.1.4" maxsize="90%" minsize="90%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.T4.13.13.1.1.1.m1.1c">(\times\ell)</annotation><annotation encoding="application/x-llamapun" id="A4.T4.13.13.1.1.1.m1.1d">( × roman_ℓ )</annotation></semantics></math></span> </span> </td> <td class="ltx_td ltx_align_left" id="A4.T4.14.14.2"><math alttext="d\times 4d" class="ltx_Math" display="inline" id="A4.T4.14.14.2.m1.1"><semantics id="A4.T4.14.14.2.m1.1a"><mrow id="A4.T4.14.14.2.m1.1.1" xref="A4.T4.14.14.2.m1.1.1.cmml"><mrow id="A4.T4.14.14.2.m1.1.1.2" xref="A4.T4.14.14.2.m1.1.1.2.cmml"><mi id="A4.T4.14.14.2.m1.1.1.2.2" mathsize="90%" xref="A4.T4.14.14.2.m1.1.1.2.2.cmml">d</mi><mo id="A4.T4.14.14.2.m1.1.1.2.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.14.14.2.m1.1.1.2.1.cmml">×</mo><mn id="A4.T4.14.14.2.m1.1.1.2.3" mathsize="90%" xref="A4.T4.14.14.2.m1.1.1.2.3.cmml">4</mn></mrow><mo id="A4.T4.14.14.2.m1.1.1.1" xref="A4.T4.14.14.2.m1.1.1.1.cmml">⁢</mo><mi id="A4.T4.14.14.2.m1.1.1.3" mathsize="90%" xref="A4.T4.14.14.2.m1.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.14.14.2.m1.1b"><apply id="A4.T4.14.14.2.m1.1.1.cmml" xref="A4.T4.14.14.2.m1.1.1"><times id="A4.T4.14.14.2.m1.1.1.1.cmml" xref="A4.T4.14.14.2.m1.1.1.1"></times><apply id="A4.T4.14.14.2.m1.1.1.2.cmml" xref="A4.T4.14.14.2.m1.1.1.2"><times id="A4.T4.14.14.2.m1.1.1.2.1.cmml" xref="A4.T4.14.14.2.m1.1.1.2.1"></times><ci id="A4.T4.14.14.2.m1.1.1.2.2.cmml" xref="A4.T4.14.14.2.m1.1.1.2.2">𝑑</ci><cn id="A4.T4.14.14.2.m1.1.1.2.3.cmml" type="integer" xref="A4.T4.14.14.2.m1.1.1.2.3">4</cn></apply><ci id="A4.T4.14.14.2.m1.1.1.3.cmml" xref="A4.T4.14.14.2.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.14.14.2.m1.1c">d\times 4d</annotation><annotation encoding="application/x-llamapun" id="A4.T4.14.14.2.m1.1d">italic_d × 4 italic_d</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="A4.T4.15.15"> <td class="ltx_td ltx_align_justify ltx_border_bb" id="A4.T4.15.15.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T4.15.15.2.1"> <span class="ltx_p" id="A4.T4.15.15.2.1.1"><span class="ltx_text" id="A4.T4.15.15.2.1.1.1" style="font-size:90%;">transformer.ln_f</span></span> </span> </td> <td class="ltx_td ltx_align_left ltx_border_bb" id="A4.T4.15.15.1"><math alttext="d\times 1" class="ltx_Math" display="inline" id="A4.T4.15.15.1.m1.1"><semantics id="A4.T4.15.15.1.m1.1a"><mrow id="A4.T4.15.15.1.m1.1.1" xref="A4.T4.15.15.1.m1.1.1.cmml"><mi id="A4.T4.15.15.1.m1.1.1.2" mathsize="90%" xref="A4.T4.15.15.1.m1.1.1.2.cmml">d</mi><mo id="A4.T4.15.15.1.m1.1.1.1" lspace="0.222em" mathsize="90%" rspace="0.222em" xref="A4.T4.15.15.1.m1.1.1.1.cmml">×</mo><mn id="A4.T4.15.15.1.m1.1.1.3" mathsize="90%" xref="A4.T4.15.15.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.T4.15.15.1.m1.1b"><apply id="A4.T4.15.15.1.m1.1.1.cmml" xref="A4.T4.15.15.1.m1.1.1"><times id="A4.T4.15.15.1.m1.1.1.1.cmml" xref="A4.T4.15.15.1.m1.1.1.1"></times><ci id="A4.T4.15.15.1.m1.1.1.2.cmml" xref="A4.T4.15.15.1.m1.1.1.2">𝑑</ci><cn id="A4.T4.15.15.1.m1.1.1.3.cmml" type="integer" xref="A4.T4.15.15.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T4.15.15.1.m1.1c">d\times 1</annotation><annotation encoding="application/x-llamapun" id="A4.T4.15.15.1.m1.1d">italic_d × 1</annotation></semantics></math></td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="A4.T5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A4.T5.15.1.1" style="font-size:90%;">Table 5</span>: </span><span class="ltx_text" id="A4.T5.16.2" style="font-size:90%;">Settings and parameters for the transformer model used in the experiments.</span></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A4.T5.13"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A4.T5.