CINXE.COM

<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806.</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> <meta content=" Multiple Radar System, Variational Message Passing, MIMO Radar, Bayesian Learning " lang="en" name="keywords"/> <base href="/html/2503.16236v1/"/></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/2503.16236v1#S1" title="In Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S2" title="In Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Signal Model</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S3" title="In Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Bayesian Network</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4" title="In Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Variational Message Passing</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5" title="In Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Simulations</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S6" title="In Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars <span class="ltx_note ltx_role_thanks" id="id1.1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">thanks: </span><sup class="ltx_sup" id="id1.1.1">∗</sup> These authors contributed equally. <br class="ltx_break"/>This work is funded by the Thomas B. Thriges Foundation grant 7538-1806.</span></span></span> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Astrid Holm Filtenborg Kitchen<sup class="ltx_sup" id="id5.4.id1">∗</sup>, Mikkel Sebastian Lundsgaard Brøndt<sup class="ltx_sup" id="id6.5.id2">∗</sup>, Marie Saugstrup Jensen<sup class="ltx_sup" id="id7.6.id3">∗</sup>, Troels Pedersen, <br class="ltx_break"/>and Anders Malthe Westerkam </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Aalborg University, Aalborg, Denmark. Email: {troels, amw}@es.aau.dk </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id8.id1">We propose a distributed joint localization and tracking algorithm using a message passing framework, for multiple-input multiple-output radars. We employ the mean field approach to derive an iterative algorithm. The obtained algorithm features a small communication overhead that scales linearly with the number of radars in the system. The proposed algorithm shows good estimation accuracy in two simulated scenarios even below <span class="ltx_ERROR undefined" id="id8.id1.1">\qty</span>0dB signal to noise ratio. In both cases the ground truth falls within the <span class="ltx_ERROR undefined" id="id8.id1.2">\qty</span>95 confidence interval of the estimated posterior for the majority of the track.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Multiple Radar System, Variational Message Passing, MIMO Radar, Bayesian Learning </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para ltx_noindent" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">In recent years, drones have become more and more present in public and private air spaces due to their advancing technology. When used with malicious intent, they raise concerns regarding public, national, and private safety <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib1" title="">1</a>]</cite>. Drones can breach restricted air spaces, e.g. at airports where a drone-plane collision could have fatal consequences <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib2" title="">2</a>]</cite>. Despite regulations prohibiting drone operations in certain areas, numerous instances of these laws being breached continue to occur <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib3" title="">3</a>]</cite>. As a result, security systems must be able to detect unauthorized drone activity to respond to these intrusions.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">There is a wide variety of drone detection and tracking methods relying on sensors such as radars or cameras with varying motion and measurement models, followed by efficient estimation models that handles uncertainties <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib4" title="">4</a>]</cite>. Of these sensors, radars are one of the most effective for aerial target surveillance as they operate in all-day, all-weather conditions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib5" title="">5</a>]</cite>, whereas vision-based systems suffer in partial occlusion conditions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib7" title="">7</a>]</cite>. Drones are generally small, high maneuvering, and fly with low altitude and velocity compared to large aircraft. In relation to radars, drones have a low radar cross section (RCS) and quickly changing kinematic parameters, which makes drones challenging to detect and track accurately with conventional radar systems. To increase the time on target, multiple-input multiple-output (MIMO) radar systems can be employed as they illuminate the entire scene of interest continuously with the trade-off being the power density leading to low signal to noise ratio (SNR) conditions. Hence, there is a pressing need for reliable methods to detect and track targets in low SNR conditions for MIMO radars.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Different approaches have been taken to enhance tracking performance in low SNR conditions using MIMO radars <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib11" title="">11</a>]</cite>. A commonly used method for tracking moving objects are different variations of Kalman filters (KFs). Additionally, KFs are being incorporated with machine learning models to enhance the performance and add more adaptability to the tracking process <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib12" title="">12</a>]</cite>. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib13" title="">13</a>]</cite>, the extended KF was replaced with an adaptive Monte Carlo method and combined with the joint probabilistic data association filter (JPDAF) for multi-target tracking. The JPDAF determines target presence by thresholding, making it susceptible to missed detections and false alarms, especially in cluttered environments. As an alternative, track-before-detect (TBD) algorithms have access to all radar measurements. Several articles have proposed TBD algorithms, based on recursive Bayesian theory. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib8" title="">8</a>]</cite> a sequential Monte Carlo method was presented, jointly detecting and tracking a target with constant velocity in <span class="ltx_ERROR undefined" id="S1.p3.1.1">\qty</span>-10dB SNR using a 4D MIMO radar, whereas <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib11" title="">11</a>]</cite> uses the mean field approach for Bayesian localization and tracking (BLaT) in <span class="ltx_ERROR undefined" id="S1.p3.1.2">\qty</span>-1dB with a MIMO radar. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib14" title="">14</a>]</cite>, the evolution of the probability of a present target was tracked for each range-Doppler cell, but the complexity of this approach is tied to the specific radar settings, as well as relying on a discretized target vector. The contributions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib16" title="">16</a>]</cite> use bistatic MIMO radars with widely separated antennas for joint multi-target detection and localization. By separating the antennas, a target can be observed from multiple angles, enhancing spatial diversity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib17" title="">17</a>]</cite>. To enhance performance at low SNR, raw measurements of all receivers are directly sent to a fusion center and jointly processed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib21" title="">21</a>]</cite>. However, this approach incurs a higher signaling overhead and increased computational cost.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">In this paper, we extend the BLaT algorithm from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib11" title="">11</a>]</cite> to support multiple communicating MIMO radars and incorporate backwards smoothing with a recursive Bayesian filter. The resulting multiple radar BLaT (MRBLaT) algorithm, estimates target parameters in Cartesian coordinates, allowing the posteriors from each MIMO radar to be combined in a global coordinate system. The performance is evaluated by using simulations of multiple 3<math alttext="\times" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mo id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><times id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">×</annotation></semantics></math>3 MIMO radars using time division multiplexing (TDM) and is compared to a multi input KF with backwards smoothing utilizing all available data and without thresholding. By using a network of MIMO radars, this approach improves the performance of jointly localizing and tracking a single target in a low SNR environment.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="260" id="S1.F1.g1" src="x1.png" width="352"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Overview of the scenario under consideration with <math alttext="N_{\text{radar}}" class="ltx_Math" display="inline" id="S1.F1.5.m1.1"><semantics id="S1.F1.5.m1.1b"><msub id="S1.F1.5.m1.1.1" xref="S1.F1.5.m1.1.1.cmml"><mi id="S1.F1.5.m1.1.1.2" xref="S1.F1.5.m1.1.1.2.cmml">N</mi><mtext id="S1.F1.5.m1.1.1.3" xref="S1.F1.5.m1.1.1.3a.cmml">radar</mtext></msub><annotation-xml encoding="MathML-Content" id="S1.F1.5.m1.1c"><apply id="S1.F1.5.m1.1.1.cmml" xref="S1.F1.5.m1.1.1"><csymbol cd="ambiguous" id="S1.F1.5.m1.1.1.1.cmml" xref="S1.F1.5.m1.1.1">subscript</csymbol><ci id="S1.F1.5.m1.1.1.2.cmml" xref="S1.F1.5.m1.1.1.2">𝑁</ci><ci id="S1.F1.5.m1.1.1.3a.cmml" xref="S1.F1.5.m1.1.1.3"><mtext id="S1.F1.5.m1.1.1.3.cmml" mathsize="70%" xref="S1.F1.5.m1.1.1.3">radar</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.5.m1.1d">N_{\text{radar}}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.5.m1.1e">italic_N start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT</annotation></semantics></math> radars are placed throughout the scene. Each radar consists of <math alttext="N_{T}" class="ltx_Math" display="inline" id="S1.F1.6.m2.1"><semantics id="S1.F1.6.m2.1b"><msub id="S1.F1.6.m2.1.1" xref="S1.F1.6.m2.1.1.cmml"><mi id="S1.F1.6.m2.1.1.2" xref="S1.F1.6.m2.1.1.2.cmml">N</mi><mi id="S1.F1.6.m2.1.1.3" xref="S1.F1.6.m2.1.1.3.cmml">T</mi></msub><annotation-xml encoding="MathML-Content" id="S1.F1.6.m2.1c"><apply id="S1.F1.6.m2.1.1.cmml" xref="S1.F1.6.m2.1.1"><csymbol cd="ambiguous" id="S1.F1.6.m2.1.1.1.cmml" xref="S1.F1.6.m2.1.1">subscript</csymbol><ci id="S1.F1.6.m2.1.1.2.cmml" xref="S1.F1.6.m2.1.1.2">𝑁</ci><ci id="S1.F1.6.m2.1.1.3.cmml" xref="S1.F1.6.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.6.m2.1d">N_{T}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.6.m2.1e">italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math> transmitters transmitting orthogonal signals, that are reflected by the target which is governed by its kinematic parameters and collected by <math alttext="N_{R}" class="ltx_Math" display="inline" id="S1.F1.7.m3.1"><semantics id="S1.F1.7.m3.1b"><msub id="S1.F1.7.m3.1.1" xref="S1.F1.7.m3.1.1.cmml"><mi id="S1.F1.7.m3.1.1.2" xref="S1.F1.7.m3.1.1.2.cmml">N</mi><mi id="S1.F1.7.m3.1.1.3" xref="S1.F1.7.m3.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="S1.F1.7.m3.1c"><apply id="S1.F1.7.m3.1.1.cmml" xref="S1.F1.7.m3.1.1"><csymbol cd="ambiguous" id="S1.F1.7.m3.1.1.1.cmml" xref="S1.F1.7.m3.1.1">subscript</csymbol><ci id="S1.F1.7.m3.1.1.2.cmml" xref="S1.F1.7.m3.1.1.2">𝑁</ci><ci id="S1.F1.7.m3.1.1.3.cmml" xref="S1.F1.7.m3.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.7.m3.1d">N_{R}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.7.m3.1e">italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> receivers. At each radar the collected signals are matched filtered resulting in <math alttext="N_{T}N_{R}" class="ltx_Math" display="inline" id="S1.F1.8.m4.1"><semantics id="S1.F1.8.m4.1b"><mrow id="S1.F1.8.m4.1.1" xref="S1.F1.8.m4.1.1.cmml"><msub id="S1.F1.8.m4.1.1.2" xref="S1.F1.8.m4.1.1.2.cmml"><mi id="S1.F1.8.m4.1.1.2.2" xref="S1.F1.8.m4.1.1.2.2.cmml">N</mi><mi id="S1.F1.8.m4.1.1.2.3" xref="S1.F1.8.m4.1.1.2.3.cmml">T</mi></msub><mo id="S1.F1.8.m4.1.1.1" xref="S1.F1.8.m4.1.1.1.cmml">⁢</mo><msub id="S1.F1.8.m4.1.1.3" xref="S1.F1.8.m4.1.1.3.cmml"><mi id="S1.F1.8.m4.1.1.3.2" xref="S1.F1.8.m4.1.1.3.2.cmml">N</mi><mi id="S1.F1.8.m4.1.1.3.3" xref="S1.F1.8.m4.1.1.3.3.cmml">R</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.8.m4.1c"><apply id="S1.F1.8.m4.1.1.cmml" xref="S1.F1.8.m4.1.1"><times id="S1.F1.8.m4.1.1.1.cmml" xref="S1.F1.8.m4.1.1.1"></times><apply id="S1.F1.8.m4.1.1.2.cmml" xref="S1.F1.8.m4.1.1.2"><csymbol cd="ambiguous" id="S1.F1.8.m4.1.1.2.1.cmml" xref="S1.F1.8.m4.1.1.2">subscript</csymbol><ci id="S1.F1.8.m4.1.1.2.2.cmml" xref="S1.F1.8.m4.1.1.2.2">𝑁</ci><ci id="S1.F1.8.m4.1.1.2.3.cmml" xref="S1.F1.8.m4.1.1.2.3">𝑇</ci></apply><apply id="S1.F1.8.m4.1.1.3.cmml" xref="S1.F1.8.m4.1.1.3"><csymbol cd="ambiguous" id="S1.F1.8.m4.1.1.3.1.cmml" xref="S1.F1.8.m4.1.1.3">subscript</csymbol><ci id="S1.F1.8.m4.1.1.3.2.cmml" xref="S1.F1.8.m4.1.1.3.2">𝑁</ci><ci id="S1.F1.8.m4.1.1.3.3.cmml" xref="S1.F1.8.m4.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.8.m4.1d">N_{T}N_{R}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.8.m4.1e">italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> outputs.</figcaption> </figure> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Signal Model</span> </h2> <div class="ltx_para ltx_noindent" id="S2.p1"> <p class="ltx_p" id="S2.p1.22">Following Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">1</span></a>, we consider a single target in a clutter-free environment. The target can be described by its kinematic parameters: <math alttext="\bm{\phi}(t)=[x(t),y(t),v^{(x)}\!(t),v^{(y)}\!(t)]^{\top}" class="ltx_Math" display="inline" id="S2.p1.1.m1.11"><semantics id="S2.p1.1.m1.11a"><mrow id="S2.p1.1.m1.11.11" xref="S2.p1.1.m1.11.11.cmml"><mrow id="S2.p1.1.m1.11.11.6" xref="S2.p1.1.m1.11.11.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p1.1.m1.11.11.6.2" mathvariant="bold-italic" xref="S2.p1.1.m1.11.11.6.2.cmml">ϕ</mi><mo id="S2.p1.1.m1.11.11.6.1" xref="S2.p1.1.m1.11.11.6.1.cmml">⁢</mo><mrow id="S2.p1.1.m1.11.11.6.3.2" xref="S2.p1.1.m1.11.11.6.cmml"><mo id="S2.p1.1.m1.11.11.6.3.2.1" stretchy="false" xref="S2.p1.1.m1.11.11.6.cmml">(</mo><mi id="S2.p1.1.m1.3.3" xref="S2.p1.1.m1.3.3.cmml">t</mi><mo id="S2.p1.1.m1.11.11.6.3.2.2" stretchy="false" xref="S2.p1.1.m1.11.11.6.cmml">)</mo></mrow></mrow><mo id="S2.p1.1.m1.11.11.5" xref="S2.p1.1.m1.11.11.5.cmml">=</mo><msup id="S2.p1.1.m1.11.11.4" xref="S2.p1.1.m1.11.11.4.cmml"><mrow id="S2.p1.1.m1.11.11.4.4.4" xref="S2.p1.1.m1.11.11.4.4.5.cmml"><mo id="S2.p1.1.m1.11.11.4.4.4.5" stretchy="false" xref="S2.p1.1.m1.11.11.4.4.5.cmml">[</mo><mrow id="S2.p1.1.m1.8.8.1.1.1.1" xref="S2.p1.1.m1.8.8.1.1.1.1.cmml"><mi id="S2.p1.1.m1.8.8.1.1.1.1.2" xref="S2.p1.1.m1.8.8.1.1.1.1.2.cmml">x</mi><mo id="S2.p1.1.m1.8.8.1.1.1.1.1" xref="S2.p1.1.m1.8.8.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.p1.1.m1.8.8.1.1.1.1.3.2" xref="S2.p1.1.m1.8.8.1.1.1.1.cmml"><mo id="S2.p1.1.m1.8.8.1.1.1.1.3.2.1" stretchy="false" xref="S2.p1.1.m1.8.8.1.1.1.1.cmml">(</mo><mi id="S2.p1.1.m1.4.4" xref="S2.p1.1.m1.4.4.cmml">t</mi><mo id="S2.p1.1.m1.8.8.1.1.1.1.3.2.2" stretchy="false" xref="S2.p1.1.m1.8.8.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.1.m1.11.11.4.4.4.6" xref="S2.p1.1.m1.11.11.4.4.5.cmml">,</mo><mrow id="S2.p1.1.m1.9.9.2.2.2.2" xref="S2.p1.1.m1.9.9.2.2.2.2.cmml"><mi id="S2.p1.1.m1.9.9.2.2.2.2.2" xref="S2.p1.1.m1.9.9.2.2.2.2.2.cmml">y</mi><mo id="S2.p1.1.m1.9.9.2.2.2.2.1" xref="S2.p1.1.m1.9.9.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.p1.1.m1.9.9.2.2.2.2.3.2" xref="S2.p1.1.m1.9.9.2.2.2.2.cmml"><mo id="S2.p1.1.m1.9.9.2.2.2.2.3.2.1" stretchy="false" xref="S2.p1.1.m1.9.9.2.2.2.2.cmml">(</mo><mi id="S2.p1.1.m1.5.5" xref="S2.p1.1.m1.5.5.cmml">t</mi><mo id="S2.p1.1.m1.9.9.2.2.2.2.3.2.2" stretchy="false" xref="S2.p1.1.m1.9.9.