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="A4.T5.1.1.2"><span class="ltx_text" id="A4.T5.1.1.2.1" style="font-size:90%;">Dataset</span></th> <td class="ltx_td ltx_align_justify ltx_border_tt" id="A4.T5.1.1.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.1.1.1"> <span class="ltx_p" id="A4.T5.1.1.1.1.1"><math alttext="k" class="ltx_Math" display="inline" id="A4.T5.1.1.1.1.1.m1.1"><semantics id="A4.T5.1.1.1.1.1.m1.1a"><mi id="A4.T5.1.1.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.1.1.1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A4.T5.1.1.1.1.1.m1.1b"><ci id="A4.T5.1.1.1.1.1.m1.1.1.cmml" xref="A4.T5.1.1.1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.1.1.1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="A4.T5.1.1.1.1.1.m1.1d">italic_k</annotation></semantics></math><span class="ltx_text" id="A4.T5.1.1.1.1.1.1" style="font-size:90%;">-th order binary Markov source</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.13.14.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.13.14.1.1"><span class="ltx_text" id="A4.T5.13.14.1.1.1" style="font-size:90%;">Architecture</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.13.14.1.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.13.14.1.2.1"> <span class="ltx_p" id="A4.T5.13.14.1.2.1.1"><span class="ltx_text" id="A4.T5.13.14.1.2.1.1.1" style="font-size:90%;">Based on the </span><span class="ltx_text ltx_font_smallcaps" id="A4.T5.13.14.1.2.1.1.2" style="font-size:90%;">GPT-2</span><span class="ltx_text" id="A4.T5.13.14.1.2.1.1.3" style="font-size:90%;"> architecture as implemented in </span><cite class="ltx_cite ltx_citemacro_cite"><span class="ltx_text" id="A4.T5.13.14.1.2.1.1.4.1" style="font-size:90%;">[</span><span class="ltx_ref ltx_missing_citation ltx_ref_self">pagliardini-llm</span><span class="ltx_text" id="A4.T5.13.14.1.2.1.1.5.2" style="font-size:90%;">]</span></cite></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.2.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A4.T5.2.2.2"><span class="ltx_text" id="A4.T5.2.2.2.1" style="font-size:90%;">Batch size</span></th> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T5.2.2.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.2.2.1.1"> <span class="ltx_p" id="A4.T5.2.2.1.1.1"><span class="ltx_text" id="A4.T5.2.2.1.1.1.1" style="font-size:90%;">Grid-searched in </span><math alttext="\{16,32,64,128,256\}" class="ltx_Math" display="inline" id="A4.T5.2.2.1.1.1.m1.5"><semantics id="A4.T5.2.2.1.1.1.m1.5a"><mrow id="A4.T5.2.2.1.1.1.m1.5.6.2" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml"><mo id="A4.T5.2.2.1.1.1.m1.5.6.2.1" maxsize="90%" minsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml">{</mo><mn id="A4.T5.2.2.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.1.1.cmml">16</mn><mo id="A4.T5.2.2.1.1.1.m1.5.6.2.2" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T5.2.2.1.1.1.m1.2.2" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.2.2.cmml">32</mn><mo id="A4.T5.2.2.1.1.1.m1.5.6.2.3" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T5.2.2.1.1.1.m1.3.3" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.3.3.cmml">64</mn><mo id="A4.T5.2.2.1.1.1.m1.5.6.2.4" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T5.2.2.1.1.1.m1.4.4" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.4.4.cmml">128</mn><mo id="A4.T5.2.2.1.1.1.m1.5.6.2.5" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml">,</mo><mn id="A4.T5.2.2.1.1.1.m1.5.5" mathsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.5.cmml">256</mn><mo id="A4.T5.2.2.1.1.1.m1.5.6.2.6" maxsize="90%" minsize="90%" xref="A4.T5.2.2.1.1.1.m1.5.6.