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.1.m1.11.11.4.4.4.7" xref="S2.p1.1.m1.11.11.4.4.5.cmml">,</mo><mrow id="S2.p1.1.m1.10.10.3.3.3.3" xref="S2.p1.1.m1.10.10.3.3.3.3.cmml"><msup id="S2.p1.1.m1.10.10.3.3.3.3.2" xref="S2.p1.1.m1.10.10.3.3.3.3.2.cmml"><mi id="S2.p1.1.m1.10.10.3.3.3.3.2.2" xref="S2.p1.1.m1.10.10.3.3.3.3.2.2.cmml">v</mi><mrow id="S2.p1.1.m1.1.1.1.3" xref="S2.p1.1.m1.10.10.3.3.3.3.2.cmml"><mo id="S2.p1.1.m1.1.1.1.3.1" stretchy="false" xref="S2.p1.1.m1.10.10.3.3.3.3.2.cmml">(</mo><mi id="S2.p1.1.m1.1.1.1.1" xref="S2.p1.1.m1.1.1.1.1.cmml">x</mi><mo id="S2.p1.1.m1.1.1.1.3.2" stretchy="false" xref="S2.p1.1.m1.10.10.3.3.3.3.2.cmml">)</mo></mrow></msup><mo id="S2.p1.1.m1.10.10.3.3.3.3.1" xref="S2.p1.1.m1.10.10.3.3.3.3.1.cmml">⁢</mo><mrow id="S2.p1.1.m1.10.10.3.3.3.3.3.2" xref="S2.p1.1.m1.10.10.3.3.3.3.cmml"><mo id="S2.p1.1.m1.10.10.3.3.3.3.3.2.1" stretchy="false" xref="S2.p1.1.m1.10.10.3.3.3.3.cmml">(</mo><mi id="S2.p1.1.m1.6.6" xref="S2.p1.1.m1.6.6.cmml">t</mi><mo id="S2.p1.1.m1.10.10.3.3.3.3.3.2.2" stretchy="false" xref="S2.p1.1.m1.10.10.3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.p1.1.m1.11.11.4.4.4.8" xref="S2.p1.1.m1.11.11.4.4.5.cmml">,</mo><mrow id="S2.p1.1.m1.11.11.4.4.4.4" xref="S2.p1.1.m1.11.11.4.4.4.4.cmml"><msup id="S2.p1.1.m1.11.11.4.4.4.4.2" xref="S2.p1.1.m1.11.11.4.4.4.4.2.cmml"><mi id="S2.p1.1.m1.11.11.4.4.4.4.2.2" xref="S2.p1.1.m1.11.11.4.4.4.4.2.2.cmml">v</mi><mrow id="S2.p1.1.m1.2.2.1.3" xref="S2.p1.1.m1.11.11.4.4.4.4.2.cmml"><mo id="S2.p1.1.m1.2.2.1.3.1" stretchy="false" xref="S2.p1.1.m1.11.11.4.4.4.4.2.cmml">(</mo><mi id="S2.p1.1.m1.2.2.1.1" xref="S2.p1.1.m1.2.2.1.1.cmml">y</mi><mo id="S2.p1.1.m1.2.2.1.3.2" stretchy="false" xref="S2.p1.1.m1.11.11.4.4.4.4.2.cmml">)</mo></mrow></msup><mo id="S2.p1.1.m1.11.11.4.4.4.4.1" xref="S2.p1.1.m1.11.11.4.4.4.4.1.cmml">⁢</mo><mrow id="S2.p1.1.m1.11.11.4.4.4.4.3.2" xref="S2.p1.1.m1.11.11.4.4.4.4.cmml"><mo id="S2.p1.1.m1.11.11.4.4.4.4.3.2.1" stretchy="false" xref="S2.p1.1.m1.11.11.4.4.4.4.cmml">(</mo><mi id="S2.p1.1.m1.7.7" xref="S2.p1.1.m1.7.7.cmml">t</mi><mo id="S2.p1.1.m1.11.11.4.4.4.4.3.2.2" stretchy="false" xref="S2.p1.1.m1.11.11.4.4.4.4.cmml">)</mo></mrow></mrow><mo id="S2.p1.1.m1.11.11.4.4.4.9" stretchy="false" xref="S2.p1.1.m1.11.11.4.4.5.cmml">]</mo></mrow><mo id="S2.p1.1.m1.11.11.4.6" xref="S2.p1.1.m1.11.11.4.6.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.11b"><apply id="S2.p1.1.m1.11.11.cmml" xref="S2.p1.1.m1.11.11"><eq id="S2.p1.1.m1.11.11.5.cmml" xref="S2.p1.1.m1.11.11.5"></eq><apply id="S2.p1.1.m1.11.11.6.cmml" xref="S2.p1.1.m1.11.11.6"><times id="S2.p1.1.m1.11.11.6.1.cmml" xref="S2.p1.1.m1.11.11.6.1"></times><ci id="S2.p1.1.m1.11.11.6.2.cmml" xref="S2.p1.1.m1.11.11.6.2">bold-italic-ϕ</ci><ci id="S2.p1.1.m1.3.3.cmml" xref="S2.p1.1.m1.3.3">𝑡</ci></apply><apply id="S2.p1.1.m1.11.11.4.cmml" xref="S2.p1.1.m1.11.11.4"><csymbol cd="ambiguous" id="S2.p1.1.m1.11.11.4.5.cmml" xref="S2.p1.1.m1.11.11.4">superscript</csymbol><list id="S2.p1.1.m1.11.11.4.4.5.cmml" xref="S2.p1.1.m1.11.11.4.4.4"><apply id="S2.p1.1.m1.8.8.1.1.1.1.cmml" xref="S2.p1.1.m1.8.8.1.1.1.1"><times id="S2.p1.1.m1.8.8.1.1.1.1.1.cmml" xref="S2.p1.1.m1.8.8.1.1.1.1.1"></times><ci id="S2.p1.1.m1.8.8.1.1.1.1.2.cmml" xref="S2.p1.1.m1.8.8.1.1.1.1.2">𝑥</ci><ci id="S2.p1.1.m1.4.4.cmml" xref="S2.p1.1.m1.4.4">𝑡</ci></apply><apply id="S2.p1.1.m1.9.9.2.2.2.2.cmml" xref="S2.p1.1.m1.9.9.2.2.2.2"><times id="S2.p1.1.m1.9.9.2.2.2.2.1.cmml" xref="S2.p1.1.m1.9.9.2.2.2.2.1"></times><ci id="S2.p1.1.m1.9.9.2.2.2.2.2.cmml" xref="S2.p1.1.m1.9.9.2.2.2.2.2">𝑦</ci><ci id="S2.p1.1.m1.5.5.cmml" xref="S2.p1.1.m1.5.5">𝑡</ci></apply><apply id="S2.p1.1.m1.10.10.3.3.3.3.cmml" xref="S2.p1.1.m1.10.10.3.3.3.3"><times id="S2.p1.1.m1.10.10.3.3.3.3.1.cmml" xref="S2.p1.1.m1.10.10.3.3.3.3.1"></times><apply id="S2.p1.1.m1.10.10.3.3.3.3.2.cmml" xref="S2.p1.1.m1.10.10.3.3.3.3.2"><csymbol cd="ambiguous" id="S2.p1.1.m1.10.10.3.3.3.3.2.1.cmml" xref="S2.p1.1.m1.10.10.3.3.3.3.2">superscript</csymbol><ci id="S2.p1.1.m1.10.10.3.3.3.3.2.2.cmml" xref="S2.p1.1.m1.10.10.3.3.3.3.2.2">𝑣</ci><ci id="S2.p1.1.m1.1.1.1.1.cmml" xref="S2.p1.1.m1.1.1.1.1">𝑥</ci></apply><ci id="S2.p1.1.m1.6.6.cmml" xref="S2.p1.1.m1.6.6">𝑡</ci></apply><apply id="S2.p1.1.m1.11.11.4.4.4.4.cmml" xref="S2.p1.1.m1.11.11.4.4.4.4"><times id="S2.p1.1.m1.11.11.4.4.4.4.1.cmml" xref="S2.p1.1.m1.11.11.4.4.4.4.1"></times><apply id="S2.p1.1.m1.11.11.4.4.4.4.2.cmml" xref="S2.p1.1.m1.11.11.4.4.4.4.2"><csymbol cd="ambiguous" id="S2.p1.1.m1.11.11.4.4.4.4.2.1.cmml" xref="S2.p1.1.m1.11.11.4.4.4.4.2">superscript</csymbol><ci id="S2.p1.1.m1.11.11.4.4.4.4.2.2.cmml" xref="S2.p1.1.m1.11.11.4.4.4.4.2.2">𝑣</ci><ci id="S2.p1.1.m1.2.2.1.1.cmml" xref="S2.p1.1.m1.2.2.1.1">𝑦</ci></apply><ci id="S2.p1.1.m1.7.7.cmml" xref="S2.p1.1.m1.7.7">𝑡</ci></apply></list><csymbol cd="latexml" id="S2.p1.1.m1.11.11.4.6.cmml" xref="S2.p1.1.m1.11.11.4.6">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.11c">\bm{\phi}(t)=[x(t),y(t),v^{(x)}\!(t),v^{(y)}\!(t)]^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.11d">bold_italic_ϕ ( italic_t ) = [ italic_x ( italic_t ) , italic_y ( italic_t ) , italic_v start_POSTSUPERSCRIPT ( italic_x ) end_POSTSUPERSCRIPT ( italic_t ) , italic_v start_POSTSUPERSCRIPT ( italic_y ) end_POSTSUPERSCRIPT ( italic_t ) ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="(x(t),y(t))" class="ltx_Math" display="inline" id="S2.p1.2.m2.4"><semantics id="S2.p1.2.m2.4a"><mrow id="S2.p1.2.m2.4.4.2" xref="S2.p1.2.m2.4.4.3.cmml"><mo id="S2.p1.2.m2.4.4.2.3" stretchy="false" xref="S2.p1.2.m2.4.4.3.cmml">(</mo><mrow id="S2.p1.2.m2.3.3.1.1" xref="S2.p1.2.m2.3.3.1.1.cmml"><mi id="S2.p1.2.m2.3.3.1.1.2" xref="S2.p1.2.m2.3.3.1.1.2.cmml">x</mi><mo id="S2.p1.2.m2.3.3.1.1.1" xref="S2.p1.2.m2.3.3.1.1.1.cmml">⁢</mo><mrow id="S2.p1.2.m2.3.3.1.1.3.2" xref="S2.p1.2.m2.3.3.1.1.cmml"><mo id="S2.p1.2.m2.3.3.1.1.3.2.1" stretchy="false" xref="S2.p1.2.m2.3.3.1.1.cmml">(</mo><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">t</mi><mo id="S2.p1.2.m2.3.3.1.1.3.2.2" stretchy="false" xref="S2.p1.2.m2.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.2.m2.4.4.2.4" xref="S2.p1.2.m2.4.4.3.cmml">,</mo><mrow id="S2.p1.2.m2.4.4.2.2" xref="S2.p1.2.m2.4.4.2.2.cmml"><mi id="S2.p1.2.m2.4.4.2.2.2" xref="S2.p1.2.m2.4.4.2.2.2.cmml">y</mi><mo id="S2.p1.2.m2.4.4.2.2.1" xref="S2.p1.2.m2.4.4.2.2.1.cmml">⁢</mo><mrow id="S2.p1.2.m2.4.4.2.2.3.2" xref="S2.p1.2.m2.4.4.2.2.cmml"><mo id="S2.p1.2.m2.4.4.2.2.3.2.1" stretchy="false" xref="S2.p1.2.m2.4.4.2.2.cmml">(</mo><mi id="S2.p1.2.m2.2.2" xref="S2.p1.2.m2.2.2.cmml">t</mi><mo id="S2.p1.2.m2.4.4.2.2.3.2.2" stretchy="false" xref="S2.p1.2.m2.4.4.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.2.m2.4.4.2.5" stretchy="false" xref="S2.p1.2.m2.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.4b"><interval closure="open" id="S2.p1.2.m2.4.4.3.cmml" xref="S2.p1.2.m2.4.4.2"><apply id="S2.p1.2.m2.3.3.1.1.cmml" xref="S2.p1.2.m2.3.3.1.1"><times id="S2.p1.2.m2.3.3.1.1.1.cmml" xref="S2.p1.2.m2.3.3.1.1.1"></times><ci id="S2.p1.2.m2.3.3.1.1.2.cmml" xref="S2.p1.2.m2.3.3.1.1.2">𝑥</ci><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝑡</ci></apply><apply id="S2.p1.2.m2.4.4.2.2.cmml" xref="S2.p1.2.m2.4.4.2.2"><times id="S2.p1.2.m2.4.4.2.2.1.cmml" xref="S2.p1.2.m2.4.4.2.2.1"></times><ci id="S2.p1.2.m2.4.4.2.2.2.cmml" xref="S2.p1.2.m2.4.4.2.2.2">𝑦</ci><ci id="S2.p1.2.m2.2.2.cmml" xref="S2.p1.2.m2.2.2">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.4c">(x(t),y(t))</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.4d">( italic_x ( italic_t ) , italic_y ( italic_t ) )</annotation></semantics></math> denote the position of the target in a global Cartesian coordinate system, while <math alttext="v^{(x)}\!(t)" class="ltx_Math" display="inline" id="S2.p1.3.m3.2"><semantics id="S2.p1.3.m3.2a"><mrow id="S2.p1.3.m3.2.3" xref="S2.p1.3.m3.2.3.cmml"><msup id="S2.p1.3.m3.2.3.2" xref="S2.p1.3.m3.2.3.2.cmml"><mi id="S2.p1.3.m3.2.3.2.2" xref="S2.p1.3.m3.2.3.2.2.cmml">v</mi><mrow id="S2.p1.3.m3.1.1.1.3" xref="S2.p1.3.m3.2.3.2.cmml"><mo id="S2.p1.3.m3.1.1.1.3.1" stretchy="false" xref="S2.p1.3.m3.2.3.2.cmml">(</mo><mi id="S2.p1.3.m3.1.1.1.1" xref="S2.p1.3.m3.1.1.1.1.cmml">x</mi><mo id="S2.p1.3.m3.1.1.1.3.2" stretchy="false" xref="S2.p1.3.m3.2.3.2.cmml">)</mo></mrow></msup><mo id="S2.p1.3.m3.2.3.1" xref="S2.p1.3.m3.2.3.1.cmml">⁢</mo><mrow id="S2.p1.3.m3.2.3.3.2" xref="S2.p1.3.m3.2.3.cmml"><mo id="S2.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S2.p1.3.m3.2.3.cmml">(</mo><mi id="S2.p1.3.m3.2.2" xref="S2.p1.3.m3.2.2.cmml">t</mi><mo id="S2.p1.3.m3.2.3.3.2.2" stretchy="false" xref="S2.p1.3.m3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.2b"><apply id="S2.p1.3.m3.2.3.cmml" xref="S2.p1.3.m3.2.3"><times id="S2.p1.3.m3.2.3.1.cmml" xref="S2.p1.3.m3.2.3.1"></times><apply id="S2.p1.3.m3.2.3.2.cmml" xref="S2.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="S2.p1.3.m3.2.3.2.1.cmml" xref="S2.p1.3.m3.2.3.2">superscript</csymbol><ci id="S2.p1.3.m3.2.3.2.2.cmml" xref="S2.p1.3.m3.2.3.2.2">𝑣</ci><ci id="S2.p1.3.m3.1.1.1.1.cmml" xref="S2.p1.3.m3.1.1.1.1">𝑥</ci></apply><ci id="S2.p1.3.m3.2.2.cmml" xref="S2.p1.3.m3.2.2">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.2c">v^{(x)}\!(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.2d">italic_v start_POSTSUPERSCRIPT ( italic_x ) end_POSTSUPERSCRIPT ( italic_t )</annotation></semantics></math> and <math alttext="v^{(y)}\!(t)" class="ltx_Math" display="inline" id="S2.p1.4.m4.2"><semantics id="S2.p1.4.m4.2a"><mrow id="S2.p1.4.m4.2.3" xref="S2.p1.4.m4.2.3.cmml"><msup id="S2.p1.4.m4.2.3.2" xref="S2.p1.4.m4.2.3.2.cmml"><mi id="S2.p1.4.m4.2.3.2.2" xref="S2.p1.4.m4.2.3.2.2.cmml">v</mi><mrow id="S2.p1.4.m4.1.1.1.3" xref="S2.p1.4.m4.2.3.2.cmml"><mo id="S2.p1.4.m4.1.1.1.3.1" stretchy="false" xref="S2.p1.4.m4.2.3.2.cmml">(</mo><mi id="S2.p1.4.m4.1.1.1.1" xref="S2.p1.4.m4.1.1.1.1.cmml">y</mi><mo id="S2.p1.4.m4.1.1.1.3.2" stretchy="false" xref="S2.p1.4.m4.2.3.2.cmml">)</mo></mrow></msup><mo id="S2.p1.4.m4.2.3.1" xref="S2.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S2.p1.4.m4.2.3.3.2" xref="S2.p1.4.m4.2.3.cmml"><mo id="S2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S2.p1.4.m4.2.3.cmml">(</mo><mi id="S2.p1.4.m4.2.2" xref="S2.p1.4.m4.2.2.cmml">t</mi><mo id="S2.p1.4.m4.2.3.3.2.2" stretchy="false" xref="S2.p1.4.m4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.2b"><apply id="S2.p1.4.m4.2.3.cmml" xref="S2.p1.4.m4.2.3"><times id="S2.p1.4.m4.2.3.1.cmml" xref="S2.p1.4.m4.2.3.1"></times><apply id="S2.p1.4.m4.2.3.2.cmml" xref="S2.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S2.p1.4.m4.2.3.2.1.cmml" xref="S2.p1.4.m4.2.3.2">superscript</csymbol><ci id="S2.p1.4.m4.2.3.2.2.cmml" xref="S2.p1.4.m4.2.3.2.2">𝑣</ci><ci id="S2.p1.4.m4.1.1.1.1.cmml" xref="S2.p1.4.m4.1.1.1.1">𝑦</ci></apply><ci id="S2.p1.4.m4.2.2.cmml" xref="S2.p1.4.m4.2.2">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.2c">v^{(y)}\!(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.2d">italic_v start_POSTSUPERSCRIPT ( italic_y ) end_POSTSUPERSCRIPT ( italic_t )</annotation></semantics></math> denote the velocity along each of the coordinate axes. The target is illuminated by <math alttext="N_{\text{radar}}" class="ltx_Math" display="inline" id="S2.p1.5.m5.1"><semantics id="S2.p1.5.m5.1a"><msub id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml"><mi id="S2.p1.5.m5.1.1.2" xref="S2.p1.5.m5.1.1.2.cmml">N</mi><mtext id="S2.p1.5.m5.1.1.3" xref="S2.p1.5.m5.1.1.3a.cmml">radar</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.1b"><apply id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.p1.5.m5.1.1.1.cmml" xref="S2.p1.5.m5.1.1">subscript</csymbol><ci id="S2.p1.5.m5.1.1.2.cmml" xref="S2.p1.5.m5.1.1.2">𝑁</ci><ci id="S2.p1.5.m5.1.1.3a.cmml" xref="S2.p1.5.m5.1.1.3"><mtext id="S2.p1.5.m5.1.1.3.cmml" mathsize="70%" xref="S2.p1.5.m5.1.1.3">radar</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.1c">N_{\text{radar}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.1d">italic_N start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT</annotation></semantics></math> monostatic MIMO radars each with <math alttext="N_{T}" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><msub id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml"><mi id="S2.p1.6.m6.1.1.2" xref="S2.p1.6.m6.1.1.2.cmml">N</mi><mi id="S2.p1.6.m6.1.1.3" xref="S2.p1.6.m6.1.1.3.cmml">T</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><apply id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p1.6.m6.1.1.1.cmml" xref="S2.p1.6.m6.1.1">subscript</csymbol><ci id="S2.p1.6.m6.1.1.2.cmml" xref="S2.p1.6.m6.1.1.2">𝑁</ci><ci id="S2.p1.6.m6.1.1.3.cmml" xref="S2.p1.6.m6.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">N_{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math> transmitting antennas and <math alttext="N_{R}" class="ltx_Math" display="inline" id="S2.p1.7.m7.1"><semantics id="S2.p1.7.m7.1a"><msub id="S2.p1.7.m7.1.1" xref="S2.p1.7.m7.1.1.cmml"><mi id="S2.p1.7.m7.1.1.2" xref="S2.p1.7.m7.1.1.2.cmml">N</mi><mi id="S2.p1.7.m7.1.1.3" xref="S2.p1.7.m7.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.7.m7.1b"><apply id="S2.p1.7.m7.1.1.cmml" xref="S2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.p1.7.m7.1.1.1.cmml" xref="S2.p1.7.m7.1.1">subscript</csymbol><ci id="S2.p1.7.m7.1.1.2.cmml" xref="S2.p1.7.m7.1.1.2">𝑁</ci><ci id="S2.p1.7.m7.1.1.3.cmml" xref="S2.p1.7.m7.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m7.1c">N_{R}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m7.1d">italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> receiving antennas. All antennas are considered isotropic. Transmitter <math alttext="m" class="ltx_Math" display="inline" id="S2.p1.8.m8.1"><semantics id="S2.p1.8.m8.1a"><mi id="S2.p1.8.m8.1.1" xref="S2.p1.8.m8.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m8.1b"><ci id="S2.p1.8.m8.1.1.cmml" xref="S2.p1.8.m8.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m8.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m8.1d">italic_m</annotation></semantics></math> on radar <math alttext="k" class="ltx_Math" display="inline" id="S2.p1.9.m9.1"><semantics id="S2.p1.9.m9.1a"><mi id="S2.p1.9.m9.1.1" xref="S2.p1.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p1.9.m9.1b"><ci id="S2.p1.9.m9.1.1.cmml" xref="S2.p1.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m9.1d">italic_k</annotation></semantics></math> denoted as <math alttext="m^{(k)}" class="ltx_Math" display="inline" id="S2.p1.10.m10.1"><semantics id="S2.p1.10.m10.1a"><msup id="S2.p1.10.m10.1.2" xref="S2.p1.10.m10.1.2.cmml"><mi id="S2.p1.10.m10.1.2.2" xref="S2.p1.10.m10.1.2.2.cmml">m</mi><mrow id="S2.p1.10.m10.1.1.1.3" xref="S2.p1.10.m10.1.2.cmml"><mo id="S2.p1.10.m10.1.1.1.3.1" stretchy="false" xref="S2.p1.10.m10.1.2.cmml">(</mo><mi id="S2.p1.10.m10.1.1.1.1" xref="S2.p1.10.m10.1.1.1.1.cmml">k</mi><mo id="S2.p1.10.m10.1.1.1.3.2" stretchy="false" xref="S2.p1.10.m10.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p1.10.m10.1b"><apply id="S2.p1.10.m10.1.2.cmml" xref="S2.p1.10.m10.1.2"><csymbol cd="ambiguous" id="S2.p1.10.m10.1.2.1.cmml" xref="S2.p1.10.m10.1.2">superscript</csymbol><ci id="S2.p1.10.m10.1.2.2.cmml" xref="S2.p1.10.m10.1.2.2">𝑚</ci><ci id="S2.p1.10.m10.1.1.1.1.cmml" xref="S2.p1.10.m10.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m10.1c">m^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m10.1d">italic_m start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> emits a signal <math alttext="\text{Re}\{u^{\left(m^{(k)}\right)}\!(t)\mathrm{e}^{i\omega_{c}t}\}" class="ltx_Math" display="inline" id="S2.p1.11.m11.4"><semantics id="S2.p1.11.m11.4a"><mrow id="S2.p1.11.m11.4.4" xref="S2.p1.11.m11.4.4.cmml"><mtext id="S2.p1.11.m11.4.4.3" xref="S2.p1.11.m11.4.4.3a.cmml">Re</mtext><mo id="S2.p1.11.m11.4.4.2" xref="S2.p1.11.m11.4.4.2.cmml">⁢</mo><mrow id="S2.p1.11.m11.4.4.1.1" xref="S2.p1.11.m11.4.4.1.2.cmml"><mo id="S2.p1.11.m11.4.4.1.1.2" stretchy="false" xref="S2.p1.11.m11.4.4.1.2.cmml">{</mo><mrow id="S2.p1.11.m11.4.4.1.1.1" xref="S2.p1.11.m11.4.4.1.1.1.cmml"><msup id="S2.p1.11.m11.4.4.1.1.1.2" xref="S2.p1.11.m11.4.4.1.1.1.2.cmml"><mi id="S2.p1.11.m11.4.4.1.1.1.2.2" xref="S2.p1.11.m11.4.4.1.1.1.2.2.cmml">u</mi><mrow id="S2.p1.11.m11.2.2.2.2" xref="S2.p1.11.m11.2.2.2.2.1.cmml"><mo id="S2.p1.11.m11.2.2.2.2.2" xref="S2.p1.11.m11.2.2.2.2.1.cmml">(</mo><msup id="S2.p1.11.m11.2.2.2.2.1" xref="S2.p1.11.m11.2.2.2.2.1.cmml"><mi id="S2.p1.11.m11.2.2.2.2.1.2" xref="S2.p1.11.m11.2.2.2.2.1.2.cmml">m</mi><mrow id="S2.p1.11.m11.1.1.1.1.1.3" xref="S2.p1.11.m11.2.2.2.2.1.cmml"><mo id="S2.p1.11.m11.1.1.1.1.1.3.1" stretchy="false" xref="S2.p1.11.m11.2.2.2.2.1.cmml">(</mo><mi id="S2.p1.11.m11.1.1.1.1.1.1" xref="S2.p1.11.m11.1.1.1.1.1.1.cmml">k</mi><mo id="S2.p1.11.m11.1.1.1.1.1.3.2" stretchy="false" xref="S2.p1.11.m11.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S2.p1.11.m11.2.2.2.2.3" xref="S2.p1.11.m11.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S2.p1.11.m11.4.4.1.1.1.1" xref="S2.p1.11.m11.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S2.p1.11.m11.4.4.1.1.1.3.2" xref="S2.p1.11.m11.4.4.1.1.1.cmml"><mo id="S2.p1.11.m11.4.4.1.1.1.3.2.1" stretchy="false" xref="S2.p1.11.m11.4.4.1.1.1.cmml">(</mo><mi id="S2.p1.11.m11.3.3" xref="S2.p1.11.m11.3.3.cmml">t</mi><mo id="S2.p1.11.m11.4.4.1.1.1.3.2.2" stretchy="false" xref="S2.p1.11.m11.4.4.1.1.1.cmml">)</mo></mrow><mo id="S2.p1.11.m11.4.4.1.1.1.1a" xref="S2.p1.11.m11.4.4.1.1.1.1.cmml">⁢</mo><msup id="S2.p1.11.m11.4.4.1.1.1.4" xref="S2.p1.11.m11.4.4.1.1.1.4.cmml"><mi id="S2.p1.11.m11.4.4.1.1.1.4.2" mathvariant="normal" xref="S2.p1.11.m11.4.4.1.1.1.4.2.cmml">e</mi><mrow id="S2.p1.11.m11.4.4.1.1.1.4.3" xref="S2.p1.11.m11.4.4.1.1.1.4.3.cmml"><mi id="S2.p1.11.m11.4.4.1.1.1.4.3.2" xref="S2.p1.11.m11.4.4.1.1.1.4.3.2.cmml">i</mi><mo id="S2.p1.11.m11.4.4.1.1.1.4.3.1" xref="S2.p1.11.m11.4.4.1.1.1.4.3.1.cmml">⁢</mo><msub id="S2.p1.11.m11.4.4.1.1.1.4.3.3" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3.cmml"><mi id="S2.p1.11.m11.4.4.1.1.1.4.3.3.2" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3.2.cmml">ω</mi><mi id="S2.p1.11.m11.4.4.1.1.1.4.3.3.3" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3.3.cmml">c</mi></msub><mo id="S2.p1.11.m11.4.4.1.1.1.4.3.1a" xref="S2.p1.11.m11.4.4.1.1.1.4.3.1.cmml">⁢</mo><mi id="S2.p1.11.m11.4.4.1.1.1.4.3.4" xref="S2.p1.11.m11.4.4.1.1.1.4.3.4.cmml">t</mi></mrow></msup></mrow><mo id="S2.p1.11.m11.4.4.1.1.3" stretchy="false" xref="S2.p1.11.m11.4.4.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.11.m11.4b"><apply id="S2.p1.11.m11.4.4.cmml" xref="S2.p1.11.m11.4.4"><times id="S2.p1.11.m11.4.4.2.cmml" xref="S2.p1.11.m11.4.4.2"></times><ci id="S2.p1.11.m11.4.4.3a.cmml" xref="S2.p1.11.m11.4.4.3"><mtext id="S2.p1.11.m11.4.4.3.cmml" xref="S2.p1.11.m11.4.4.3">Re</mtext></ci><set id="S2.p1.11.m11.4.4.1.2.cmml" xref="S2.p1.11.m11.4.4.1.1"><apply id="S2.p1.11.m11.4.4.1.1.1.cmml" xref="S2.p1.11.m11.4.4.1.1.1"><times id="S2.p1.11.m11.4.4.1.1.1.1.cmml" xref="S2.p1.11.m11.4.4.1.1.1.1"></times><apply id="S2.p1.11.m11.4.4.1.1.1.2.cmml" xref="S2.p1.11.m11.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.11.m11.4.4.1.1.1.2.1.cmml" xref="S2.p1.11.m11.4.4.1.1.1.2">superscript</csymbol><ci id="S2.p1.11.m11.4.4.1.1.1.2.2.cmml" xref="S2.p1.11.m11.4.4.1.1.1.2.2">𝑢</ci><apply id="S2.p1.11.m11.2.2.2.2.1.cmml" xref="S2.p1.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.11.m11.2.2.2.2.1.1.cmml" xref="S2.p1.11.m11.2.2.2.2">superscript</csymbol><ci id="S2.p1.11.m11.2.2.2.2.1.2.cmml" xref="S2.p1.11.m11.2.2.2.2.1.2">𝑚</ci><ci id="S2.p1.11.m11.1.1.1.1.1.1.cmml" xref="S2.p1.11.m11.1.1.1.1.1.1">𝑘</ci></apply></apply><ci id="S2.p1.11.m11.3.3.cmml" xref="S2.p1.11.m11.3.3">𝑡</ci><apply id="S2.p1.11.m11.4.4.1.1.1.4.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4"><csymbol cd="ambiguous" id="S2.p1.11.m11.4.4.1.1.1.4.1.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4">superscript</csymbol><ci id="S2.p1.11.m11.4.4.1.1.1.4.2.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.2">e</ci><apply id="S2.p1.11.m11.4.4.1.1.1.4.3.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3"><times id="S2.p1.11.m11.4.4.1.1.1.4.3.1.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.1"></times><ci id="S2.p1.11.m11.4.4.1.1.1.4.3.2.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.2">𝑖</ci><apply id="S2.p1.11.m11.4.4.1.1.1.4.3.3.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3"><csymbol cd="ambiguous" id="S2.p1.11.m11.4.4.1.1.1.4.3.3.1.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3">subscript</csymbol><ci id="S2.p1.11.m11.4.4.1.1.1.4.3.3.2.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3.2">𝜔</ci><ci id="S2.p1.11.m11.4.4.1.1.1.4.3.3.3.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.3.3">𝑐</ci></apply><ci id="S2.p1.11.m11.4.4.1.1.1.4.3.4.cmml" xref="S2.p1.11.m11.4.4.1.1.1.4.3.4">𝑡</ci></apply></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m11.4c">\text{Re}\{u^{\left(m^{(k)}\right)}\!(t)\mathrm{e}^{i\omega_{c}t}\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m11.4d">Re { italic_u start_POSTSUPERSCRIPT ( italic_m start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ( italic_t ) roman_e start_POSTSUPERSCRIPT italic_i italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT }</annotation></semantics></math>, where <math alttext="u^{\left(m^{(k)}\right)}\!(t)" class="ltx_Math" display="inline" id="S2.p1.12.m12.3"><semantics id="S2.p1.12.m12.3a"><mrow id="S2.p1.12.m12.3.4" xref="S2.p1.12.m12.3.4.cmml"><msup id="S2.p1.12.m12.3.4.2" xref="S2.p1.12.m12.3.4.2.cmml"><mi id="S2.p1.12.m12.3.4.2.2" xref="S2.p1.12.m12.3.4.2.2.cmml">u</mi><mrow id="S2.p1.12.m12.2.2.2.2" xref="S2.p1.12.m12.2.2.2.2.1.cmml"><mo id="S2.p1.12.m12.2.2.2.2.2" xref="S2.p1.12.m12.2.2.2.2.1.cmml">(</mo><msup id="S2.p1.12.m12.2.2.2.2.1" xref="S2.p1.12.m12.2.2.2.2.1.cmml"><mi id="S2.p1.12.m12.2.2.2.2.1.2" xref="S2.p1.12.m12.2.2.2.2.1.2.cmml">m</mi><mrow id="S2.p1.12.m12.1.1.1.1.1.3" xref="S2.p1.12.m12.2.2.2.2.1.cmml"><mo id="S2.p1.12.m12.1.1.1.1.1.3.1" stretchy="false" xref="S2.p1.12.m12.2.2.2.2.1.cmml">(</mo><mi id="S2.p1.12.m12.1.1.1.1.1.1" xref="S2.p1.12.m12.1.1.1.1.1.1.cmml">k</mi><mo id="S2.p1.12.m12.1.1.1.1.1.3.2" stretchy="false" xref="S2.p1.12.m12.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S2.p1.12.m12.2.2.2.2.3" xref="S2.p1.12.m12.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S2.p1.12.m12.3.4.1" xref="S2.p1.12.m12.3.4.1.cmml">⁢</mo><mrow id="S2.p1.12.m12.3.4.3.2" xref="S2.p1.12.m12.3.4.cmml"><mo id="S2.p1.12.m12.3.4.3.2.1" stretchy="false" xref="S2.p1.12.m12.3.4.cmml">(</mo><mi id="S2.p1.12.m12.3.3" xref="S2.p1.12.m12.3.3.cmml">t</mi><mo id="S2.p1.12.m12.3.4.3.2.2" stretchy="false" xref="S2.p1.12.m12.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m12.3b"><apply id="S2.p1.12.m12.3.4.cmml" xref="S2.p1.12.m12.3.4"><times id="S2.p1.12.m12.3.4.1.cmml" xref="S2.p1.12.m12.3.4.1"></times><apply id="S2.p1.12.m12.3.4.2.cmml" xref="S2.p1.12.m12.3.4.2"><csymbol cd="ambiguous" id="S2.p1.12.m12.3.4.2.1.cmml" xref="S2.p1.12.m12.3.4.2">superscript</csymbol><ci id="S2.p1.12.m12.3.4.2.2.cmml" xref="S2.p1.12.m12.3.4.2.2">𝑢</ci><apply id="S2.p1.12.m12.2.2.2.2.1.cmml" xref="S2.p1.12.m12.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.12.m12.2.2.2.2.1.1.cmml" xref="S2.p1.12.m12.2.2.2.2">superscript</csymbol><ci id="S2.p1.12.m12.2.2.2.2.1.2.cmml" xref="S2.p1.12.m12.2.2.2.2.1.2">𝑚</ci><ci id="S2.p1.12.m12.1.1.1.1.1.1.cmml" xref="S2.p1.12.m12.1.1.1.1.1.1">𝑘</ci></apply></apply><ci id="S2.p1.12.m12.3.3.cmml" xref="S2.p1.12.m12.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m12.3c">u^{\left(m^{(k)}\right)}\!(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m12.3d">italic_u start_POSTSUPERSCRIPT ( italic_m start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ( italic_t )</annotation></semantics></math> denotes the complex baseband signal with angular carrier frequency <math alttext="\omega_{c}=2\pi f_{c}" class="ltx_Math" display="inline" id="S2.p1.13.m13.1"><semantics id="S2.p1.13.m13.1a"><mrow id="S2.p1.13.m13.1.1" xref="S2.p1.13.m13.1.1.cmml"><msub id="S2.p1.13.m13.1.1.2" xref="S2.p1.13.m13.1.1.2.cmml"><mi id="S2.p1.13.m13.1.1.2.2" xref="S2.p1.13.m13.1.1.2.2.cmml">ω</mi><mi id="S2.p1.13.m13.1.1.2.3" xref="S2.p1.13.m13.1.1.2.3.cmml">c</mi></msub><mo id="S2.p1.13.m13.1.1.1" xref="S2.p1.13.m13.1.1.1.cmml">=</mo><mrow id="S2.p1.13.m13.1.1.3" xref="S2.p1.13.m13.1.1.3.cmml"><mn id="S2.p1.13.m13.1.1.3.2" xref="S2.p1.13.m13.1.1.3.2.cmml">2</mn><mo id="S2.p1.13.m13.1.1.3.1" xref="S2.p1.13.m13.1.1.3.1.cmml">⁢</mo><mi id="S2.p1.13.m13.1.1.3.3" xref="S2.p1.13.m13.1.1.3.3.cmml">π</mi><mo id="S2.p1.13.m13.1.1.3.1a" xref="S2.p1.13.m13.1.1.3.1.cmml">⁢</mo><msub id="S2.p1.13.m13.1.1.3.4" xref="S2.p1.13.m13.1.1.3.4.cmml"><mi id="S2.p1.13.m13.1.1.3.4.2" xref="S2.p1.13.m13.1.1.3.4.2.cmml">f</mi><mi id="S2.p1.13.m13.1.1.3.4.3" xref="S2.p1.13.m13.1.1.3.4.3.cmml">c</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.13.m13.1b"><apply id="S2.p1.13.m13.1.1.cmml" xref="S2.p1.13.m13.1.1"><eq id="S2.p1.13.m13.1.1.1.cmml" xref="S2.p1.13.m13.1.1.1"></eq><apply id="S2.p1.13.m13.1.1.2.cmml" xref="S2.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S2.p1.13.m13.1.1.2.1.cmml" xref="S2.p1.13.m13.1.1.2">subscript</csymbol><ci id="S2.p1.13.m13.1.1.2.2.cmml" xref="S2.p1.13.m13.1.1.2.2">𝜔</ci><ci id="S2.p1.13.m13.1.1.2.3.cmml" xref="S2.p1.13.m13.1.1.2.3">𝑐</ci></apply><apply id="S2.p1.13.m13.1.1.3.cmml" xref="S2.p1.13.m13.1.1.3"><times id="S2.p1.13.m13.1.1.3.1.cmml" xref="S2.p1.13.m13.1.1.3.1"></times><cn id="S2.p1.13.m13.1.1.3.2.cmml" type="integer" xref="S2.p1.13.m13.1.1.3.2">2</cn><ci id="S2.p1.13.m13.1.1.3.3.cmml" xref="S2.p1.13.m13.1.1.3.3">𝜋</ci><apply id="S2.p1.13.m13.1.1.3.4.cmml" xref="S2.p1.13.m13.1.1.3.4"><csymbol cd="ambiguous" id="S2.p1.13.m13.1.1.3.4.1.cmml" xref="S2.p1.13.m13.1.1.3.4">subscript</csymbol><ci id="S2.p1.13.m13.1.1.3.4.2.cmml" xref="S2.p1.13.m13.1.1.3.4.2">𝑓</ci><ci id="S2.p1.13.m13.1.1.3.4.3.cmml" xref="S2.p1.13.m13.1.1.3.4.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m13.1c">\omega_{c}=2\pi f_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m13.1d">italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 2 italic_π italic_f start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="i" class="ltx_Math" display="inline" id="S2.p1.14.m14.1"><semantics id="S2.p1.14.m14.1a"><mi id="S2.p1.14.m14.1.1" xref="S2.p1.14.m14.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.p1.14.m14.1b"><ci id="S2.p1.14.m14.1.1.cmml" xref="S2.p1.14.m14.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m14.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m14.1d">italic_i</annotation></semantics></math> the imaginary unit. For each transmitter the transmitted signals are assumed mutually orthogonal. We assume no mutual interference, which could be achieved by e.g. frequency division of the radars. For simplicity, each radar has the same transmission scheme resulting in <math alttext="u^{\left(m^{(k)}\right)}\!(t)=u^{(m)}\!(t)" class="ltx_Math" display="inline" id="S2.p1.15.m15.5"><semantics id="S2.p1.15.m15.5a"><mrow id="S2.p1.15.m15.5.6" xref="S2.p1.15.m15.5.6.cmml"><mrow id="S2.p1.15.m15.5.6.2" xref="S2.p1.15.m15.5.6.2.cmml"><msup id="S2.p1.15.m15.5.6.2.2" xref="S2.p1.15.m15.5.6.2.2.cmml"><mi id="S2.p1.15.m15.5.6.2.2.2" xref="S2.p1.15.m15.5.6.2.2.2.cmml">u</mi><mrow id="S2.p1.15.m15.2.2.2.2" xref="S2.p1.15.m15.2.2.2.2.1.cmml"><mo id="S2.p1.15.m15.2.2.2.2.2" xref="S2.p1.15.m15.2.2.2.2.1.cmml">(</mo><msup id="S2.p1.15.m15.2.2.2.2.1" xref="S2.p1.15.m15.2.2.2.2.1.cmml"><mi id="S2.p1.15.m15.2.2.2.2.1.2" xref="S2.p1.15.m15.2.2.2.2.1.2.cmml">m</mi><mrow id="S2.p1.15.m15.1.1.1.1.1.3" xref="S2.p1.15.m15.2.2.2.2.1.cmml"><mo id="S2.p1.15.m15.1.1.1.1.1.3.1" stretchy="false" xref="S2.p1.15.m15.2.2.2.2.1.cmml">(</mo><mi id="S2.p1.15.m15.1.1.1.1.1.1" xref="S2.p1.15.m15.1.1.1.1.1.1.cmml">k</mi><mo id="S2.p1.15.m15.1.1.1.1.1.3.2" stretchy="false" xref="S2.p1.15.m15.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S2.p1.15.m15.2.2.2.2.3" xref="S2.p1.15.m15.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S2.p1.15.m15.5.6.2.1" xref="S2.p1.15.m15.5.6.2.1.cmml">⁢</mo><mrow id="S2.p1.15.m15.5.6.2.3.2" xref="S2.p1.15.m15.5.6.2.cmml"><mo id="S2.p1.15.m15.5.6.2.3.2.1" stretchy="false" xref="S2.p1.15.m15.5.6.2.cmml">(</mo><mi id="S2.p1.15.m15.4.4" xref="S2.p1.15.m15.4.4.cmml">t</mi><mo id="S2.p1.15.m15.5.6.2.3.2.2" stretchy="false" xref="S2.p1.15.m15.5.6.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.15.m15.5.6.1" xref="S2.p1.15.m15.5.6.1.cmml">=</mo><mrow id="S2.p1.15.m15.5.6.3" xref="S2.p1.15.m15.5.6.3.cmml"><msup id="S2.p1.15.m15.5.6.3.2" xref="S2.p1.15.m15.5.6.3.2.cmml"><mi id="S2.p1.15.m15.5.6.3.2.2" xref="S2.p1.15.m15.5.6.3.2.2.cmml">u</mi><mrow id="S2.p1.15.m15.3.3.1.3" xref="S2.p1.15.m15.5.6.3.2.cmml"><mo id="S2.p1.15.m15.3.3.1.3.1" stretchy="false" xref="S2.p1.15.m15.5.6.3.2.cmml">(</mo><mi id="S2.p1.15.m15.3.3.1.1" xref="S2.p1.15.m15.3.3.1.1.cmml">m</mi><mo id="S2.p1.15.m15.3.3.1.3.2" stretchy="false" xref="S2.p1.15.m15.5.6.3.2.cmml">)</mo></mrow></msup><mo id="S2.p1.15.m15.5.6.3.1" xref="S2.p1.15.m15.5.6.3.1.cmml">⁢</mo><mrow id="S2.p1.15.m15.5.6.3.3.2" xref="S2.p1.15.m15.5.6.3.cmml"><mo id="S2.p1.15.m15.5.6.3.3.2.1" stretchy="false" xref="S2.p1.15.m15.5.6.3.cmml">(</mo><mi id="S2.p1.15.m15.5.5" xref="S2.p1.15.m15.5.5.cmml">t</mi><mo id="S2.p1.15.m15.5.6.3.3.2.2" stretchy="false" xref="S2.p1.15.m15.5.6.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.15.m15.5b"><apply id="S2.p1.15.m15.5.6.cmml" xref="S2.p1.15.m15.5.6"><eq id="S2.p1.15.m15.5.6.1.cmml" xref="S2.p1.15.m15.5.6.1"></eq><apply id="S2.p1.15.m15.5.6.2.cmml" xref="S2.p1.15.m15.5.6.2"><times id="S2.p1.15.m15.5.6.2.1.cmml" xref="S2.p1.15.m15.5.6.2.1"></times><apply id="S2.p1.15.m15.5.6.2.2.cmml" xref="S2.p1.15.m15.5.6.2.2"><csymbol cd="ambiguous" id="S2.p1.15.m15.5.6.2.2.1.cmml" xref="S2.p1.15.m15.5.6.2.2">superscript</csymbol><ci id="S2.p1.15.m15.5.6.2.2.2.cmml" xref="S2.p1.15.m15.5.6.2.2.2">𝑢</ci><apply id="S2.p1.15.m15.2.2.2.2.1.cmml" xref="S2.p1.15.m15.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.15.m15.2.2.2.2.1.1.cmml" xref="S2.p1.15.m15.2.2.2.2">superscript</csymbol><ci id="S2.p1.15.m15.2.2.2.2.1.2.cmml" xref="S2.p1.15.m15.2.2.2.2.1.2">𝑚</ci><ci id="S2.p1.15.m15.1.1.1.1.1.1.cmml" xref="S2.p1.15.m15.1.1.1.1.1.1">𝑘</ci></apply></apply><ci id="S2.p1.15.m15.4.4.cmml" xref="S2.p1.15.m15.4.4">𝑡</ci></apply><apply id="S2.p1.15.m15.5.6.3.cmml" xref="S2.p1.15.m15.5.6.3"><times id="S2.p1.15.m15.5.6.3.1.cmml" xref="S2.p1.15.m15.5.6.3.1"></times><apply id="S2.p1.15.m15.5.6.3.2.cmml" xref="S2.p1.15.m15.5.6.3.2"><csymbol cd="ambiguous" id="S2.p1.15.m15.5.6.3.2.1.cmml" xref="S2.p1.15.m15.5.6.3.2">superscript</csymbol><ci id="S2.p1.15.m15.5.6.3.2.2.cmml" xref="S2.p1.15.m15.5.6.3.2.2">𝑢</ci><ci id="S2.p1.15.m15.3.3.1.1.cmml" xref="S2.p1.15.m15.3.3.1.1">𝑚</ci></apply><ci id="S2.p1.15.m15.5.5.cmml" xref="S2.p1.15.m15.5.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m15.5c">u^{\left(m^{(k)}\right)}\!(t)=u^{(m)}\!(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m15.5d">italic_u start_POSTSUPERSCRIPT ( italic_m start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ( italic_t ) = italic_u start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT ( italic_t )</annotation></semantics></math>. We consider the transmission of a MIMO pulse as the transmission from all <math alttext="N_{T}" class="ltx_Math" display="inline" id="S2.p1.16.m16.1"><semantics id="S2.p1.16.m16.1a"><msub id="S2.p1.16.m16.1.1" xref="S2.p1.16.m16.1.1.cmml"><mi id="S2.p1.16.m16.1.1.2" xref="S2.p1.16.m16.1.1.2.cmml">N</mi><mi id="S2.p1.16.m16.1.1.3" xref="S2.p1.16.m16.1.1.3.cmml">T</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.16.m16.1b"><apply id="S2.p1.16.m16.1.1.cmml" xref="S2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.p1.16.m16.1.1.1.cmml" xref="S2.p1.16.m16.1.1">subscript</csymbol><ci id="S2.p1.16.m16.1.1.2.cmml" xref="S2.p1.16.m16.1.1.2">𝑁</ci><ci id="S2.p1.16.m16.1.1.3.cmml" xref="S2.p1.16.m16.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m16.1c">N_{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m16.1d">italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math> transmitters, resulting in a total transmission time of <math alttext="T_{Tx}N_{T}" class="ltx_Math" display="inline" id="S2.p1.17.m17.1"><semantics id="S2.p1.17.m17.1a"><mrow id="S2.p1.17.m17.1.1" xref="S2.p1.17.m17.1.1.cmml"><msub id="S2.p1.17.m17.1.1.2" xref="S2.p1.17.m17.1.1.2.cmml"><mi id="S2.p1.17.m17.1.1.2.2" xref="S2.p1.17.m17.1.1.2.2.cmml">T</mi><mrow id="S2.p1.17.m17.1.1.2.3" xref="S2.p1.17.m17.1.1.2.3.cmml"><mi id="S2.p1.17.m17.1.1.2.3.2" xref="S2.p1.17.m17.1.1.2.3.2.cmml">T</mi><mo id="S2.p1.17.m17.1.1.2.3.1" xref="S2.p1.17.m17.