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.T5.2.2.1.1.1.m1.5b"><set id="A4.T5.2.2.1.1.1.m1.5.6.1.cmml" xref="A4.T5.2.2.1.1.1.m1.5.6.2"><cn id="A4.T5.2.2.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.2.2.1.1.1.m1.1.1">16</cn><cn id="A4.T5.2.2.1.1.1.m1.2.2.cmml" type="integer" xref="A4.T5.2.2.1.1.1.m1.2.2">32</cn><cn id="A4.T5.2.2.1.1.1.m1.3.3.cmml" type="integer" xref="A4.T5.2.2.1.1.1.m1.3.3">64</cn><cn id="A4.T5.2.2.1.1.1.m1.4.4.cmml" type="integer" xref="A4.T5.2.2.1.1.1.m1.4.4">128</cn><cn id="A4.T5.2.2.1.1.1.m1.5.5.cmml" type="integer" xref="A4.T5.2.2.1.1.1.m1.5.5">256</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.2.2.1.1.1.m1.5c">\{16,32,64,128,256\}</annotation><annotation encoding="application/x-llamapun" id="A4.T5.2.2.1.1.1.m1.5d">{ 16 , 32 , 64 , 128 , 256 }</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.13.15.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.13.15.2.1"><span class="ltx_text" id="A4.T5.13.15.2.1.1" style="font-size:90%;">Accumulation steps</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.13.15.2.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.13.15.2.2.1"> <span class="ltx_p" id="A4.T5.13.15.2.2.1.1"><span class="ltx_text" id="A4.T5.13.15.2.2.1.1.1" style="font-size:90%;">1</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A4.T5.3.3.2"><span class="ltx_text" id="A4.T5.3.3.2.1" style="font-size:90%;">Optimizer</span></th> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T5.3.3.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.3.3.1.1"> <span class="ltx_p" id="A4.T5.3.3.1.1.1"><span class="ltx_text" id="A4.T5.3.3.1.1.1.1" style="font-size:90%;">AdamW (</span><math alttext="\beta_{1}=0.9,\beta_{2}=0.95" class="ltx_Math" display="inline" id="A4.T5.3.3.1.1.1.m1.2"><semantics id="A4.T5.3.3.1.1.1.m1.2a"><mrow id="A4.T5.3.3.1.1.1.m1.2.2.2" xref="A4.T5.3.3.1.1.1.m1.2.2.3.cmml"><mrow id="A4.T5.3.3.1.1.1.m1.1.1.1.1" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.cmml"><msub id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.cmml"><mi id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.2" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.2.cmml">β</mi><mn id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.3" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="A4.T5.3.3.1.1.1.m1.1.1.1.1.1" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.1.cmml">=</mo><mn id="A4.T5.3.3.1.1.1.m1.1.1.1.1.3" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.3.cmml">0.9</mn></mrow><mo id="A4.T5.3.3.1.1.1.m1.2.2.2.3" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.2.2.3a.cmml">,</mo><mrow id="A4.T5.3.3.1.1.1.m1.2.2.2.2" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.cmml"><msub id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.cmml"><mi id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.2" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.2.cmml">β</mi><mn id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.3" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.3.cmml">2</mn></msub><mo id="A4.T5.3.3.1.1.1.m1.2.2.2.2.1" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.1.cmml">=</mo><mn id="A4.T5.3.3.1.1.1.m1.2.2.2.2.3" mathsize="90%" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.3.cmml">0.95</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.T5.3.3.1.1.1.m1.2b"><apply id="A4.T5.3.3.1.1.1.m1.2.2.3.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2"><csymbol cd="ambiguous" id="A4.T5.3.3.1.1.1.m1.2.2.3a.