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p1.17.m17.1.1.2.3.3" xref="S2.p1.17.m17.1.1.2.3.3.cmml">x</mi></mrow></msub><mo id="S2.p1.17.m17.1.1.1" xref="S2.p1.17.m17.1.1.1.cmml">⁢</mo><msub id="S2.p1.17.m17.1.1.3" xref="S2.p1.17.m17.1.1.3.cmml"><mi id="S2.p1.17.m17.1.1.3.2" xref="S2.p1.17.m17.1.1.3.2.cmml">N</mi><mi id="S2.p1.17.m17.1.1.3.3" xref="S2.p1.17.m17.1.1.3.3.cmml">T</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.17.m17.1b"><apply id="S2.p1.17.m17.1.1.cmml" xref="S2.p1.17.m17.1.1"><times id="S2.p1.17.m17.1.1.1.cmml" xref="S2.p1.17.m17.1.1.1"></times><apply id="S2.p1.17.m17.1.1.2.cmml" xref="S2.p1.17.m17.1.1.2"><csymbol cd="ambiguous" id="S2.p1.17.m17.1.1.2.1.cmml" xref="S2.p1.17.m17.1.1.2">subscript</csymbol><ci id="S2.p1.17.m17.1.1.2.2.cmml" xref="S2.p1.17.m17.1.1.2.2">𝑇</ci><apply id="S2.p1.17.m17.1.1.2.3.cmml" xref="S2.p1.17.m17.1.1.2.3"><times id="S2.p1.17.m17.1.1.2.3.1.cmml" xref="S2.p1.17.m17.1.1.2.3.1"></times><ci id="S2.p1.17.m17.1.1.2.3.2.cmml" xref="S2.p1.17.m17.1.1.2.3.2">𝑇</ci><ci id="S2.p1.17.m17.1.1.2.3.3.cmml" xref="S2.p1.17.m17.1.1.2.3.3">𝑥</ci></apply></apply><apply id="S2.p1.17.m17.1.1.3.cmml" xref="S2.p1.17.m17.1.1.3"><csymbol cd="ambiguous" id="S2.p1.17.m17.1.1.3.1.cmml" xref="S2.p1.17.m17.1.1.3">subscript</csymbol><ci id="S2.p1.17.m17.1.1.3.2.cmml" xref="S2.p1.17.m17.1.1.3.2">𝑁</ci><ci id="S2.p1.17.m17.1.1.3.3.cmml" xref="S2.p1.17.m17.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m17.1c">T_{Tx}N_{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m17.1d">italic_T start_POSTSUBSCRIPT italic_T italic_x end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math>. The transmission repeats with an interval <math alttext="\Delta t" class="ltx_Math" display="inline" id="S2.p1.18.m18.1"><semantics id="S2.p1.18.m18.1a"><mrow id="S2.p1.18.m18.1.1" xref="S2.p1.18.m18.1.1.cmml"><mi id="S2.p1.18.m18.1.1.2" mathvariant="normal" xref="S2.p1.18.m18.1.1.2.cmml">Δ</mi><mo id="S2.p1.18.m18.1.1.1" xref="S2.p1.18.m18.1.1.1.cmml">⁢</mo><mi id="S2.p1.18.m18.1.1.3" xref="S2.p1.18.m18.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.18.m18.1b"><apply id="S2.p1.18.m18.1.1.cmml" xref="S2.p1.18.m18.1.1"><times id="S2.p1.18.m18.1.1.1.cmml" xref="S2.p1.18.m18.1.1.1"></times><ci id="S2.p1.18.m18.1.1.2.cmml" xref="S2.p1.18.m18.1.1.2">Δ</ci><ci id="S2.p1.18.m18.1.1.3.cmml" xref="S2.p1.18.m18.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.18.m18.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.18.m18.1d">roman_Δ italic_t</annotation></semantics></math>, i.e. the MIMO radars have a pulse repetition frequency (PRF) of <math alttext="1/{\Delta t}" class="ltx_Math" display="inline" id="S2.p1.19.m19.1"><semantics id="S2.p1.19.m19.1a"><mrow id="S2.p1.19.m19.1.1" xref="S2.p1.19.m19.1.1.cmml"><mrow id="S2.p1.19.m19.1.1.2" xref="S2.p1.19.m19.1.1.2.cmml"><mn id="S2.p1.19.m19.1.1.2.2" xref="S2.p1.19.m19.1.1.2.2.cmml">1</mn><mo id="S2.p1.19.m19.1.1.2.1" xref="S2.p1.19.m19.1.1.2.1.cmml">/</mo><mi id="S2.p1.19.m19.1.1.2.3" mathvariant="normal" xref="S2.p1.19.m19.1.1.2.3.cmml">Δ</mi></mrow><mo id="S2.p1.19.m19.1.1.1" xref="S2.p1.19.m19.1.1.1.cmml">⁢</mo><mi id="S2.p1.19.m19.1.1.3" xref="S2.p1.19.m19.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.19.m19.1b"><apply id="S2.p1.19.m19.1.1.cmml" xref="S2.p1.19.m19.1.1"><times id="S2.p1.19.m19.1.1.1.cmml" xref="S2.p1.19.m19.1.1.1"></times><apply id="S2.p1.19.m19.1.1.2.cmml" xref="S2.p1.19.m19.1.1.2"><divide id="S2.p1.19.m19.1.1.2.1.cmml" xref="S2.p1.19.m19.1.1.2.1"></divide><cn id="S2.p1.19.m19.1.1.2.2.cmml" type="integer" xref="S2.p1.19.m19.1.1.2.2">1</cn><ci id="S2.p1.19.m19.1.1.2.3.cmml" xref="S2.p1.19.m19.1.1.2.3">Δ</ci></apply><ci id="S2.p1.19.m19.1.1.3.cmml" xref="S2.p1.19.m19.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.19.m19.1c">1/{\Delta t}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.19.m19.1d">1 / roman_Δ italic_t</annotation></semantics></math>. Each MIMO pulse is reflected by the target and collected by the receiver on the radar from which it originated. It is further assumed that the target is slow moving w.r.t. <math alttext="T_{Tx}N_{T}" 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"><msub id="S2.p1.20.m20.1.1.2" xref="S2.p1.20.m20.1.1.2.cmml"><mi id="S2.p1.20.m20.1.1.2.2" xref="S2.p1.20.m20.1.1.2.2.cmml">T</mi><mrow id="S2.p1.20.m20.1.1.2.3" xref="S2.p1.20.m20.1.1.2.3.cmml"><mi id="S2.p1.20.m20.1.1.2.3.2" xref="S2.p1.20.m20.1.1.2.3.2.cmml">T</mi><mo id="S2.p1.20.m20.1.1.2.3.1" xref="S2.p1.20.m20.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p1.20.m20.1.1.2.3.3" xref="S2.p1.20.m20.1.1.2.3.3.cmml">x</mi></mrow></msub><mo id="S2.p1.20.m20.1.1.1" xref="S2.p1.20.m20.1.1.1.cmml">⁢</mo><msub id="S2.p1.20.m20.1.1.3" xref="S2.p1.20.m20.1.1.3.cmml"><mi id="S2.p1.20.m20.1.1.3.2" xref="S2.p1.20.m20.1.1.3.2.cmml">N</mi><mi id="S2.p1.20.m20.1.1.3.3" xref="S2.p1.20.m20.1.1.3.3.cmml">T</mi></msub></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"><times id="S2.p1.20.m20.1.1.1.cmml" xref="S2.p1.20.m20.1.1.1"></times><apply id="S2.p1.20.m20.1.1.2.cmml" xref="S2.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S2.p1.20.m20.1.1.2.1.cmml" xref="S2.p1.20.m20.1.1.2">subscript</csymbol><ci id="S2.p1.20.m20.1.1.2.2.cmml" xref="S2.p1.20.m20.1.1.2.2">𝑇</ci><apply id="S2.p1.20.m20.1.1.2.3.cmml" xref="S2.p1.20.m20.1.1.2.3"><times id="S2.p1.20.m20.1.1.2.3.1.cmml" xref="S2.p1.20.m20.1.1.2.3.1"></times><ci id="S2.p1.20.m20.1.1.2.3.2.cmml" xref="S2.p1.20.m20.1.1.2.3.2">𝑇</ci><ci id="S2.p1.20.m20.1.1.2.3.3.cmml" xref="S2.p1.20.m20.1.1.2.3.3">𝑥</ci></apply></apply><apply id="S2.p1.20.m20.1.1.3.cmml" xref="S2.p1.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.p1.20.m20.1.1.3.1.cmml" xref="S2.p1.20.m20.1.1.3">subscript</csymbol><ci id="S2.p1.20.m20.1.1.3.2.cmml" xref="S2.p1.20.m20.1.1.3.2">𝑁</ci><ci id="S2.p1.20.m20.1.1.3.3.cmml" xref="S2.p1.20.m20.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.20.m20.1c">T_{Tx}N_{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.20.m20.1d">italic_T start_POSTSUBSCRIPT italic_T italic_x end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, such that the <span class="ltx_ERROR undefined" id="S2.p1.22.1">\say</span>stop-and-hop approach may be employed, and that the RCS of the target is constant across time and radars. The kinematic parameters are considered time invariant between pulses, i.e., <math alttext="\bm{\phi}(t)=\bm{\phi}_{n}" class="ltx_Math" display="inline" id="S2.p1.21.m21.1"><semantics id="S2.p1.21.m21.1a"><mrow id="S2.p1.21.m21.1.2" xref="S2.p1.21.m21.1.2.cmml"><mrow id="S2.p1.21.m21.1.2.2" xref="S2.p1.21.m21.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p1.21.m21.1.2.2.2" mathvariant="bold-italic" xref="S2.p1.21.m21.1.2.2.2.cmml">ϕ</mi><mo id="S2.p1.21.m21.1.2.2.1" xref="S2.p1.21.m21.1.2.2.1.cmml">⁢</mo><mrow id="S2.p1.21.m21.1.2.2.3.2" xref="S2.p1.21.m21.1.2.2.cmml"><mo id="S2.p1.21.m21.1.2.2.3.2.1" stretchy="false" xref="S2.p1.21.m21.1.2.2.cmml">(</mo><mi id="S2.p1.21.m21.1.1" xref="S2.p1.21.m21.1.1.cmml">t</mi><mo id="S2.p1.21.m21.1.2.2.3.2.2" stretchy="false" xref="S2.p1.21.m21.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.21.m21.1.2.1" xref="S2.p1.21.m21.1.2.1.cmml">=</mo><msub id="S2.p1.21.m21.1.2.3" xref="S2.p1.21.m21.1.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S2.p1.21.m21.1.2.3.2" mathvariant="bold-italic" xref="S2.p1.21.m21.1.2.3.2.cmml">ϕ</mi><mi id="S2.p1.21.m21.1.2.3.3" xref="S2.p1.21.m21.1.2.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.21.m21.1b"><apply id="S2.p1.21.m21.1.2.cmml" xref="S2.p1.21.m21.1.2"><eq id="S2.p1.21.m21.1.2.1.cmml" xref="S2.p1.21.m21.1.2.1"></eq><apply id="S2.p1.21.m21.1.2.2.cmml" xref="S2.p1.21.m21.1.2.2"><times id="S2.p1.21.m21.1.2.2.1.cmml" xref="S2.p1.21.m21.1.2.2.1"></times><ci id="S2.p1.21.m21.1.2.2.2.cmml" xref="S2.p1.21.m21.1.2.2.2">bold-italic-ϕ</ci><ci id="S2.p1.21.m21.1.1.cmml" xref="S2.p1.21.m21.1.1">𝑡</ci></apply><apply id="S2.p1.21.m21.1.2.3.cmml" xref="S2.p1.21.m21.1.2.3"><csymbol cd="ambiguous" id="S2.p1.21.m21.1.2.3.1.cmml" xref="S2.p1.21.m21.1.2.3">subscript</csymbol><ci id="S2.p1.21.m21.1.2.3.2.cmml" xref="S2.p1.21.m21.1.2.3.2">bold-italic-ϕ</ci><ci id="S2.p1.21.m21.1.2.3.3.cmml" xref="S2.p1.21.m21.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.21.m21.1c">\bm{\phi}(t)=\bm{\phi}_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.21.m21.1d">bold_italic_ϕ ( italic_t ) = bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="n\Delta t\leq t\leq(n+1)\Delta t" class="ltx_Math" display="inline" id="S2.p1.22.m22.1"><semantics id="S2.p1.22.m22.1a"><mrow id="S2.p1.22.m22.1.1" xref="S2.p1.22.m22.1.1.cmml"><mrow id="S2.p1.22.m22.1.1.3" xref="S2.p1.22.m22.1.1.3.cmml"><mi id="S2.p1.22.m22.1.1.3.2" xref="S2.p1.22.m22.1.1.3.2.cmml">n</mi><mo id="S2.p1.22.m22.1.1.3.1" xref="S2.p1.22.m22.1.1.3.1.cmml">⁢</mo><mi id="S2.p1.22.m22.1.1.3.3" mathvariant="normal" xref="S2.p1.22.m22.1.1.3.3.cmml">Δ</mi><mo id="S2.p1.22.m22.1.1.3.1a" xref="S2.p1.22.m22.1.1.3.1.cmml">⁢</mo><mi id="S2.p1.22.m22.1.1.3.4" xref="S2.p1.22.m22.1.1.3.4.cmml">t</mi></mrow><mo id="S2.p1.22.m22.1.1.4" xref="S2.p1.22.m22.1.1.4.cmml">≤</mo><mi id="S2.p1.22.m22.1.1.5" xref="S2.p1.22.m22.1.1.5.cmml">t</mi><mo id="S2.p1.22.m22.1.1.6" xref="S2.p1.22.m22.1.1.6.cmml">≤</mo><mrow id="S2.p1.22.m22.1.1.1" xref="S2.p1.22.m22.1.1.1.cmml"><mrow id="S2.p1.22.m22.1.1.1.1.1" xref="S2.p1.22.m22.1.1.1.1.1.1.cmml"><mo id="S2.p1.22.m22.1.1.1.1.1.2" stretchy="false" xref="S2.p1.22.m22.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.p1.22.m22.1.1.1.1.1.1" xref="S2.p1.22.m22.1.1.1.1.1.1.cmml"><mi id="S2.p1.22.m22.1.1.1.1.1.1.2" xref="S2.p1.22.m22.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.p1.22.m22.1.1.1.1.1.1.1" xref="S2.p1.22.m22.1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.p1.22.m22.1.1.1.1.1.1.3" xref="S2.p1.22.m22.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.p1.22.m22.1.1.1.1.1.3" stretchy="false" xref="S2.p1.22.m22.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.p1.22.m22.1.1.1.2" xref="S2.p1.22.m22.1.1.1.2.cmml">⁢</mo><mi id="S2.p1.22.m22.1.1.1.3" mathvariant="normal" xref="S2.p1.22.m22.1.1.1.3.cmml">Δ</mi><mo id="S2.p1.22.m22.1.1.1.2a" xref="S2.p1.22.m22.1.1.1.2.cmml">⁢</mo><mi id="S2.p1.22.m22.1.1.1.4" xref="S2.p1.22.m22.1.1.1.4.cmml">t</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.22.m22.1b"><apply id="S2.p1.22.m22.1.1.cmml" xref="S2.p1.22.m22.1.1"><and id="S2.p1.22.m22.1.1a.cmml" xref="S2.p1.22.m22.1.1"></and><apply id="S2.p1.22.m22.1.1b.cmml" xref="S2.p1.22.m22.1.1"><leq id="S2.p1.22.m22.1.1.4.cmml" xref="S2.p1.22.m22.1.1.4"></leq><apply id="S2.p1.22.m22.1.1.3.cmml" xref="S2.p1.22.m22.1.1.3"><times id="S2.p1.22.m22.1.1.3.1.cmml" xref="S2.p1.22.m22.1.1.3.1"></times><ci id="S2.p1.22.m22.1.1.3.2.cmml" xref="S2.p1.22.m22.1.1.3.2">𝑛</ci><ci id="S2.p1.22.m22.1.1.3.3.cmml" xref="S2.p1.22.m22.1.1.3.3">Δ</ci><ci id="S2.p1.22.m22.1.1.3.4.cmml" xref="S2.p1.22.m22.1.1.3.4">𝑡</ci></apply><ci id="S2.p1.22.m22.1.1.5.cmml" xref="S2.p1.22.m22.1.1.5">𝑡</ci></apply><apply id="S2.p1.22.m22.1.1c.cmml" xref="S2.p1.22.m22.1.1"><leq id="S2.p1.22.m22.1.1.6.cmml" xref="S2.p1.22.m22.1.1.6"></leq><share href="https://arxiv.org/html/2503.16236v1#S2.p1.22.m22.1.1.5.cmml" id="S2.p1.22.m22.1.1d.cmml" xref="S2.p1.22.m22.1.1"></share><apply id="S2.p1.22.m22.1.1.1.cmml" xref="S2.p1.22.m22.1.1.1"><times id="S2.p1.22.m22.1.1.1.2.cmml" xref="S2.p1.22.m22.1.1.1.2"></times><apply id="S2.p1.22.m22.1.1.1.1.1.1.cmml" xref="S2.p1.22.m22.1.1.1.1.1"><plus id="S2.p1.22.m22.1.1.1.1.1.1.1.cmml" xref="S2.p1.22.m22.1.1.1.1.1.1.1"></plus><ci id="S2.p1.22.m22.1.1.1.1.1.1.2.cmml" xref="S2.p1.22.m22.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.p1.22.m22.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p1.22.m22.1.1.1.1.1.1.3">1</cn></apply><ci id="S2.p1.22.m22.1.1.1.3.cmml" xref="S2.p1.22.m22.1.1.1.3">Δ</ci><ci id="S2.p1.22.m22.1.1.1.4.cmml" xref="S2.p1.22.m22.1.1.1.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.22.m22.1c">n\Delta t\leq t\leq(n+1)\Delta t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.22.m22.1d">italic_n roman_Δ italic_t ≤ italic_t ≤ ( italic_n + 1 ) roman_Δ italic_t</annotation></semantics></math>. The Doppler shift is ignored due to the low velocity.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.2">Finally, assuming only a direct path and the target being in the far-field of all radars, the signal received at each receiver <math alttext="j" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_j</annotation></semantics></math> on radar <math alttext="k" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mi id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><ci id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_k</annotation></semantics></math> after baseband conversion can be modeled as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <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="\mathcal{Y}^{(j,k)}_{n}(t)=\overbrace{\sum_{m=1}^{N_{T}}\alpha\left(\tau^{(k)}% _{n}\right)A^{(j,m,k)}\!(x_{n},y_{n})u^{(m)}\!\left(t-\tau^{(k)}_{n}\right)}^{% \tilde{s}_{n}^{(j,k)}\!(t)}\\ +w^{(j,k)}_{n}\!(t)," class="ltx_Math" display="block" id="S2.E1.m1.19"><semantics id="S2.E1.m1.19a"><mtable displaystyle="true" id="S2.E1.m1.19.19.2" rowspacing="0pt"><mtr id="S2.E1.m1.19.19.2a"><mtd class="ltx_align_left" columnalign="left" id="S2.E1.m1.19.19.2b"><mrow id="S2.E1.m1.9.9.9.9.9"><mrow id="S2.E1.m1.9.9.9.9.9.10"><msubsup id="S2.E1.m1.9.9.9.9.9.10.2"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml">𝒴</mi><mi id="S2.E1.m1.3.3.3.3.3.3.1" xref="S2.E1.m1.3.3.3.3.3.3.1.cmml">n</mi><mrow id="S2.E1.m1.2.2.2.2.2.2.1.4" xref="S2.E1.m1.2.2.2.2.2.2.1.3.cmml"><mo id="S2.E1.m1.2.2.2.2.2.2.1.4.1" stretchy="false" xref="S2.E1.m1.2.2.2.2.2.2.1.3.cmml">(</mo><mi id="S2.E1.m1.2.2.2.2.2.2.1.1" xref="S2.E1.m1.2.2.2.2.2.2.1.1.cmml">j</mi><mo id="S2.E1.m1.2.2.2.2.2.2.1.4.2" xref="S2.E1.m1.2.2.2.2.2.2.1.3.cmml">,</mo><mi id="S2.E1.m1.2.2.2.2.2.2.1.2" xref="S2.E1.m1.2.2.2.2.2.2.1.2.cmml">k</mi><mo id="S2.E1.m1.2.2.2.2.2.2.1.4.3" stretchy="false" xref="S2.E1.m1.2.2.2.2.2.2.1.3.cmml">)</mo></mrow></msubsup><mo id="S2.E1.m1.9.9.9.9.9.10.1" xref="S2.E1.m1.18.18.1.1.1.cmml">⁢</mo><mrow id="S2.E1.m1.9.9.9.9.9.10.3"><mo id="S2.E1.m1.4.4.4.4.4.4" stretchy="false" xref="S2.E1.m1.18.18.1.1.1.cmml">(</mo><mi id="S2.E1.m1.5.5.5.5.5.5" xref="S2.E1.m1.5.5.5.5.5.5.cmml">t</mi><mo id="S2.E1.m1.6.6.6.6.6.6" stretchy="false" xref="S2.E1.m1.18.18.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.7.7.7.7.7.7" rspace="0.111em" xref="S2.E1.m1.7.7.7.7.7.7.cmml">=</mo><mover id="S2.E1.m1.9.9.9.9.9.11"><mover accent="true" id="S2.E1.m1.8.8.8.8.8.8" xref="S2.E1.m1.8.8.8.8.8.8.cmml"><mrow id="S2.E1.m1.8.8.8.8.8.8.10" xref="S2.E1.m1.8.8.8.8.8.8.10.cmml"><munderover id="S2.E1.m1.8.8.8.8.8.8.10.11" xref="S2.E1.m1.8.8.8.8.8.8.10.11.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.10.11.2.2" movablelimits="false" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.2.cmml">∑</mo><mrow id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.2" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.2.cmml">m</mi><mo id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.1" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.1.cmml">=</mo><mn id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.3" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.3.cmml">1</mn></mrow><msub id="S2.E1.m1.8.8.8.8.8.8.10.11.3" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.11.3.2" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3.2.cmml">N</mi><mi id="S2.E1.m1.8.8.8.8.8.8.10.11.3.3" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3.3.cmml">T</mi></msub></munderover><mrow id="S2.E1.m1.8.8.8.8.8.8.10.10" xref="S2.E1.m1.8.8.8.8.8.8.10.10.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.10.6" xref="S2.E1.m1.8.8.8.8.8.8.10.10.6.cmml">α</mi><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.5" xref="S2.E1.m1.8.8.8.8.8.8.10.10.5.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.2" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml">(</mo><msubsup id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.2.2" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.2.2.cmml">τ</mi><mi id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.3" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.3.cmml">n</mi><mrow id="S2.E1.m1.8.8.8.8.8.8.1.1.1.3" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.1.1.1.3.1" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml">(</mo><mi id="S2.E1.m1.8.8.8.8.8.8.1.1.1.1" xref="S2.E1.m1.8.8.8.8.8.8.1.1.1.1.cmml">k</mi><mo id="S2.E1.m1.8.8.8.8.8.8.1.1.1.3.2" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.3" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml">)</mo></mrow><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.5a" xref="S2.E1.m1.8.8.8.8.8.8.10.10.5.cmml">⁢</mo><msup id="S2.E1.m1.8.8.8.8.8.8.10.10.7" xref="S2.E1.m1.8.8.8.8.8.8.10.10.7.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.10.7.2" xref="S2.E1.m1.8.8.8.8.8.8.10.10.7.2.cmml">A</mi><mrow id="S2.E1.m1.8.8.8.8.8.8.4.4.3.5" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.4.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.4.4.3.5.1" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.4.cmml">(</mo><mi id="S2.E1.m1.8.8.8.8.8.8.2.2.1.1" xref="S2.E1.m1.8.8.8.8.8.8.2.2.1.1.cmml">j</mi><mo id="S2.E1.m1.8.8.8.8.8.8.4.4.3.5.2" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.4.cmml">,</mo><mi id="S2.E1.m1.8.8.8.8.8.8.3.3.2.2" xref="S2.E1.m1.8.8.8.8.8.8.3.3.2.2.cmml">m</mi><mo id="S2.E1.m1.8.8.8.8.8.8.4.4.3.5.3" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.4.cmml">,</mo><mi id="S2.E1.m1.8.8.8.8.8.8.4.4.3.3" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.3.cmml">k</mi><mo id="S2.E1.m1.8.8.8.8.8.8.4.4.3.5.4" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.4.cmml">)</mo></mrow></msup><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.5b" xref="S2.E1.m1.8.8.8.8.8.8.10.10.5.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.3.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.3" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.3.cmml">(</mo><msub id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.2" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.2.cmml">x</mi><mi id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.3" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.3.cmml">n</mi></msub><mo id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.4" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.3.cmml">,</mo><msub id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.2" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.2.cmml">y</mi><mi id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.3" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.3.cmml">n</mi></msub><mo id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.5" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.3.cmml">)</mo></mrow><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.5c" xref="S2.E1.m1.8.8.8.8.8.8.10.10.5.cmml">⁢</mo><msup id="S2.E1.m1.8.8.8.8.8.8.10.10.8" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.10.8.2" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8.2.cmml">u</mi><mrow id="S2.E1.m1.8.8.8.8.8.8.5.5.1.3" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.5.5.1.3.1" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8.cmml">(</mo><mi id="S2.E1.m1.8.8.8.8.8.8.5.5.1.1" xref="S2.E1.m1.8.8.8.8.8.8.5.5.1.1.cmml">m</mi><mo id="S2.E1.m1.8.8.8.8.8.8.5.5.1.3.2" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8.cmml">)</mo></mrow></msup><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.5d" xref="S2.E1.m1.8.8.8.8.8.8.10.10.5.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.2" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.cmml">(</mo><mrow id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.2" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.2.cmml">t</mi><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.1" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.1.cmml">−</mo><msubsup id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.cmml"><mi id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.2.2" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.2.2.cmml">τ</mi><mi id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.3" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.3.cmml">n</mi><mrow id="S2.E1.m1.8.8.8.8.8.8.6.6.1.3" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.cmml"><mo id="S2.E1.m1.8.8.8.8.8.8.6.6.1.3.1" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.cmml">(</mo><mi id="S2.E1.m1.8.8.8.8.8.8.6.6.1.1" xref="S2.E1.m1.8.8.8.8.8.8.6.6.1.1.cmml">k</mi><mo id="S2.E1.m1.8.8.8.8.8.8.6.6.1.3.2" stretchy="false" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.3" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.8.8.8.8.8.8.11" xref="S2.E1.m1.8.8.8.8.8.8.11.cmml">⏞</mo></mover><mrow id="S2.E1.m1.9.9.9.9.9.9.1" xref="S2.E1.m1.9.9.9.9.9.9.1.cmml"><msubsup id="S2.E1.m1.9.9.9.9.9.9.1.5" xref="S2.E1.m1.9.9.9.9.9.9.1.5.cmml"><mover accent="true" id="S2.E1.m1.9.9.9.9.9.9.1.5.2.2" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.cmml"><mi id="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.2" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.2.cmml">s</mi><mo id="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.1" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.1.cmml">~</mo></mover><mi id="S2.E1.m1.9.9.9.9.9.9.1.5.2.3" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.3.cmml">n</mi><mrow id="S2.E1.m1.9.9.9.9.9.9.1.2.2.4" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.3.cmml"><mo id="S2.E1.m1.9.9.9.9.9.9.1.2.2.4.1" stretchy="false" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.3.cmml">(</mo><mi id="S2.E1.m1.9.9.9.9.9.9.1.1.1.1" xref="S2.E1.m1.9.9.9.9.9.9.1.1.1.1.cmml">j</mi><mo id="S2.E1.m1.9.9.9.9.9.9.1.2.2.4.2" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.3.cmml">,</mo><mi id="S2.E1.m1.9.9.9.9.9.9.1.2.2.2" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.2.cmml">k</mi><mo id="S2.E1.m1.9.9.9.9.9.9.1.2.2.4.3" stretchy="false" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.3.cmml">)</mo></mrow></msubsup><mo id="S2.E1.m1.9.9.9.9.9.9.1.4" xref="S2.E1.m1.9.9.9.9.9.9.1.4.cmml">⁢</mo><mrow id="S2.E1.m1.9.9.9.9.9.9.1.6.2" xref="S2.E1.m1.9.9.9.9.9.9.1.cmml"><mo id="S2.E1.m1.9.9.9.9.9.9.1.6.2.1" stretchy="false" xref="S2.E1.m1.9.9.9.9.9.9.1.cmml">(</mo><mi id="S2.E1.m1.9.9.9.9.9.9.1.3" xref="S2.E1.m1.9.9.9.9.9.9.1.3.cmml">t</mi><mo id="S2.E1.m1.9.9.9.9.9.9.1.6.2.2" stretchy="false" xref="S2.E1.m1.9.9.9.9.9.9.1.cmml">)</mo></mrow></mrow></mover></mrow></mtd></mtr><mtr id="S2.E1.m1.19.19.2c"><mtd class="ltx_align_right" columnalign="right" id="S2.E1.m1.19.19.2d"><mrow id="S2.E1.m1.19.19.2.18.9.9.9"><mrow id="S2.E1.m1.19.19.2.18.9.9.9.1"><mo id="S2.E1.m1.19.19.2.18.9.9.9.1a" xref="S2.E1.m1.18.18.1.1.1.cmml">+</mo><mrow id="S2.E1.m1.19.19.2.18.9.9.9.1.1"><msubsup id="S2.E1.m1.19.19.2.18.9.9.9.1.1.2"><mi id="S2.E1.m1.11.11.11.2.2.2" xref="S2.E1.m1.11.11.11.2.2.2.cmml">w</mi><mi id="S2.E1.m1.13.13.13.4.4.4.1" xref="S2.E1.m1.13.13.13.4.4.4.1.cmml">n</mi><mrow id="S2.E1.m1.12.12.12.3.3.3.1.4" xref="S2.E1.m1.12.12.12.3.3.3.1.3.cmml"><mo id="S2.E1.m1.12.12.12.3.3.3.1.4.1" stretchy="false" xref="S2.E1.m1.12.12.12.3.3.3.1.3.cmml">(</mo><mi id="S2.E1.m1.12.12.12.3.3.3.1.1" xref="S2.E1.m1.12.12.12.3.3.3.1.1.cmml">j</mi><mo id="S2.E1.m1.12.12.12.3.3.3.1.4.2" xref="S2.E1.m1.12.12.12.3.3.3.1.3.cmml">,</mo><mi id="S2.E1.m1.12.12.12.3.3.3.1.2" xref="S2.E1.m1.12.12.12.3.3.3.1.2.cmml">k</mi><mo id="S2.E1.m1.12.12.12.3.3.3.1.4.3" stretchy="false" xref="S2.E1.m1.12.12.12.3.3.3.1.3.cmml">)</mo></mrow></msubsup><mo id="S2.E1.m1.19.19.2.18.9.9.9.1.1.1" xref="S2.E1.m1.18.18.1.1.1.cmml">⁢</mo><mrow id="S2.E1.m1.19.19.2.18.9.9.9.1.1.3"><mo id="S2.E1.m1.14.14.14.5.5.5" stretchy="false" xref="S2.E1.m1.18.18.1.1.1.cmml">(</mo><mi id="S2.E1.m1.15.15.15.6.6.6" xref="S2.E1.m1.15.15.15.6.6.6.cmml">t</mi><mo id="S2.E1.m1.16.16.16.7.7.7" stretchy="false" xref="S2.E1.m1.18.18.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.17.17.17.8.8.8" xref="S2.E1.m1.18.18.1.1.1.cmml">,</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S2.E1.m1.19b"><apply id="S2.E1.m1.18.18.1.1.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><eq id="S2.E1.m1.7.7.7.7.7.7.cmml" xref="S2.E1.m1.7.7.7.7.7.7"></eq><apply id="S2.E1.m1.18.18.1.1.1.2.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><times id="S2.E1.m1.18.18.1.1.1.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"></times><apply id="S2.E1.m1.18.18.1.1.1.2.2.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><csymbol cd="ambiguous" id="S2.E1.m1.18.18.1.1.1.2.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1">subscript</csymbol><apply id="S2.E1.m1.18.18.1.1.1.2.2.2.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><csymbol cd="ambiguous" id="S2.E1.m1.18.18.1.1.1.2.2.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1">superscript</csymbol><ci id="S2.E1.m1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1.1">𝒴</ci><interval closure="open" id="S2.E1.m1.2.2.2.2.2.2.1.3.cmml" xref="S2.E1.m1.2.2.2.2.2.2.1.4"><ci id="S2.E1.m1.2.2.2.2.2.2.1.1.cmml" xref="S2.E1.m1.2.2.2.2.2.2.1.1">𝑗</ci><ci id="S2.E1.m1.2.2.2.2.2.2.1.2.cmml" xref="S2.E1.m1.2.2.2.2.2.2.1.2">𝑘</ci></interval></apply><ci id="S2.E1.m1.3.3.3.3.3.3.1.cmml" xref="S2.E1.m1.3.3.3.3.3.3.1">𝑛</ci></apply><ci id="S2.E1.m1.5.5.5.5.5.5.cmml" xref="S2.E1.m1.5.5.5.5.5.5">𝑡</ci></apply><apply id="S2.E1.m1.18.18.1.1.1.3.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><plus id="S2.E1.m1.10.10.10.1.1.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"></plus><apply id="S2.E1.m1.18.18.1.1.1.3.2.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><csymbol cd="ambiguous" id="S2.E1.m1.18.18.1.1.1.3.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1">superscript</csymbol><apply id="S2.E1.m1.8.8.8.8.8.8.cmml" xref="S2.E1.m1.8.8.8.8.8.8"><ci id="S2.E1.m1.8.8.8.8.8.8.11.cmml" xref="S2.E1.m1.8.8.8.8.8.8.11">⏞</ci><apply id="S2.E1.m1.8.8.8.8.8.8.10.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10"><apply id="S2.E1.m1.8.8.8.8.8.8.10.11.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.11.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11">superscript</csymbol><apply id="S2.E1.m1.8.8.8.8.8.8.10.11.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.11.2.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11">subscript</csymbol><sum id="S2.E1.m1.8.8.8.8.8.8.10.11.2.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.2"></sum><apply id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3"><eq id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.1"></eq><ci id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.2">𝑚</ci><cn id="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.3.cmml" type="integer" xref="S2.E1.m1.8.8.8.8.8.8.10.11.2.3.3">1</cn></apply></apply><apply id="S2.E1.m1.8.8.8.8.8.8.10.11.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.11.3.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3">subscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.10.11.3.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3.2">𝑁</ci><ci id="S2.E1.m1.8.8.8.8.8.8.10.11.3.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.11.3.3">𝑇</ci></apply></apply><apply id="S2.E1.m1.8.8.8.8.8.8.10.10.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10"><times id="S2.E1.m1.8.8.8.8.8.8.10.10.5.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.5"></times><ci id="S2.E1.m1.8.8.8.8.8.8.10.10.6.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.6">𝛼</ci><apply id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1">subscript</csymbol><apply id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.2.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1">superscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.2.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.2.2">𝜏</ci><ci id="S2.E1.m1.8.8.8.8.8.8.1.1.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.1.1.1.1">𝑘</ci></apply><ci id="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.7.7.1.1.1.3">𝑛</ci></apply><apply id="S2.E1.m1.8.8.8.8.8.8.10.10.7.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.7"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.10.7.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.7">superscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.10.10.7.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.7.2">𝐴</ci><vector id="S2.E1.m1.8.8.8.8.8.8.4.4.3.4.cmml" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.5"><ci id="S2.E1.m1.8.8.8.8.8.8.2.2.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.2.2.1.1">𝑗</ci><ci id="S2.E1.m1.8.8.8.8.8.8.3.3.2.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.3.3.2.2">𝑚</ci><ci id="S2.E1.m1.8.8.8.8.8.8.4.4.3.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.4.4.3.3">𝑘</ci></vector></apply><interval closure="open" id="S2.E1.m1.8.8.8.8.8.8.9.9.3.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2"><apply id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1">subscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.2">𝑥</ci><ci id="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.8.8.2.1.1.3">𝑛</ci></apply><apply id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2">subscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.2">𝑦</ci><ci id="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.9.9.3.2.2.3">𝑛</ci></apply></interval><apply id="S2.E1.m1.8.8.8.8.8.8.10.10.8.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.10.8.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8">superscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.10.10.8.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.8.2">𝑢</ci><ci id="S2.E1.m1.8.8.8.8.8.8.5.5.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.5.5.1.1">𝑚</ci></apply><apply id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1"><minus id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.1"></minus><ci id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.2">𝑡</ci><apply id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3">subscript</csymbol><apply id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3"><csymbol cd="ambiguous" id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.2.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3">superscript</csymbol><ci id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.2.2.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.2.2">𝜏</ci><ci id="S2.E1.m1.8.8.8.8.8.8.6.6.1.1.cmml" xref="S2.E1.m1.8.8.8.8.8.8.6.6.1.1">𝑘</ci></apply><ci id="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.3.cmml" xref="S2.E1.m1.8.8.8.8.8.8.10.10.4.1.1.3.3">𝑛</ci></apply></apply></apply></apply></apply><apply id="S2.E1.m1.9.9.9.9.9.9.1.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1"><times id="S2.E1.m1.9.9.9.9.9.9.1.4.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.4"></times><apply id="S2.E1.m1.9.9.9.9.9.9.1.5.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5"><csymbol cd="ambiguous" id="S2.E1.m1.9.9.9.9.9.9.1.5.1.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5">superscript</csymbol><apply id="S2.E1.m1.9.9.9.9.9.9.1.5.2.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5"><csymbol cd="ambiguous" id="S2.E1.m1.9.9.9.9.9.9.1.5.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5">subscript</csymbol><apply id="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.2"><ci id="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.1">~</ci><ci id="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.2.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.2.2">𝑠</ci></apply><ci id="S2.E1.m1.9.9.9.9.9.9.1.5.2.3.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.5.2.3">𝑛</ci></apply><interval closure="open" id="S2.E1.m1.9.9.9.9.9.9.1.2.2.3.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.4"><ci id="S2.E1.m1.9.9.9.9.9.9.1.1.1.1.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.1.1.1">𝑗</ci><ci id="S2.E1.m1.9.9.9.9.9.9.1.2.2.2.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.2.2.2">𝑘</ci></interval></apply><ci id="S2.E1.m1.9.9.9.9.9.9.1.3.cmml" xref="S2.E1.m1.9.9.9.9.9.9.1.3">𝑡</ci></apply></apply><apply id="S2.E1.m1.18.18.1.1.1.3.3.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><times id="S2.E1.m1.18.18.1.1.1.3.3.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"></times><apply id="S2.E1.m1.18.18.1.1.1.3.3.2.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><csymbol cd="ambiguous" id="S2.E1.m1.18.18.1.1.1.3.3.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1">subscript</csymbol><apply id="S2.E1.m1.18.18.1.1.1.3.3.2.2.