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="A4.T5.3.3.1.1.1.m1.1.1.1.1.cmml" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1"><eq id="A4.T5.3.3.1.1.1.m1.1.1.1.1.1.cmml" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.1"></eq><apply id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.cmml" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.1.cmml" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2">subscript</csymbol><ci id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.2.cmml" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.2">𝛽</ci><cn id="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.3.cmml" type="integer" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.2.3">1</cn></apply><cn id="A4.T5.3.3.1.1.1.m1.1.1.1.1.3.cmml" type="float" xref="A4.T5.3.3.1.1.1.m1.1.1.1.1.3">0.9</cn></apply><apply id="A4.T5.3.3.1.1.1.m1.2.2.2.2.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2"><eq id="A4.T5.3.3.1.1.1.m1.2.2.2.2.1.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.1"></eq><apply id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.1.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2">subscript</csymbol><ci id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.2.cmml" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.2">𝛽</ci><cn id="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.2.3">2</cn></apply><cn id="A4.T5.3.3.1.1.1.m1.2.2.2.2.3.cmml" type="float" xref="A4.T5.3.3.1.1.1.m1.2.2.2.2.3">0.95</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.3.3.1.1.1.m1.2c">\beta_{1}=0.9,\beta_{2}=0.95</annotation><annotation encoding="application/x-llamapun" id="A4.T5.3.3.1.1.1.m1.2d">italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.9 , italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.95</annotation></semantics></math><span class="ltx_text" id="A4.T5.3.3.1.1.1.2" style="font-size:90%;">)</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.4.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.4.4.2"><span class="ltx_text" id="A4.T5.4.4.2.1" style="font-size:90%;">Learning rate</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.4.4.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.4.4.1.1"> <span class="ltx_p" id="A4.T5.4.4.1.1.1"><math alttext="0.001" class="ltx_Math" display="inline" id="A4.T5.4.4.1.1.1.m1.1"><semantics id="A4.T5.4.4.1.1.1.m1.1a"><mn id="A4.T5.4.4.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.4.4.1.1.1.m1.1.1.cmml">0.001</mn><annotation-xml encoding="MathML-Content" id="A4.T5.4.4.1.1.1.m1.1b"><cn id="A4.T5.4.4.1.1.1.m1.1.1.cmml" type="float" xref="A4.T5.4.4.1.1.1.m1.1.1">0.001</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.4.4.1.1.1.m1.1c">0.001</annotation><annotation encoding="application/x-llamapun" id="A4.T5.4.4.1.1.1.m1.1d">0.001</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.13.16.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.13.16.3.1"><span class="ltx_text" id="A4.T5.13.16.3.1.1" style="font-size:90%;">Scheduler</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.13.16.3.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.13.16.3.2.1"> <span class="ltx_p" id="A4.T5.13.16.3.2.1.1"><span class="ltx_text" id="A4.T5.13.16.3.2.1.1.1" style="font-size:90%;">Cosine</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.5.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.5.5.2"><span class="ltx_text" id="A4.T5.5.5.2.1" style="font-size:90%;"># Iterations</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.5.5.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.5.5.1.1"> <span class="ltx_p" id="A4.T5.5.5.1.1.1"><math alttext="10000" class="ltx_Math" display="inline" id="A4.T5.5.5.1.1.1.m1.1"><semantics id="A4.T5.5.5.1.1.1.m1.1a"><mn id="A4.T5.5.5.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.5.5.1.1.1.m1.1.1.cmml">10000</mn><annotation-xml encoding="MathML-Content" id="A4.T5.5.5.1.1.1.m1.1b"><cn id="A4.T5.5.5.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.5.5.1.1.1.m1.1.1">10000</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.5.5.1.1.1.m1.1c">10000</annotation><annotation encoding="application/x-llamapun" id="A4.T5.5.5.1.1.1.m1.1d">10000</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.6.6.2"><span class="ltx_text" id="A4.T5.6.6.2.1" style="font-size:90%;">Weight decay</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.6.6.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.6.6.1.1"> <span class="ltx_p" id="A4.T5.6.6.1.1.1"><math alttext="1\text{\times}{10}^{-3}" class="ltx_markedasmath" display="inline" id="A4.T5.6.6.1.1.1.m1.1.1.m1.3"><semantics id="A4.T5.6.6.1.1.1.m1.1.1.m1.3a"><mrow id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml"><mn id="A4.T5.6.6.1.1.1.m1.1.1.m1.1.1.1.1.1.1.1" mathsize="90%" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">1</mn><mtext id="A4.T5.6.6.1.1.1.m1.1.1.m1.2.2.2.2.2.2.2" mathsize="90%" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">×</mtext><msup id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml"><mn id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.2" mathsize="90%" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">10</mn><mrow id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.3.2" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml"><mo id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.3.2a" mathsize="90%" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">−</mo><mn id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.3.3.3.3.3.2.2" mathsize="90%" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.T5.6.6.1.1.1.m1.1.1.m1.3b"><csymbol cd="latexml" id="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3.cmml" xref="A4.T5.6.6.1.1.1.m1.1.1.m1.3.3.3">1E-3</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.6.6.1.1.1.m1.1.1.m1.3c">1\text{\times}{10}^{-3}</annotation><annotation encoding="application/x-llamapun" id="A4.T5.6.6.1.1.1.m1.1.1.m1.3d">start_ARG 1 end_ARG start_ARG times end_ARG start_ARG power start_ARG 10 end_ARG start_ARG - 3 end_ARG end_ARG</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.7.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A4.T5.7.7.2"><span class="ltx_text" id="A4.T5.7.7.2.1" style="font-size:90%;">Dropout</span></th> <td class="ltx_td ltx_align_justify ltx_border_t" id="A4.T5.7.7.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.7.7.1.1"> <span class="ltx_p" id="A4.T5.7.7.1.1.1"><math alttext="0" class="ltx_Math" display="inline" id="A4.T5.7.7.1.1.1.m1.1"><semantics id="A4.T5.7.7.1.1.1.m1.1a"><mn id="A4.T5.7.7.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.7.7.1.1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A4.T5.7.7.1.1.1.m1.1b"><cn id="A4.T5.7.7.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.7.7.1.1.1.m1.1.1">0</cn></annotation-xml></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.8.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.8.8.2"><span class="ltx_text" id="A4.T5.8.8.2.1" style="font-size:90%;">Sequence length</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.8.8.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.8.8.1.1"> <span class="ltx_p" id="A4.T5.8.8.1.1.1"><span class="ltx_text" id="A4.T5.8.8.1.1.1.1" style="font-size:90%;">Grid-searched in </span><math alttext="\{128,256,512,1024\}" class="ltx_Math" display="inline" id="A4.T5.8.8.1.1.1.m1.4"><semantics id="A4.T5.8.8.1.1.1.m1.4a"><mrow id="A4.T5.8.8.1.1.1.m1.4.5.2" xref="A4.T5.8.8.1.1.1.m1.4.5.1.cmml"><mo id="A4.T5.8.8.1.1.1.m1.4.5.2.1" maxsize="90%" minsize="90%" xref="A4.T5.8.8.1.1.1.m1.4.5.1.cmml">{</mo><mn id="A4.T5.8.8.