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1"><csymbol cd="ambiguous" id="S2.E1.m1.18.18.1.1.1.3.3.2.2.1.cmml" xref="S2.E1.m1.9.9.9.9.9.10.1">superscript</csymbol><ci id="S2.E1.m1.11.11.11.2.2.2.cmml" xref="S2.E1.m1.11.11.11.2.2.2">𝑤</ci><interval closure="open" id="S2.E1.m1.12.12.12.3.3.3.1.3.cmml" xref="S2.E1.m1.12.12.12.3.3.3.1.4"><ci id="S2.E1.m1.12.12.12.3.3.3.1.1.cmml" xref="S2.E1.m1.12.12.12.3.3.3.1.1">𝑗</ci><ci id="S2.E1.m1.12.12.12.3.3.3.1.2.cmml" xref="S2.E1.m1.12.12.12.3.3.3.1.2">𝑘</ci></interval></apply><ci id="S2.E1.m1.13.13.13.4.4.4.1.cmml" xref="S2.E1.m1.13.13.13.4.4.4.1">𝑛</ci></apply><ci id="S2.E1.m1.15.15.15.6.6.6.cmml" xref="S2.E1.m1.15.15.15.6.6.6">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.19c">\mathcal{Y}^{(j,k)}_{n}(t)=\overbrace{\sum_{m=1}^{N_{T}}\alpha\left(\tau^{(k)}% _{n}\right)A^{(j,m,k)}\!(x_{n},y_{n})u^{(m)}\!\left(t-\tau^{(k)}_{n}\right)}^{% \tilde{s}_{n}^{(j,k)}\!(t)}\\ +w^{(j,k)}_{n}\!(t),</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.19d">start_ROW start_CELL caligraphic_Y start_POSTSUPERSCRIPT ( italic_j , italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = over⏞ start_ARG ∑ start_POSTSUBSCRIPT italic_m = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_α ( italic_τ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) italic_A start_POSTSUPERSCRIPT ( italic_j , italic_m , italic_k ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) italic_u start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT ( italic_t - italic_τ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_ARG start_POSTSUPERSCRIPT over~ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j , italic_k ) end_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + italic_w start_POSTSUPERSCRIPT ( italic_j , italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) , 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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p2.8">where <math alttext="w^{(j,k)}_{n}\!(t)" class="ltx_Math" display="inline" id="S2.p2.3.m1.3"><semantics id="S2.p2.3.m1.3a"><mrow id="S2.p2.3.m1.3.4" xref="S2.p2.3.m1.3.4.cmml"><msubsup id="S2.p2.3.m1.3.4.2" xref="S2.p2.3.m1.3.4.2.cmml"><mi id="S2.p2.3.m1.3.4.2.2.2" xref="S2.p2.3.m1.3.4.2.2.2.cmml">w</mi><mi id="S2.p2.3.m1.3.4.2.3" xref="S2.p2.3.m1.3.4.2.3.cmml">n</mi><mrow id="S2.p2.3.m1.2.2.2.4" xref="S2.p2.3.m1.2.2.2.3.cmml"><mo id="S2.p2.3.m1.2.2.2.4.1" stretchy="false" xref="S2.p2.3.m1.2.2.2.3.cmml">(</mo><mi id="S2.p2.3.m1.1.1.1.1" xref="S2.p2.3.m1.1.1.1.1.cmml">j</mi><mo id="S2.p2.3.m1.2.2.2.4.2" xref="S2.p2.3.m1.2.2.2.3.cmml">,</mo><mi id="S2.p2.3.m1.2.2.2.2" xref="S2.p2.3.m1.2.2.2.2.cmml">k</mi><mo id="S2.p2.3.m1.2.2.2.4.3" stretchy="false" xref="S2.p2.3.m1.2.2.2.3.cmml">)</mo></mrow></msubsup><mo id="S2.p2.3.m1.3.4.1" xref="S2.p2.3.m1.3.4.1.cmml">⁢</mo><mrow id="S2.p2.3.m1.3.4.3.2" xref="S2.p2.3.m1.3.4.cmml"><mo id="S2.p2.3.m1.3.4.3.2.1" stretchy="false" xref="S2.p2.3.m1.3.4.cmml">(</mo><mi id="S2.p2.3.m1.3.3" xref="S2.p2.3.m1.3.3.cmml">t</mi><mo id="S2.p2.3.m1.3.4.3.2.2" stretchy="false" xref="S2.p2.3.m1.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m1.3b"><apply id="S2.p2.3.m1.3.4.cmml" xref="S2.p2.3.m1.3.4"><times id="S2.p2.3.m1.3.4.1.cmml" xref="S2.p2.3.m1.3.4.1"></times><apply id="S2.p2.3.m1.3.4.2.cmml" xref="S2.p2.3.m1.3.4.2"><csymbol cd="ambiguous" id="S2.p2.3.m1.3.4.2.1.cmml" xref="S2.p2.3.m1.3.4.2">subscript</csymbol><apply id="S2.p2.3.m1.3.4.2.2.cmml" xref="S2.p2.3.m1.3.4.2"><csymbol cd="ambiguous" id="S2.p2.3.m1.3.4.2.2.1.cmml" xref="S2.p2.3.m1.3.4.2">superscript</csymbol><ci id="S2.p2.3.m1.3.4.2.2.2.cmml" xref="S2.p2.3.m1.3.4.2.2.2">𝑤</ci><interval closure="open" id="S2.p2.3.m1.2.2.2.3.cmml" xref="S2.p2.3.m1.2.2.2.4"><ci id="S2.p2.3.m1.1.1.1.1.cmml" xref="S2.p2.3.m1.1.1.1.1">𝑗</ci><ci id="S2.p2.3.m1.2.2.2.2.cmml" xref="S2.p2.3.m1.2.2.2.2">𝑘</ci></interval></apply><ci id="S2.p2.3.m1.3.4.2.3.cmml" xref="S2.p2.3.m1.3.4.2.3">𝑛</ci></apply><ci id="S2.p2.3.m1.3.3.cmml" xref="S2.p2.3.m1.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m1.3c">w^{(j,k)}_{n}\!(t)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m1.3d">italic_w start_POSTSUPERSCRIPT ( italic_j , italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is complex circularly symmetric white Gaussian noise with zero mean and variance <math alttext="\sigma_{w}^{2}" class="ltx_Math" display="inline" id="S2.p2.4.m2.1"><semantics id="S2.p2.4.m2.1a"><msubsup id="S2.p2.4.m2.1.1" xref="S2.p2.4.m2.1.1.cmml"><mi id="S2.p2.4.m2.1.1.2.2" xref="S2.p2.4.m2.1.1.2.2.cmml">σ</mi><mi id="S2.p2.4.m2.1.1.2.3" xref="S2.p2.4.m2.1.1.2.3.cmml">w</mi><mn id="S2.p2.4.m2.1.1.3" xref="S2.p2.4.m2.1.1.3.cmml">2</mn></msubsup><annotation-xml encoding="MathML-Content" id="S2.p2.4.m2.1b"><apply id="S2.p2.4.m2.1.1.cmml" xref="S2.p2.4.m2.1.1"><csymbol cd="ambiguous" id="S2.p2.4.m2.1.1.1.cmml" xref="S2.p2.4.m2.1.1">superscript</csymbol><apply id="S2.p2.4.m2.1.1.2.cmml" xref="S2.p2.4.m2.1.1"><csymbol cd="ambiguous" id="S2.p2.4.m2.1.1.2.1.cmml" xref="S2.p2.4.m2.1.1">subscript</csymbol><ci id="S2.p2.4.m2.1.1.2.2.cmml" xref="S2.p2.4.m2.1.1.2.2">𝜎</ci><ci id="S2.p2.4.m2.1.1.2.3.cmml" xref="S2.p2.4.m2.1.1.2.3">𝑤</ci></apply><cn id="S2.p2.4.m2.1.1.3.cmml" type="integer" xref="S2.p2.4.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m2.1c">\sigma_{w}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m2.1d">italic_σ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="A^{(j,m,k)}" class="ltx_Math" display="inline" id="S2.p2.5.m3.3"><semantics id="S2.p2.5.m3.3a"><msup id="S2.p2.5.m3.3.4" xref="S2.p2.5.m3.3.4.cmml"><mi id="S2.p2.5.m3.3.4.2" xref="S2.p2.5.m3.3.4.2.cmml">A</mi><mrow id="S2.p2.5.m3.3.3.3.5" xref="S2.p2.5.m3.3.3.3.4.cmml"><mo id="S2.p2.5.m3.3.3.3.5.1" stretchy="false" xref="S2.p2.5.m3.3.3.3.4.cmml">(</mo><mi id="S2.p2.5.m3.1.1.1.1" xref="S2.p2.5.m3.1.1.1.1.cmml">j</mi><mo id="S2.p2.5.m3.3.3.3.5.2" xref="S2.p2.5.m3.3.3.3.4.cmml">,</mo><mi id="S2.p2.5.m3.2.2.2.2" xref="S2.p2.5.m3.2.2.2.2.cmml">m</mi><mo id="S2.p2.5.m3.3.3.3.5.3" xref="S2.p2.5.m3.3.3.3.4.cmml">,</mo><mi id="S2.p2.5.m3.3.3.3.3" xref="S2.p2.5.m3.3.3.3.3.cmml">k</mi><mo id="S2.p2.5.m3.3.3.3.5.4" stretchy="false" xref="S2.p2.5.m3.3.3.3.4.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p2.5.m3.3b"><apply id="S2.p2.5.m3.3.4.cmml" xref="S2.p2.5.m3.3.4"><csymbol cd="ambiguous" id="S2.p2.5.m3.3.4.1.cmml" xref="S2.p2.5.m3.3.4">superscript</csymbol><ci id="S2.p2.5.m3.3.4.2.cmml" xref="S2.p2.5.m3.3.4.2">𝐴</ci><vector id="S2.p2.5.m3.3.3.3.4.cmml" xref="S2.p2.5.m3.3.3.3.5"><ci id="S2.p2.5.m3.1.1.1.1.cmml" xref="S2.p2.5.m3.1.1.1.1">𝑗</ci><ci id="S2.p2.5.m3.2.2.2.2.cmml" xref="S2.p2.5.m3.2.2.2.2">𝑚</ci><ci id="S2.p2.5.m3.3.3.3.3.cmml" xref="S2.p2.5.m3.3.3.3.3">𝑘</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m3.3c">A^{(j,m,k)}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m3.3d">italic_A start_POSTSUPERSCRIPT ( italic_j , italic_m , italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> is the steering matrix elements, and <math alttext="\tau^{(k)}_{n}" class="ltx_Math" display="inline" id="S2.p2.6.m4.1"><semantics id="S2.p2.6.m4.1a"><msubsup id="S2.p2.6.m4.1.2" xref="S2.p2.6.m4.1.2.cmml"><mi id="S2.p2.6.m4.1.2.2.2" xref="S2.p2.6.m4.1.2.2.2.cmml">τ</mi><mi id="S2.p2.6.m4.1.2.3" xref="S2.p2.6.m4.1.2.3.cmml">n</mi><mrow id="S2.p2.6.m4.1.1.1.3" xref="S2.p2.6.m4.1.2.cmml"><mo id="S2.p2.6.m4.1.1.1.3.1" stretchy="false" xref="S2.p2.6.m4.1.2.cmml">(</mo><mi id="S2.p2.6.m4.1.1.1.1" xref="S2.p2.6.m4.1.1.1.1.cmml">k</mi><mo id="S2.p2.6.m4.1.1.1.3.2" stretchy="false" xref="S2.p2.6.m4.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.p2.6.m4.1b"><apply id="S2.p2.6.m4.1.2.cmml" xref="S2.p2.6.m4.1.2"><csymbol cd="ambiguous" id="S2.p2.6.m4.1.2.1.cmml" xref="S2.p2.6.m4.1.2">subscript</csymbol><apply id="S2.p2.6.m4.1.2.2.cmml" xref="S2.p2.6.m4.1.2"><csymbol cd="ambiguous" id="S2.p2.6.m4.1.2.2.1.cmml" xref="S2.p2.6.m4.1.2">superscript</csymbol><ci id="S2.p2.6.m4.1.2.2.2.cmml" xref="S2.p2.6.m4.1.2.2.2">𝜏</ci><ci id="S2.p2.6.m4.1.1.1.1.cmml" xref="S2.p2.6.m4.1.1.1.1">𝑘</ci></apply><ci id="S2.p2.6.m4.1.2.3.cmml" xref="S2.p2.6.m4.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m4.1c">\tau^{(k)}_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m4.1d">italic_τ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is the two way time delay between radar <math alttext="k" class="ltx_Math" display="inline" id="S2.p2.7.m5.1"><semantics id="S2.p2.7.m5.1a"><mi id="S2.p2.7.m5.1.1" xref="S2.p2.7.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p2.7.m5.1b"><ci id="S2.p2.7.m5.1.1.cmml" xref="S2.p2.7.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.7.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p2.7.m5.1d">italic_k</annotation></semantics></math> and the target. The path loss <math alttext="\alpha(\tau_{n}^{(k)})" class="ltx_Math" display="inline" id="S2.p2.8.m6.2"><semantics id="S2.p2.8.m6.2a"><mrow id="S2.p2.8.m6.2.2" xref="S2.p2.8.m6.2.2.cmml"><mi id="S2.p2.8.m6.2.2.3" xref="S2.p2.8.m6.2.2.3.cmml">α</mi><mo id="S2.p2.8.m6.2.2.2" xref="S2.p2.8.m6.2.2.2.cmml">⁢</mo><mrow id="S2.p2.8.m6.2.2.1.1" xref="S2.p2.8.m6.2.2.1.1.1.cmml"><mo id="S2.p2.8.m6.2.2.1.1.2" stretchy="false" xref="S2.p2.8.m6.2.2.1.1.1.cmml">(</mo><msubsup id="S2.p2.8.m6.2.2.1.1.1" xref="S2.p2.8.m6.2.2.1.1.1.cmml"><mi id="S2.p2.8.m6.2.2.1.1.1.2.2" xref="S2.p2.8.m6.2.2.1.1.1.2.2.cmml">τ</mi><mi id="S2.p2.8.m6.2.2.1.1.1.2.3" xref="S2.p2.8.m6.2.2.1.1.1.2.3.cmml">n</mi><mrow id="S2.p2.8.m6.1.1.1.3" xref="S2.p2.8.m6.2.2.1.1.1.cmml"><mo id="S2.p2.8.m6.1.1.1.3.1" stretchy="false" xref="S2.p2.8.m6.2.2.1.1.1.cmml">(</mo><mi id="S2.p2.8.m6.1.1.1.1" xref="S2.p2.8.m6.1.1.1.1.cmml">k</mi><mo id="S2.p2.8.m6.1.1.1.3.2" stretchy="false" xref="S2.p2.8.m6.2.2.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S2.p2.8.m6.2.2.1.1.3" stretchy="false" xref="S2.p2.8.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.8.m6.2b"><apply id="S2.p2.8.m6.2.2.cmml" xref="S2.p2.8.m6.2.2"><times id="S2.p2.8.m6.2.2.2.cmml" xref="S2.p2.8.m6.2.2.2"></times><ci id="S2.p2.8.m6.2.2.3.cmml" xref="S2.p2.8.m6.2.2.3">𝛼</ci><apply id="S2.p2.8.m6.2.2.1.1.1.cmml" xref="S2.p2.8.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.p2.8.m6.2.2.1.1.1.1.cmml" xref="S2.p2.8.m6.2.2.1.1">superscript</csymbol><apply id="S2.p2.8.m6.2.2.1.1.1.2.cmml" xref="S2.p2.8.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.p2.8.m6.2.2.1.1.1.2.1.cmml" xref="S2.p2.8.m6.2.2.1.1">subscript</csymbol><ci id="S2.p2.8.m6.2.2.1.1.1.2.2.cmml" xref="S2.p2.8.m6.2.2.1.1.1.2.2">𝜏</ci><ci id="S2.p2.8.m6.2.2.1.1.1.2.3.cmml" xref="S2.p2.8.m6.2.2.1.1.1.2.3">𝑛</ci></apply><ci id="S2.p2.8.m6.1.1.1.1.cmml" xref="S2.p2.8.m6.1.1.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.8.m6.2c">\alpha(\tau_{n}^{(k)})</annotation><annotation encoding="application/x-llamapun" id="S2.p2.8.m6.2d">italic_α ( italic_τ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math> is calculated using the radar range equation.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.3">The received signal is sampled and matched filtered which provides <math alttext="N_{T}N_{R}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><msub id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml"><mi id="S2.p3.1.m1.1.1.2.2" xref="S2.p3.1.m1.1.1.2.2.cmml">N</mi><mi id="S2.p3.1.m1.1.1.2.3" xref="S2.p3.1.m1.1.1.2.3.cmml">T</mi></msub><mo id="S2.p3.1.m1.1.1.1" xref="S2.p3.1.m1.1.1.1.cmml">⁢</mo><msub id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml"><mi id="S2.p3.1.m1.1.1.3.2" xref="S2.p3.1.m1.1.1.3.2.cmml">N</mi><mi id="S2.p3.1.m1.1.1.3.3" xref="S2.p3.1.m1.1.1.3.3.cmml">R</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><times id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1"></times><apply id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.2.1.cmml" xref="S2.p3.1.m1.1.1.2">subscript</csymbol><ci id="S2.p3.1.m1.1.1.2.2.cmml" xref="S2.p3.1.m1.1.1.2.2">𝑁</ci><ci id="S2.p3.1.m1.1.1.2.3.cmml" xref="S2.p3.1.m1.1.1.2.3">𝑇</ci></apply><apply id="S2.p3.1.m1.1.1.3.cmml" xref="S2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.3.1.cmml" xref="S2.p3.1.m1.1.1.3">subscript</csymbol><ci id="S2.p3.1.m1.1.1.3.2.cmml" xref="S2.p3.1.m1.1.1.3.2">𝑁</ci><ci id="S2.p3.1.m1.1.1.3.3.cmml" xref="S2.p3.1.m1.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">N_{T}N_{R}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> complex signal vectors of length <math alttext="N_{s}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><msub id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">N</mi><mi id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1">subscript</csymbol><ci id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">𝑁</ci><ci id="S2.p3.2.m2.1.1.3.cmml" xref="S2.p3.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">N_{s}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_N start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, which is the number of samples. In the frequency domain the signal at radar <math alttext="k" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><mi id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><ci id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_k</annotation></semantics></math> reads</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="\bm{Z}_{n}^{(k)}=\overbrace{\tilde{\bm{S}}_{n}\otimes\bm{U}^{*}}^{\bm{S}_{n}(% \phi_{n})}+\tilde{\bm{W}}_{n}^{(k)}\in\mathbb{C}^{N_{R}N_{T}\times N_{s}}," class="ltx_Math" display="block" id="S2.E2.m1.4"><semantics id="S2.E2.m1.4a"><mrow id="S2.E2.m1.4.4.1" xref="S2.E2.m1.4.4.1.1.cmml"><mrow id="S2.E2.m1.4.4.1.1" xref="S2.E2.m1.4.4.1.1.cmml"><msubsup id="S2.E2.m1.4.4.1.1.2" xref="S2.E2.m1.4.4.1.1.2.cmml"><mi id="S2.E2.m1.4.4.1.1.2.2.2" xref="S2.E2.m1.4.4.1.1.2.2.2.cmml">𝒁</mi><mi id="S2.E2.m1.4.4.1.1.2.2.3" xref="S2.E2.m1.4.4.1.1.2.2.3.cmml">n</mi><mrow id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.4.4.1.1.2.cmml"><mo id="S2.E2.m1.1.1.1.3.1" stretchy="false" xref="S2.E2.m1.4.4.1.1.2.cmml">(</mo><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">k</mi><mo id="S2.E2.m1.1.1.1.3.2" stretchy="false" xref="S2.E2.m1.4.4.1.1.2.cmml">)</mo></mrow></msubsup><mo id="S2.E2.m1.4.4.1.1.3" xref="S2.E2.m1.4.4.1.1.3.cmml">=</mo><mrow id="S2.E2.m1.4.4.1.1.4" xref="S2.E2.m1.4.4.1.1.4.cmml"><mover id="S2.E2.m1.4.4.1.1.4.2" xref="S2.E2.m1.4.4.1.1.4.2.cmml"><mover accent="true" id="S2.E2.m1.4.4.1.1.4.2.2" xref="S2.E2.m1.4.4.1.1.4.2.2.cmml"><mrow id="S2.E2.m1.4.4.1.1.4.2.2.2" xref="S2.E2.m1.4.4.1.1.4.2.2.2.cmml"><msub id="S2.E2.m1.4.4.1.1.4.2.2.2.2" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.cmml"><mover accent="true" id="S2.E2.m1.4.4.1.1.4.2.2.2.2.2" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.cmml"><mi id="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.2" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.2.cmml">𝑺</mi><mo id="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.1" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.1.cmml">~</mo></mover><mi id="S2.E2.m1.4.4.1.1.4.2.2.2.2.3" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.3.cmml">n</mi></msub><mo id="S2.E2.m1.4.4.1.1.4.2.2.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E2.m1.4.4.1.1.4.2.2.2.1.cmml">⊗</mo><msup id="S2.E2.m1.4.4.1.1.4.2.2.2.3" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3.cmml"><mi id="S2.E2.m1.4.4.1.1.4.2.2.2.3.2" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3.2.cmml">𝑼</mi><mo id="S2.E2.m1.4.4.1.1.4.2.2.2.3.3" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3.3.cmml">∗</mo></msup></mrow><mo id="S2.E2.m1.4.4.1.1.4.2.2.1" xref="S2.E2.m1.4.4.1.1.4.2.2.1.cmml">⏞</mo></mover><mrow id="S2.E2.m1.2.2.1" xref="S2.E2.m1.2.2.1.cmml"><msub id="S2.E2.m1.2.2.1.3" xref="S2.E2.m1.2.2.1.3.cmml"><mi id="S2.E2.m1.2.2.1.3.2" xref="S2.E2.m1.2.2.1.3.2.cmml">𝑺</mi><mi id="S2.E2.m1.2.2.1.3.3" xref="S2.E2.m1.2.2.1.3.3.cmml">n</mi></msub><mo id="S2.E2.m1.2.2.1.2" xref="S2.E2.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.2.2.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.cmml"><mo id="S2.E2.m1.2.2.1.1.1.2" stretchy="false" xref="S2.E2.m1.2.2.1.1.1.1.cmml">(</mo><msub id="S2.E2.m1.2.2.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.cmml"><mi id="S2.E2.m1.2.2.1.1.1.1.2" xref="S2.E2.m1.2.2.1.1.1.1.2.cmml">ϕ</mi><mi id="S2.E2.m1.2.2.1.1.1.1.3" xref="S2.E2.m1.2.2.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.E2.m1.2.2.1.1.1.3" stretchy="false" xref="S2.E2.