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.1.1.cmml">128</mn><mo id="A4.T5.8.8.1.1.1.m1.4.5.2.2" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.4.5.1.cmml">,</mo><mn id="A4.T5.8.8.1.1.1.m1.2.2" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.2.2.cmml">256</mn><mo id="A4.T5.8.8.1.1.1.m1.4.5.2.3" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.4.5.1.cmml">,</mo><mn id="A4.T5.8.8.1.1.1.m1.3.3" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.3.3.cmml">512</mn><mo id="A4.T5.8.8.1.1.1.m1.4.5.2.4" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.4.5.1.cmml">,</mo><mn id="A4.T5.8.8.1.1.1.m1.4.4" mathsize="90%" xref="A4.T5.8.8.1.1.1.m1.4.4.cmml">1024</mn><mo id="A4.T5.8.8.1.1.1.m1.4.5.2.5" maxsize="90%" minsize="90%" xref="A4.T5.8.8.1.1.1.m1.4.5.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.T5.8.8.1.1.1.m1.4b"><set id="A4.T5.8.8.1.1.1.m1.4.5.1.cmml" xref="A4.T5.8.8.1.1.1.m1.4.5.2"><cn id="A4.T5.8.8.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.8.8.1.1.1.m1.1.1">128</cn><cn id="A4.T5.8.8.1.1.1.m1.2.2.cmml" type="integer" xref="A4.T5.8.8.1.1.1.m1.2.2">256</cn><cn id="A4.T5.8.8.1.1.1.m1.3.3.cmml" type="integer" xref="A4.T5.8.8.1.1.1.m1.3.3">512</cn><cn id="A4.T5.8.8.1.1.1.m1.4.4.cmml" type="integer" xref="A4.T5.8.8.1.1.1.m1.4.4">1024</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.8.8.1.1.1.m1.4c">\{128,256,512,1024\}</annotation><annotation encoding="application/x-llamapun" id="A4.T5.8.8.1.1.1.m1.4d">{ 128 , 256 , 512 , 1024 }</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.9.9"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.9.9.2"><span class="ltx_text" id="A4.T5.9.9.2.1" style="font-size:90%;">Embedding dimension</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.9.9.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.9.9.1.1"> <span class="ltx_p" id="A4.T5.9.9.1.1.1"><span class="ltx_text" id="A4.T5.9.9.1.1.1.1" style="font-size:90%;">Grid-searched in </span><math alttext="\{4,8,16,32\}" class="ltx_Math" display="inline" id="A4.T5.9.9.1.1.1.m1.4"><semantics id="A4.T5.9.9.1.1.1.m1.4a"><mrow id="A4.T5.9.9.1.1.1.m1.4.5.2" xref="A4.T5.9.9.1.1.1.m1.4.5.1.cmml"><mo id="A4.T5.9.9.1.1.1.m1.4.5.2.1" maxsize="90%" minsize="90%" xref="A4.T5.9.9.1.1.1.m1.4.5.1.cmml">{</mo><mn id="A4.T5.9.9.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.1.1.cmml">4</mn><mo id="A4.T5.9.9.1.1.1.m1.4.5.2.2" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.4.5.1.cmml">,</mo><mn id="A4.T5.9.9.1.1.1.m1.2.2" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.2.2.cmml">8</mn><mo id="A4.T5.9.9.1.1.1.m1.4.5.2.3" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.4.5.1.cmml">,</mo><mn id="A4.T5.9.9.1.1.1.m1.3.3" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.3.3.cmml">16</mn><mo id="A4.T5.9.9.1.1.1.m1.4.5.2.4" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.4.5.1.cmml">,</mo><mn id="A4.T5.9.9.1.1.1.m1.4.4" mathsize="90%" xref="A4.T5.9.9.1.1.1.m1.4.4.cmml">32</mn><mo id="A4.T5.9.9.1.1.1.m1.4.5.2.5" maxsize="90%" minsize="90%" xref="A4.T5.9.9.1.1.1.m1.4.5.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.T5.9.9.1.1.1.m1.4b"><set id="A4.T5.9.9.1.1.1.m1.4.5.1.cmml" xref="A4.T5.9.9.1.1.1.m1.4.5.2"><cn id="A4.T5.9.9.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.9.9.1.1.1.m1.1.1">4</cn><cn id="A4.T5.9.9.1.1.1.m1.2.2.cmml" type="integer" xref="A4.T5.9.9.1.1.1.m1.2.2">8</cn><cn id="A4.T5.9.9.1.1.1.m1.3.3.cmml" type="integer" xref="A4.T5.9.9.1.1.1.m1.3.3">16</cn><cn id="A4.T5.9.9.1.1.1.m1.4.4.cmml" type="integer" xref="A4.T5.9.9.1.1.1.m1.4.4">32</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.9.9.1.1.1.m1.4c">\{4,8,16,32\}</annotation><annotation encoding="application/x-llamapun" id="A4.T5.9.9.1.1.1.m1.4d">{ 4 , 8 , 16 , 32 }</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.11.11"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.11.11.3"><span class="ltx_text" id="A4.T5.11.11.3.1" style="font-size:90%;">Transformer layers</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.11.11.