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mover><mo id="S2.E2.m1.4.4.1.1.4.1" xref="S2.E2.m1.4.4.1.1.4.1.cmml">+</mo><msubsup id="S2.E2.m1.4.4.1.1.4.3" xref="S2.E2.m1.4.4.1.1.4.3.cmml"><mover accent="true" id="S2.E2.m1.4.4.1.1.4.3.2.2" xref="S2.E2.m1.4.4.1.1.4.3.2.2.cmml"><mi id="S2.E2.m1.4.4.1.1.4.3.2.2.2" xref="S2.E2.m1.4.4.1.1.4.3.2.2.2.cmml">𝑾</mi><mo id="S2.E2.m1.4.4.1.1.4.3.2.2.1" xref="S2.E2.m1.4.4.1.1.4.3.2.2.1.cmml">~</mo></mover><mi id="S2.E2.m1.4.4.1.1.4.3.2.3" xref="S2.E2.m1.4.4.1.1.4.3.2.3.cmml">n</mi><mrow id="S2.E2.m1.3.3.1.3" xref="S2.E2.m1.4.4.1.1.4.3.cmml"><mo id="S2.E2.m1.3.3.1.3.1" stretchy="false" xref="S2.E2.m1.4.4.1.1.4.3.cmml">(</mo><mi id="S2.E2.m1.3.3.1.1" xref="S2.E2.m1.3.3.1.1.cmml">k</mi><mo id="S2.E2.m1.3.3.1.3.2" stretchy="false" xref="S2.E2.m1.4.4.1.1.4.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S2.E2.m1.4.4.1.1.5" xref="S2.E2.m1.4.4.1.1.5.cmml">∈</mo><msup id="S2.E2.m1.4.4.1.1.6" xref="S2.E2.m1.4.4.1.1.6.cmml"><mi id="S2.E2.m1.4.4.1.1.6.2" xref="S2.E2.m1.4.4.1.1.6.2.cmml">ℂ</mi><mrow id="S2.E2.m1.4.4.1.1.6.3" xref="S2.E2.m1.4.4.1.1.6.3.cmml"><mrow id="S2.E2.m1.4.4.1.1.6.3.2" xref="S2.E2.m1.4.4.1.1.6.3.2.cmml"><msub id="S2.E2.m1.4.4.1.1.6.3.2.2" xref="S2.E2.m1.4.4.1.1.6.3.2.2.cmml"><mi id="S2.E2.m1.4.4.1.1.6.3.2.2.2" xref="S2.E2.m1.4.4.1.1.6.3.2.2.2.cmml">N</mi><mi id="S2.E2.m1.4.4.1.1.6.3.2.2.3" xref="S2.E2.m1.4.4.1.1.6.3.2.2.3.cmml">R</mi></msub><mo id="S2.E2.m1.4.4.1.1.6.3.2.1" xref="S2.E2.m1.4.4.1.1.6.3.2.1.cmml">⁢</mo><msub id="S2.E2.m1.4.4.1.1.6.3.2.3" xref="S2.E2.m1.4.4.1.1.6.3.2.3.cmml"><mi id="S2.E2.m1.4.4.1.1.6.3.2.3.2" xref="S2.E2.m1.4.4.1.1.6.3.2.3.2.cmml">N</mi><mi id="S2.E2.m1.4.4.1.1.6.3.2.3.3" xref="S2.E2.m1.4.4.1.1.6.3.2.3.3.cmml">T</mi></msub></mrow><mo id="S2.E2.m1.4.4.1.1.6.3.1" lspace="0.222em" rspace="0.222em" xref="S2.E2.m1.4.4.1.1.6.3.1.cmml">×</mo><msub id="S2.E2.m1.4.4.1.1.6.3.3" xref="S2.E2.m1.4.4.1.1.6.3.3.cmml"><mi id="S2.E2.m1.4.4.1.1.6.3.3.2" xref="S2.E2.m1.4.4.1.1.6.3.3.2.cmml">N</mi><mi id="S2.E2.m1.4.4.1.1.6.3.3.3" xref="S2.E2.m1.4.4.1.1.6.3.3.3.cmml">s</mi></msub></mrow></msup></mrow><mo id="S2.E2.m1.4.4.1.2" xref="S2.E2.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.4b"><apply id="S2.E2.m1.4.4.1.1.cmml" xref="S2.E2.m1.4.4.1"><and id="S2.E2.m1.4.4.1.1a.cmml" xref="S2.E2.m1.4.4.1"></and><apply id="S2.E2.m1.4.4.1.1b.cmml" xref="S2.E2.m1.4.4.1"><eq id="S2.E2.m1.4.4.1.1.3.cmml" xref="S2.E2.m1.4.4.1.1.3"></eq><apply id="S2.E2.m1.4.4.1.1.2.cmml" xref="S2.E2.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.2.1.cmml" xref="S2.E2.m1.4.4.1.1.2">superscript</csymbol><apply id="S2.E2.m1.4.4.1.1.2.2.cmml" xref="S2.E2.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.2">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.2.2.2">𝒁</ci><ci id="S2.E2.m1.4.4.1.1.2.2.3.cmml" xref="S2.E2.m1.4.4.1.1.2.2.3">𝑛</ci></apply><ci id="S2.E2.m1.1.1.1.1.cmml" xref="S2.E2.m1.1.1.1.1">𝑘</ci></apply><apply id="S2.E2.m1.4.4.1.1.4.cmml" xref="S2.E2.m1.4.4.1.1.4"><plus id="S2.E2.m1.4.4.1.1.4.1.cmml" xref="S2.E2.m1.4.4.1.1.4.1"></plus><apply id="S2.E2.m1.4.4.1.1.4.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.4.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2">superscript</csymbol><apply id="S2.E2.m1.4.4.1.1.4.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2"><ci id="S2.E2.m1.4.4.1.1.4.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.1">⏞</ci><apply id="S2.E2.m1.4.4.1.1.4.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2"><csymbol cd="latexml" id="S2.E2.m1.4.4.1.1.4.2.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.1">tensor-product</csymbol><apply id="S2.E2.m1.4.4.1.1.4.2.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.4.2.2.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2">subscript</csymbol><apply id="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.2"><ci id="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.1">~</ci><ci id="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.2.2">𝑺</ci></apply><ci id="S2.E2.m1.4.4.1.1.4.2.2.2.2.3.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.2.3">𝑛</ci></apply><apply id="S2.E2.m1.4.4.1.1.4.2.2.2.3.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.4.2.2.2.3.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3">superscript</csymbol><ci id="S2.E2.m1.4.4.1.1.4.2.2.2.3.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3.2">𝑼</ci><times id="S2.E2.m1.4.4.1.1.4.2.2.2.3.3.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2.2.3.3"></times></apply></apply></apply><apply id="S2.E2.m1.2.2.1.cmml" xref="S2.E2.m1.2.2.1"><times id="S2.E2.m1.2.2.1.2.cmml" xref="S2.E2.m1.2.2.1.2"></times><apply id="S2.E2.m1.2.2.1.3.cmml" xref="S2.E2.m1.2.2.1.3"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.1.3.1.cmml" xref="S2.E2.m1.2.2.1.3">subscript</csymbol><ci id="S2.E2.m1.2.2.1.3.2.cmml" xref="S2.E2.m1.2.2.1.3.2">𝑺</ci><ci id="S2.E2.m1.2.2.1.3.3.cmml" xref="S2.E2.m1.2.2.1.3.3">𝑛</ci></apply><apply id="S2.E2.m1.2.2.1.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.1.1.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.E2.m1.2.2.1.1.1.1.2.cmml" xref="S2.E2.m1.2.2.1.1.1.1.2">italic-ϕ</ci><ci id="S2.E2.m1.2.2.1.1.1.1.3.cmml" xref="S2.E2.m1.2.2.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="S2.E2.m1.4.4.1.1.4.3.cmml" xref="S2.E2.m1.4.4.1.1.4.3"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.4.3.1.cmml" xref="S2.E2.m1.4.4.1.1.4.3">superscript</csymbol><apply id="S2.E2.m1.4.4.1.1.4.3.2.cmml" xref="S2.E2.m1.4.4.1.1.4.3"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.4.3.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.3">subscript</csymbol><apply id="S2.E2.m1.4.4.1.1.4.3.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.3.2.2"><ci id="S2.E2.m1.4.4.1.1.4.3.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.3.2.2.1">~</ci><ci id="S2.E2.m1.4.4.1.1.4.3.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.3.2.2.2">𝑾</ci></apply><ci id="S2.E2.m1.4.4.1.1.4.3.2.3.cmml" xref="S2.E2.m1.4.4.1.1.4.3.2.3">𝑛</ci></apply><ci id="S2.E2.m1.3.3.1.1.cmml" xref="S2.E2.m1.3.3.1.1">𝑘</ci></apply></apply></apply><apply id="S2.E2.m1.4.4.1.1c.cmml" xref="S2.E2.m1.4.4.1"><in id="S2.E2.m1.4.4.1.1.5.cmml" xref="S2.E2.m1.4.4.1.1.5"></in><share href="https://arxiv.org/html/2503.16236v1#S2.E2.m1.4.4.1.1.4.cmml" id="S2.E2.m1.4.4.1.1d.cmml" xref="S2.E2.m1.4.4.1"></share><apply id="S2.E2.m1.4.4.1.1.6.cmml" xref="S2.E2.m1.4.4.1.1.6"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.6.1.cmml" xref="S2.E2.m1.4.4.1.1.6">superscript</csymbol><ci id="S2.E2.m1.4.4.1.1.6.2.cmml" xref="S2.E2.m1.4.4.1.1.6.2">ℂ</ci><apply id="S2.E2.m1.4.4.1.1.6.3.cmml" xref="S2.E2.m1.4.4.1.1.6.3"><times id="S2.E2.m1.4.4.1.1.6.3.1.cmml" xref="S2.E2.m1.4.4.1.1.6.3.1"></times><apply id="S2.E2.m1.4.4.1.1.6.3.2.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2"><times id="S2.E2.m1.4.4.1.1.6.3.2.1.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.1"></times><apply id="S2.E2.m1.4.4.1.1.6.3.2.2.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.6.3.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.2">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.6.3.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.2.2">𝑁</ci><ci id="S2.E2.m1.4.4.1.1.6.3.2.2.3.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.2.3">𝑅</ci></apply><apply id="S2.E2.m1.4.4.1.1.6.3.2.3.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.3"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.6.3.2.3.1.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.3">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.6.3.2.3.2.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.3.2">𝑁</ci><ci id="S2.E2.m1.4.4.1.1.6.3.2.3.3.cmml" xref="S2.E2.m1.4.4.1.1.6.3.2.3.3">𝑇</ci></apply></apply><apply id="S2.E2.m1.4.4.1.1.6.3.3.cmml" xref="S2.E2.m1.4.4.1.1.6.3.3"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.6.3.3.1.cmml" xref="S2.E2.m1.4.4.1.1.6.3.3">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.6.3.3.2.cmml" xref="S2.E2.m1.4.4.1.1.6.3.3.2">𝑁</ci><ci id="S2.E2.m1.4.4.1.1.6.3.3.3.cmml" xref="S2.E2.m1.4.4.1.1.6.3.3.3">𝑠</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.4c">\bm{Z}_{n}^{(k)}=\overbrace{\tilde{\bm{S}}_{n}\otimes\bm{U}^{*}}^{\bm{S}_{n}(% \phi_{n})}+\tilde{\bm{W}}_{n}^{(k)}\in\mathbb{C}^{N_{R}N_{T}\times N_{s}},</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.4d">bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = over⏞ start_ARG over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⊗ bold_italic_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG start_POSTSUPERSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + over~ start_ARG bold_italic_W end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ blackboard_C start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT × italic_N start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT 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.p3.9">with <math alttext="\tilde{\bm{S}}_{n}\in\mathbb{C}^{N_{R}\times N_{s}}" class="ltx_Math" display="inline" id="S2.p3.4.m1.1"><semantics id="S2.p3.4.m1.1a"><mrow id="S2.p3.4.m1.1.1" xref="S2.p3.4.m1.1.1.cmml"><msub id="S2.p3.4.m1.1.1.2" xref="S2.p3.4.m1.1.1.2.cmml"><mover accent="true" id="S2.p3.4.m1.1.1.2.2" xref="S2.p3.4.m1.1.1.2.2.cmml"><mi id="S2.p3.4.m1.1.1.2.2.2" xref="S2.p3.4.m1.1.1.2.2.2.cmml">𝑺</mi><mo id="S2.p3.4.m1.1.1.2.2.1" xref="S2.p3.4.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="S2.p3.4.m1.1.1.2.3" xref="S2.p3.4.m1.1.1.2.3.cmml">n</mi></msub><mo id="S2.p3.4.m1.1.1.1" xref="S2.p3.4.m1.1.1.1.cmml">∈</mo><msup id="S2.p3.4.m1.1.1.3" xref="S2.p3.4.m1.1.1.3.cmml"><mi id="S2.p3.4.m1.1.1.3.2" xref="S2.p3.4.m1.1.1.3.2.cmml">ℂ</mi><mrow id="S2.p3.4.m1.1.1.3.3" xref="S2.p3.4.m1.1.1.3.3.cmml"><msub id="S2.p3.4.m1.1.1.3.3.2" xref="S2.p3.4.m1.1.1.3.3.2.cmml"><mi id="S2.p3.4.m1.1.1.3.3.2.2" xref="S2.p3.4.m1.1.1.3.3.2.2.cmml">N</mi><mi id="S2.p3.4.m1.1.1.3.3.2.3" xref="S2.p3.4.m1.1.1.3.3.2.3.cmml">R</mi></msub><mo id="S2.p3.4.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p3.4.m1.1.1.3.3.1.cmml">×</mo><msub id="S2.p3.4.m1.1.1.3.3.3" xref="S2.p3.4.m1.1.1.3.3.3.cmml"><mi id="S2.p3.4.m1.1.1.3.3.3.2" xref="S2.p3.4.m1.1.1.3.3.3.2.cmml">N</mi><mi id="S2.p3.4.m1.1.1.3.3.3.3" xref="S2.p3.4.m1.1.1.3.3.3.3.cmml">s</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m1.1b"><apply id="S2.p3.4.m1.1.1.cmml" xref="S2.p3.4.m1.1.1"><in id="S2.p3.4.m1.1.1.1.cmml" xref="S2.p3.4.m1.1.1.1"></in><apply id="S2.p3.4.m1.1.1.2.cmml" xref="S2.p3.4.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.4.m1.1.1.2.1.cmml" xref="S2.p3.4.m1.1.1.2">subscript</csymbol><apply id="S2.p3.4.m1.1.1.2.2.cmml" xref="S2.p3.4.m1.1.1.2.2"><ci id="S2.p3.4.m1.1.1.2.2.1.cmml" xref="S2.p3.4.m1.1.1.2.2.1">~</ci><ci id="S2.p3.4.m1.1.1.2.2.2.cmml" xref="S2.p3.4.m1.1.1.2.2.2">𝑺</ci></apply><ci id="S2.p3.4.m1.1.1.2.3.cmml" xref="S2.p3.4.m1.1.1.2.3">𝑛</ci></apply><apply id="S2.p3.4.m1.1.1.3.cmml" xref="S2.p3.4.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p3.4.m1.1.1.3.1.cmml" xref="S2.p3.4.m1.1.1.3">superscript</csymbol><ci id="S2.p3.4.m1.1.1.3.2.cmml" xref="S2.p3.4.m1.1.1.3.2">ℂ</ci><apply id="S2.p3.4.m1.1.1.3.3.cmml" xref="S2.p3.4.m1.1.1.3.3"><times id="S2.p3.4.m1.1.1.3.3.1.cmml" xref="S2.p3.4.m1.1.1.3.3.1"></times><apply id="S2.p3.4.m1.1.1.3.3.2.cmml" xref="S2.p3.4.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.p3.4.m1.1.1.3.3.2.1.cmml" xref="S2.p3.4.m1.1.1.3.3.2">subscript</csymbol><ci id="S2.p3.4.m1.1.1.3.3.2.2.cmml" xref="S2.p3.4.m1.1.1.3.3.2.2">𝑁</ci><ci id="S2.p3.4.m1.1.1.3.3.2.3.cmml" xref="S2.p3.4.m1.1.1.3.3.2.3">𝑅</ci></apply><apply id="S2.p3.4.m1.1.1.3.3.3.cmml" xref="S2.p3.4.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.p3.4.m1.1.1.3.3.3.1.cmml" xref="S2.p3.4.m1.1.1.3.3.3">subscript</csymbol><ci id="S2.p3.4.m1.1.1.3.3.3.2.cmml" xref="S2.p3.4.m1.1.1.3.3.3.2">𝑁</ci><ci id="S2.p3.4.m1.1.1.3.3.3.3.cmml" xref="S2.p3.4.m1.1.1.3.3.3.3">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m1.1c">\tilde{\bm{S}}_{n}\in\mathbb{C}^{N_{R}\times N_{s}}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m1.1d">over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ blackboard_C start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT × italic_N start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\bm{U}\in\mathbb{C}^{N_{T}\times N_{s}}" class="ltx_Math" display="inline" id="S2.p3.5.m2.1"><semantics id="S2.p3.5.m2.1a"><mrow id="S2.p3.5.m2.1.1" xref="S2.p3.5.m2.1.1.cmml"><mi id="S2.p3.5.m2.1.1.2" xref="S2.p3.5.m2.1.1.2.cmml">𝑼</mi><mo id="S2.p3.5.m2.1.1.1" xref="S2.p3.5.m2.1.1.1.cmml">∈</mo><msup id="S2.p3.5.m2.1.1.3" xref="S2.p3.5.m2.1.1.3.cmml"><mi id="S2.p3.5.m2.1.1.3.2" xref="S2.p3.5.m2.1.1.3.2.cmml">ℂ</mi><mrow id="S2.p3.5.m2.1.1.3.3" xref="S2.p3.5.m2.1.1.3.3.cmml"><msub id="S2.p3.5.m2.1.1.3.3.2" xref="S2.p3.5.m2.1.1.3.3.2.cmml"><mi id="S2.p3.5.m2.1.1.3.3.2.2" xref="S2.p3.5.m2.1.1.3.3.2.2.cmml">N</mi><mi id="S2.p3.5.m2.1.1.3.3.2.3" xref="S2.p3.5.m2.1.1.3.3.2.3.cmml">T</mi></msub><mo id="S2.p3.5.m2.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p3.5.m2.1.1.3.3.1.cmml">×</mo><msub id="S2.p3.5.m2.1.1.3.3.3" xref="S2.p3.5.m2.1.1.3.3.3.cmml"><mi id="S2.p3.5.m2.1.1.3.3.3.2" xref="S2.p3.5.m2.1.1.3.3.3.2.cmml">N</mi><mi id="S2.p3.5.m2.1.1.3.3.3.3" xref="S2.p3.5.m2.1.1.3.3.3.3.cmml">s</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.5.m2.1b"><apply id="S2.p3.5.m2.1.1.cmml" xref="S2.p3.5.m2.1.1"><in id="S2.p3.5.m2.1.1.1.cmml" xref="S2.p3.5.m2.1.1.1"></in><ci id="S2.p3.5.m2.1.1.2.cmml" xref="S2.p3.5.m2.1.1.2">𝑼</ci><apply id="S2.p3.5.m2.1.1.3.cmml" xref="S2.p3.5.m2.1.1.3"><csymbol cd="ambiguous" id="S2.p3.5.m2.1.1.3.1.cmml" xref="S2.p3.5.m2.1.1.3">superscript</csymbol><ci id="S2.p3.5.m2.1.1.3.2.cmml" xref="S2.p3.5.m2.1.1.3.2">ℂ</ci><apply id="S2.p3.5.m2.1.1.3.3.cmml" xref="S2.p3.5.m2.1.1.3.3"><times id="S2.p3.5.m2.1.1.3.3.1.cmml" xref="S2.p3.5.m2.1.1.3.3.1"></times><apply id="S2.p3.5.m2.1.1.3.3.2.cmml" xref="S2.p3.5.m2.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.p3.5.m2.1.1.3.3.2.1.cmml" xref="S2.p3.5.m2.1.1.3.3.2">subscript</csymbol><ci id="S2.p3.5.m2.1.1.3.3.2.2.cmml" xref="S2.p3.5.m2.1.1.3.3.2.2">𝑁</ci><ci id="S2.p3.5.m2.1.1.3.3.2.3.cmml" xref="S2.p3.5.m2.1.1.3.3.2.3">𝑇</ci></apply><apply id="S2.p3.5.m2.1.1.3.3.3.cmml" xref="S2.p3.5.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.p3.5.m2.1.1.3.3.3.1.cmml" xref="S2.p3.5.m2.1.1.3.3.3">subscript</csymbol><ci id="S2.p3.5.m2.1.1.3.3.3.2.cmml" xref="S2.p3.5.m2.1.1.3.3.3.2">𝑁</ci><ci id="S2.p3.5.m2.1.1.3.3.3.3.cmml" xref="S2.p3.5.m2.1.1.3.3.3.3">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m2.1c">\bm{U}\in\mathbb{C}^{N_{T}\times N_{s}}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m2.1d">bold_italic_U ∈ blackboard_C start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT × italic_N start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> being the collection of samples into a matrix, <math alttext="\otimes" class="ltx_Math" display="inline" id="S2.p3.6.m3.1"><semantics id="S2.p3.6.m3.1a"><mo id="S2.p3.6.m3.1.1" xref="S2.p3.6.m3.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S2.p3.6.m3.1b"><csymbol cd="latexml" id="S2.p3.6.m3.1.1.cmml" xref="S2.p3.6.m3.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m3.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m3.1d">⊗</annotation></semantics></math> being the column-wise Kronecker product, and <math alttext="*" class="ltx_Math" display="inline" id="S2.p3.7.m4.1"><semantics id="S2.p3.7.m4.1a"><mo id="S2.p3.7.m4.1.1" xref="S2.p3.7.m4.1.1.cmml">∗</mo><annotation-xml encoding="MathML-Content" id="S2.p3.7.m4.1b"><times id="S2.p3.7.m4.1.1.cmml" xref="S2.p3.7.m4.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.7.m4.1c">*</annotation><annotation encoding="application/x-llamapun" id="S2.p3.7.m4.1d">∗</annotation></semantics></math> the complex conjugate. The post match filtered noise <math alttext="\tilde{\bm{W}}_{n}^{(k)}" class="ltx_Math" display="inline" id="S2.p3.8.m5.1"><semantics id="S2.p3.8.m5.1a"><msubsup id="S2.p3.8.m5.1.2" xref="S2.p3.8.m5.1.2.cmml"><mover accent="true" id="S2.p3.8.m5.1.2.2.2" xref="S2.p3.8.m5.1.2.2.2.cmml"><mi id="S2.p3.8.m5.1.2.2.2.2" xref="S2.p3.8.m5.1.2.2.2.2.cmml">𝑾</mi><mo id="S2.p3.8.m5.1.2.2.2.1" xref="S2.p3.8.m5.1.2.2.2.1.cmml">~</mo></mover><mi id="S2.p3.8.m5.1.2.2.3" xref="S2.p3.8.m5.1.2.2.3.cmml">n</mi><mrow id="S2.p3.8.m5.1.1.1.3" xref="S2.p3.8.m5.1.2.cmml"><mo id="S2.p3.8.m5.1.1.1.3.1" stretchy="false" xref="S2.p3.8.m5.1.2.cmml">(</mo><mi id="S2.p3.8.m5.1.1.1.1" xref="S2.p3.8.m5.1.1.1.1.cmml">k</mi><mo id="S2.p3.8.m5.1.1.1.3.2" stretchy="false" xref="S2.p3.8.m5.