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.11.11.2.2"> <span class="ltx_p" id="A4.T5.11.11.2.2.2"><span class="ltx_text" id="A4.T5.11.11.2.2.2.1" style="font-size:90%;">Between </span><math alttext="1" class="ltx_Math" display="inline" id="A4.T5.10.10.1.1.1.m1.1"><semantics id="A4.T5.10.10.1.1.1.m1.1a"><mn id="A4.T5.10.10.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.10.10.1.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.T5.10.10.1.1.1.m1.1b"><cn id="A4.T5.10.10.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.10.10.1.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.10.10.1.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A4.T5.10.10.1.1.1.m1.1d">1</annotation></semantics></math><span class="ltx_text" id="A4.T5.11.11.2.2.2.2" style="font-size:90%;"> and </span><math alttext="2" class="ltx_Math" display="inline" id="A4.T5.11.11.2.2.2.m2.1"><semantics id="A4.T5.11.11.2.2.2.m2.1a"><mn id="A4.T5.11.11.2.2.2.m2.1.1" mathsize="90%" xref="A4.T5.11.11.2.2.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="A4.T5.11.11.2.2.2.m2.1b"><cn id="A4.T5.11.11.2.2.2.m2.1.1.cmml" type="integer" xref="A4.T5.11.11.2.2.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.11.11.2.2.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="A4.T5.11.11.2.2.2.m2.1d">2</annotation></semantics></math><span class="ltx_text" id="A4.T5.11.11.2.2.2.3" style="font-size:90%;"> depending on the experiment</span></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.12.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T5.12.12.2"><span class="ltx_text" id="A4.T5.12.12.2.1" style="font-size:90%;">Attention heads</span></th> <td class="ltx_td ltx_align_justify" id="A4.T5.12.12.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.12.12.1.1"> <span class="ltx_p" id="A4.T5.12.12.1.1.1"><math alttext="1" class="ltx_Math" display="inline" id="A4.T5.12.12.1.1.1.m1.1"><semantics id="A4.T5.12.12.1.1.1.m1.1a"><mn id="A4.T5.12.12.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.12.12.1.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.T5.12.12.1.1.1.m1.1b"><cn id="A4.T5.12.12.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.12.12.1.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.12.12.1.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A4.T5.12.12.1.1.1.m1.1d">1</annotation></semantics></math></span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.13.13"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="A4.T5.13.13.2"><span class="ltx_text" id="A4.T5.13.13.2.1" style="font-size:90%;">Repetitions</span></th> <td class="ltx_td ltx_align_justify ltx_border_bb ltx_border_t" id="A4.T5.13.13.1"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.13.13.1.1"> <span class="ltx_p" id="A4.T5.13.13.1.1.1"><math alttext="5" class="ltx_Math" display="inline" id="A4.T5.13.13.1.1.1.m1.1"><semantics id="A4.T5.13.13.1.1.1.m1.1a"><mn id="A4.T5.13.13.1.1.1.m1.1.1" mathsize="90%" xref="A4.T5.13.13.1.1.1.m1.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="A4.T5.13.13.1.1.1.m1.1b"><cn id="A4.T5.13.13.1.1.1.m1.1.1.cmml" type="integer" xref="A4.T5.13.13.1.1.1.m1.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.13.13.1.1.1.m1.1c">5</annotation><annotation encoding="application/x-llamapun" id="A4.T5.13.13.1.1.1.m1.1d">5</annotation></semantics></math></span> </span> </td> </tr> </tbody> </table> </figure> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Fri Feb 14 14:07:20 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" 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