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.p3.8.m5.1b"><apply id="S2.p3.8.m5.1.2.cmml" xref="S2.p3.8.m5.1.2"><csymbol cd="ambiguous" id="S2.p3.8.m5.1.2.1.cmml" xref="S2.p3.8.m5.1.2">superscript</csymbol><apply id="S2.p3.8.m5.1.2.2.cmml" xref="S2.p3.8.m5.1.2"><csymbol cd="ambiguous" id="S2.p3.8.m5.1.2.2.1.cmml" xref="S2.p3.8.m5.1.2">subscript</csymbol><apply id="S2.p3.8.m5.1.2.2.2.cmml" xref="S2.p3.8.m5.1.2.2.2"><ci id="S2.p3.8.m5.1.2.2.2.1.cmml" xref="S2.p3.8.m5.1.2.2.2.1">~</ci><ci id="S2.p3.8.m5.1.2.2.2.2.cmml" xref="S2.p3.8.m5.1.2.2.2.2">𝑾</ci></apply><ci id="S2.p3.8.m5.1.2.2.3.cmml" xref="S2.p3.8.m5.1.2.2.3">𝑛</ci></apply><ci id="S2.p3.8.m5.1.1.1.1.cmml" xref="S2.p3.8.m5.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.8.m5.1c">\tilde{\bm{W}}_{n}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.8.m5.1d">over~ start_ARG bold_italic_W end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> is colored circular Gaussian noise with zero mean and precision matrix <math alttext="\bm{\Lambda}_{Z}" class="ltx_Math" display="inline" id="S2.p3.9.m6.1"><semantics id="S2.p3.9.m6.1a"><msub id="S2.p3.9.m6.1.1" xref="S2.p3.9.m6.1.1.cmml"><mi id="S2.p3.9.m6.1.1.2" xref="S2.p3.9.m6.1.1.2.cmml">𝚲</mi><mi id="S2.p3.9.m6.1.1.3" xref="S2.p3.9.m6.1.1.3.cmml">Z</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p3.9.m6.1b"><apply id="S2.p3.9.m6.1.1.cmml" xref="S2.p3.9.m6.1.1"><csymbol cd="ambiguous" id="S2.p3.9.m6.1.1.1.cmml" xref="S2.p3.9.m6.1.1">subscript</csymbol><ci id="S2.p3.9.m6.1.1.2.cmml" xref="S2.p3.9.m6.1.1.2">𝚲</ci><ci id="S2.p3.9.m6.1.1.3.cmml" xref="S2.p3.9.m6.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.9.m6.1c">\bm{\Lambda}_{Z}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.9.m6.1d">bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT</annotation></semantics></math> independent for each virtual element.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Bayesian Network</span> </h2> <div class="ltx_para ltx_noindent" id="S3.p1"> <p class="ltx_p" id="S3.p1.4">The problem at hand is to estimate the kinematic parameters of the target at time <math alttext="n" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_n</annotation></semantics></math>, denoted as <math alttext="\bm{\phi}_{n}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><msub id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p1.2.m2.1.1.2" mathvariant="bold-italic" xref="S3.p1.2.m2.1.1.2.cmml">ϕ</mi><mi id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">bold-italic-ϕ</ci><ci id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\bm{\phi}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, from the received signals at all radars <math alttext="\{\bm{Z}_{j}^{(\forall N_{\text{radar}})}\}_{j\leq n}" class="ltx_Math" display="inline" id="S3.p1.3.m3.2"><semantics id="S3.p1.3.m3.2a"><msub id="S3.p1.3.m3.2.2" xref="S3.p1.3.m3.2.2.cmml"><mrow id="S3.p1.3.m3.2.2.1.1" xref="S3.p1.3.m3.2.2.1.2.cmml"><mo id="S3.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S3.p1.3.m3.2.2.1.2.cmml">{</mo><msubsup id="S3.p1.3.m3.2.2.1.1.1" xref="S3.p1.3.m3.2.2.1.1.1.cmml"><mi id="S3.p1.3.m3.2.2.1.1.1.2.2" xref="S3.p1.3.m3.2.2.1.1.1.2.2.cmml">𝒁</mi><mi id="S3.p1.3.m3.2.2.1.1.1.2.3" xref="S3.p1.3.m3.2.2.1.1.1.2.3.cmml">j</mi><mrow id="S3.p1.3.m3.1.1.1.1" xref="S3.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.p1.3.m3.1.1.1.1.1" xref="S3.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.p1.3.m3.1.1.1.1.1.1" rspace="0.167em" xref="S3.p1.3.m3.1.1.1.1.1.1.cmml">∀</mo><msub id="S3.p1.3.m3.1.1.1.1.1.2" xref="S3.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.p1.3.m3.1.1.1.1.1.2.2" xref="S3.p1.3.m3.1.1.1.1.1.2.2.cmml">N</mi><mtext id="S3.p1.3.m3.1.1.1.1.1.2.3" xref="S3.p1.3.m3.1.1.1.1.1.2.3a.cmml">radar</mtext></msub></mrow><mo id="S3.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p1.3.m3.2.2.1.1.3" stretchy="false" xref="S3.p1.3.m3.2.2.1.2.cmml">}</mo></mrow><mrow id="S3.p1.3.m3.2.2.3" xref="S3.p1.3.m3.2.2.3.cmml"><mi id="S3.p1.3.m3.2.2.3.2" xref="S3.p1.3.m3.2.2.3.2.cmml">j</mi><mo id="S3.p1.3.m3.2.2.3.1" xref="S3.p1.3.m3.2.2.3.1.cmml">≤</mo><mi id="S3.p1.3.m3.2.2.3.3" xref="S3.p1.3.m3.2.2.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.2b"><apply id="S3.p1.3.m3.2.2.cmml" xref="S3.p1.3.m3.2.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.2.2.2.cmml" xref="S3.p1.3.m3.2.2">subscript</csymbol><set id="S3.p1.3.m3.2.2.1.2.cmml" xref="S3.p1.3.m3.2.2.1.1"><apply id="S3.p1.3.m3.2.2.1.1.1.cmml" xref="S3.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p1.3.m3.2.2.1.1.1.1.cmml" xref="S3.p1.3.m3.2.2.1.1.1">superscript</csymbol><apply id="S3.p1.3.m3.2.2.1.1.1.2.cmml" xref="S3.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p1.3.m3.2.2.1.1.1.2.1.cmml" xref="S3.p1.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S3.p1.3.m3.2.2.1.1.1.2.2.cmml" xref="S3.p1.3.m3.2.2.1.1.1.2.2">𝒁</ci><ci id="S3.p1.3.m3.2.2.1.1.1.2.3.cmml" xref="S3.p1.3.m3.2.2.1.1.1.2.3">𝑗</ci></apply><apply id="S3.p1.3.m3.1.1.1.1.1.cmml" xref="S3.p1.3.m3.1.1.1.1"><csymbol cd="latexml" id="S3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.p1.3.m3.1.1.1.1.1.1">for-all</csymbol><apply id="S3.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.p1.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S3.p1.3.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S3.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S3.p1.3.m3.1.1.1.1.1.2.2">𝑁</ci><ci id="S3.p1.3.m3.1.1.1.1.1.2.3a.cmml" xref="S3.p1.3.m3.1.1.1.1.1.2.3"><mtext id="S3.p1.3.m3.1.1.1.1.1.2.3.cmml" mathsize="50%" xref="S3.p1.3.m3.1.1.1.1.1.2.3">radar</mtext></ci></apply></apply></apply></set><apply id="S3.p1.3.m3.2.2.3.cmml" xref="S3.p1.3.m3.2.2.3"><leq id="S3.p1.3.m3.2.2.3.1.cmml" xref="S3.p1.3.m3.2.2.3.1"></leq><ci id="S3.p1.3.m3.2.2.3.2.cmml" xref="S3.p1.3.m3.2.2.3.2">𝑗</ci><ci id="S3.p1.3.m3.2.2.3.3.cmml" xref="S3.p1.3.m3.2.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.2c">\{\bm{Z}_{j}^{(\forall N_{\text{radar}})}\}_{j\leq n}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.2d">{ bold_italic_Z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( ∀ italic_N start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_j ≤ italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. The estimation algorithm is derived based on the Bayesian network in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S3.F2" title="Figure 2 ‣ III Bayesian Network ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">2</span></a>. The evolution of the track is governed by the underlying kinematics. Assuming linear motion during <math alttext="N_{Tx}T_{Tx}" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mrow id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><msub id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml"><mi id="S3.p1.4.m4.1.1.2.2" xref="S3.p1.4.m4.1.1.2.2.cmml">N</mi><mrow id="S3.p1.4.m4.1.1.2.3" xref="S3.p1.4.m4.1.1.2.3.cmml"><mi id="S3.p1.4.m4.1.1.2.3.2" xref="S3.p1.4.m4.1.1.2.3.2.cmml">T</mi><mo id="S3.p1.4.m4.1.1.2.3.1" xref="S3.p1.4.m4.1.1.2.3.1.cmml">⁢</mo><mi id="S3.p1.4.m4.1.1.2.3.3" xref="S3.p1.4.m4.1.1.2.3.3.cmml">x</mi></mrow></msub><mo id="S3.p1.4.m4.1.1.1" xref="S3.p1.4.m4.1.1.1.cmml">⁢</mo><msub id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml"><mi id="S3.p1.4.m4.1.1.3.2" xref="S3.p1.4.m4.1.1.3.2.cmml">T</mi><mrow id="S3.p1.4.m4.1.1.3.3" xref="S3.p1.4.m4.1.1.3.3.cmml"><mi id="S3.p1.4.m4.1.1.3.3.2" xref="S3.p1.4.m4.1.1.3.3.2.cmml">T</mi><mo id="S3.p1.4.m4.1.1.3.3.1" xref="S3.p1.4.m4.1.1.3.3.1.cmml">⁢</mo><mi id="S3.p1.4.m4.1.1.3.3.3" xref="S3.p1.4.m4.1.1.3.3.3.cmml">x</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1"><times id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1"></times><apply id="S3.p1.4.m4.1.1.2.cmml" xref="S3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.2.1.cmml" xref="S3.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.p1.4.m4.1.1.2.2.cmml" xref="S3.p1.4.m4.1.1.2.2">𝑁</ci><apply id="S3.p1.4.m4.1.1.2.3.cmml" xref="S3.p1.4.m4.1.1.2.3"><times id="S3.p1.4.m4.1.1.2.3.1.cmml" xref="S3.p1.4.m4.1.1.2.3.1"></times><ci id="S3.p1.4.m4.1.1.2.3.2.cmml" xref="S3.p1.4.m4.1.1.2.3.2">𝑇</ci><ci id="S3.p1.4.m4.1.1.2.3.3.cmml" xref="S3.p1.4.m4.1.1.2.3.3">𝑥</ci></apply></apply><apply id="S3.p1.4.m4.1.1.3.cmml" xref="S3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.3.1.cmml" xref="S3.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.p1.4.m4.1.1.3.2.cmml" xref="S3.p1.4.m4.1.1.3.2">𝑇</ci><apply id="S3.p1.4.m4.1.1.3.3.cmml" xref="S3.p1.4.m4.1.1.3.3"><times id="S3.p1.4.m4.1.1.3.3.1.cmml" xref="S3.p1.4.m4.1.1.3.3.1"></times><ci id="S3.p1.4.m4.1.1.3.3.2.cmml" xref="S3.p1.4.m4.1.1.3.3.2">𝑇</ci><ci id="S3.p1.4.m4.1.1.3.3.3.cmml" xref="S3.p1.4.m4.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">N_{Tx}T_{Tx}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_N start_POSTSUBSCRIPT italic_T italic_x end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_T italic_x end_POSTSUBSCRIPT</annotation></semantics></math> we use a Markov model,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx1"> <tbody id="S3.E3"><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{\phi}_{n}=\bm{T}\bm{\phi}_{n-1}+\bm{G}\bm{a},\quad\bm{a}|\bm{% \Lambda}_{a}\sim\mathcal{N}(\bm{a};\bm{0},\bm{\Lambda}_{a})." class="ltx_Math" display="inline" id="S3.E3.m1.3"><semantics id="S3.E3.m1.3a"><mrow id="S3.E3.m1.3.3.1"><mrow id="S3.E3.m1.3.3.1.1.2" xref="S3.E3.m1.3.3.1.1.3.cmml"><mrow id="S3.E3.m1.3.3.1.1.1.1" xref="S3.E3.m1.3.3.1.1.1.1.cmml"><msub id="S3.E3.m1.3.3.1.1.1.1.2" xref="S3.E3.m1.3.3.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.E3.m1.3.3.1.1.1.1.2.2" mathvariant="bold-italic" xref="S3.E3.m1.3.3.1.1.1.1.2.2.cmml">ϕ</mi><mi id="S3.E3.m1.3.3.1.1.1.1.2.3" xref="S3.E3.m1.3.3.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.E3.m1.3.3.1.1.1.1.1" xref="S3.E3.m1.3.3.1.1.1.1.1.cmml">=</mo><mrow id="S3.E3.m1.3.3.1.1.1.1.3" xref="S3.E3.m1.3.3.1.1.1.1.3.cmml"><mrow id="S3.E3.m1.3.3.1.1.1.1.3.2" xref="S3.E3.m1.3.3.1.1.1.1.3.2.cmml"><mi id="S3.E3.m1.3.3.1.1.1.1.3.2.2" xref="S3.E3.m1.3.3.1.1.1.1.3.2.2.cmml">𝑻</mi><mo id="S3.E3.m1.3.3.1.1.1.1.3.2.1" xref="S3.E3.m1.3.3.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S3.E3.m1.3.3.1.1.1.1.3.2.3" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.E3.m1.3.3.1.1.1.1.3.2.3.2" mathvariant="bold-italic" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.2.cmml">ϕ</mi><mrow id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.cmml"><mi id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.2" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.2.cmml">n</mi><mo id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.1" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.1.cmml">−</mo><mn id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.3" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S3.E3.m1.3.3.1.1.1.1.3.1" xref="S3.E3.m1.3.3.1.1.1.1.3.1.cmml">+</mo><mrow id="S3.E3.m1.3.3.1.1.1.1.3.3" xref="S3.E3.m1.3.3.1.1.1.1.3.3.cmml"><mi id="S3.E3.m1.3.3.1.1.1.1.3.3.2" xref="S3.E3.m1.3.3.1.1.1.1.3.3.2.cmml">𝑮</mi><mo id="S3.E3.m1.3.3.1.1.1.1.3.3.1" xref="S3.E3.m1.3.3.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.E3.m1.3.3.1.1.1.1.3.3.3" xref="S3.E3.m1.3.3.1.1.1.1.3.3.3.cmml">𝒂</mi></mrow></mrow></mrow><mo id="S3.E3.m1.3.3.1.1.2.3" rspace="1.167em" xref="S3.E3.m1.3.3.1.1.3a.cmml">,</mo><mrow id="S3.E3.m1.3.3.1.1.2.2" xref="S3.E3.m1.3.3.1.1.2.2.cmml"><mrow id="S3.E3.m1.3.3.1.1.2.2.3" xref="S3.E3.m1.3.3.1.1.2.2.3.cmml"><mi id="S3.E3.m1.3.3.1.1.2.2.3.2" xref="S3.E3.m1.3.3.1.1.2.2.3.2.cmml">𝒂</mi><mo fence="false" id="S3.E3.m1.3.3.1.1.2.2.3.1" xref="S3.E3.m1.3.3.1.1.2.2.3.1.cmml">|</mo><msub id="S3.E3.m1.3.3.1.1.2.2.3.3" xref="S3.E3.m1.3.3.1.1.2.2.3.3.cmml"><mi id="S3.E3.m1.3.3.1.1.2.2.3.3.2" xref="S3.E3.m1.3.3.1.1.2.2.3.3.2.cmml">𝚲</mi><mi id="S3.E3.m1.3.3.1.1.2.2.3.3.3" xref="S3.E3.m1.3.3.1.1.2.2.3.3.3.cmml">a</mi></msub></mrow><mo id="S3.E3.m1.3.3.1.1.2.2.2" xref="S3.E3.m1.3.3.1.1.2.2.2.cmml">∼</mo><mrow id="S3.E3.m1.3.3.1.1.2.2.1" xref="S3.E3.m1.3.3.1.1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.3.3.1.1.2.2.1.3" xref="S3.E3.m1.3.3.1.1.2.2.1.3.cmml">𝒩</mi><mo id="S3.E3.m1.3.3.1.1.2.2.1.2" xref="S3.E3.m1.3.3.1.1.2.2.1.2.cmml">⁢</mo><mrow id="S3.E3.m1.3.3.1.1.2.2.1.1.1" xref="S3.E3.m1.3.3.1.1.2.2.1.1.2.cmml"><mo id="S3.E3.m1.3.3.1.1.2.2.1.1.1.2" stretchy="false" xref="S3.E3.m1.3.3.1.1.2.2.1.1.2.cmml">(</mo><mi id="S3.E3.m1.1.1" xref="S3.E3.m1.1.1.cmml">𝒂</mi><mo id="S3.E3.m1.3.3.1.1.2.2.1.1.1.3" xref="S3.E3.m1.3.3.1.1.2.2.1.1.2.cmml">;</mo><mn id="S3.E3.m1.2.2" xref="S3.E3.m1.2.2.cmml">𝟎</mn><mo id="S3.E3.m1.3.3.1.1.2.2.1.1.1.4" xref="S3.E3.m1.3.3.1.1.2.2.1.1.2.cmml">,</mo><msub id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.cmml"><mi id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.2" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.2.cmml">𝚲</mi><mi id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.3" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.3.cmml">a</mi></msub><mo id="S3.E3.m1.3.3.1.1.2.2.1.1.1.5" stretchy="false" xref="S3.E3.m1.3.3.1.1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.E3.m1.3.3.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E3.m1.3b"><apply id="S3.E3.m1.3.3.1.1.3.cmml" xref="S3.E3.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.3a.cmml" xref="S3.E3.m1.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E3.m1.3.3.1.1.1.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1"><eq id="S3.E3.m1.3.3.1.1.1.1.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.1"></eq><apply id="S3.E3.m1.3.3.1.1.1.1.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.2">subscript</csymbol><ci id="S3.E3.m1.3.3.1.1.1.1.2.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.2.2">bold-italic-ϕ</ci><ci id="S3.E3.m1.3.3.1.1.1.1.2.3.cmml" xref="S3.E3.m1.3.3.1.1.1.1.2.3">𝑛</ci></apply><apply id="S3.E3.m1.3.3.1.1.1.1.3.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3"><plus id="S3.E3.m1.3.3.1.1.1.1.3.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.1"></plus><apply id="S3.E3.m1.3.3.1.1.1.1.3.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2"><times id="S3.E3.m1.3.3.1.1.1.1.3.2.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.1"></times><ci id="S3.E3.m1.3.3.1.1.1.1.3.2.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.2">𝑻</ci><apply id="S3.E3.m1.3.3.1.1.1.1.3.2.3.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.1.1.3.2.3.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3">subscript</csymbol><ci id="S3.E3.m1.3.3.1.1.1.1.3.2.3.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.2">bold-italic-ϕ</ci><apply id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3"><minus id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.1"></minus><ci id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.2">𝑛</ci><cn id="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.3.cmml" type="integer" xref="S3.E3.m1.3.3.1.1.1.1.3.2.3.3.3">1</cn></apply></apply></apply><apply id="S3.E3.m1.3.3.1.1.1.1.3.3.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.3"><times id="S3.E3.m1.3.3.1.1.1.1.3.3.1.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.3.1"></times><ci id="S3.E3.m1.3.3.1.1.1.1.3.3.2.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.3.2">𝑮</ci><ci id="S3.E3.m1.3.3.1.1.1.1.3.3.3.cmml" xref="S3.E3.m1.3.3.1.1.1.1.3.3.3">𝒂</ci></apply></apply></apply><apply id="S3.E3.m1.3.3.1.1.2.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2"><csymbol cd="latexml" id="S3.E3.m1.3.3.1.1.2.2.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.2">similar-to</csymbol><apply id="S3.E3.m1.3.3.1.1.2.2.3.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3"><csymbol cd="latexml" id="S3.E3.m1.3.3.1.1.2.2.3.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3.1">conditional</csymbol><ci id="S3.E3.m1.3.3.1.1.2.2.3.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3.2">𝒂</ci><apply id="S3.E3.m1.3.3.1.1.2.2.3.3.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.2.2.3.3.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3.3">subscript</csymbol><ci id="S3.E3.m1.3.3.1.1.2.2.3.3.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3.3.2">𝚲</ci><ci id="S3.E3.m1.3.3.1.1.2.2.3.3.3.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3.3.3">𝑎</ci></apply></apply><apply id="S3.E3.m1.3.3.1.1.2.2.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1"><times id="S3.E3.m1.3.3.1.1.2.2.1.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.2"></times><ci id="S3.E3.m1.3.3.1.1.2.2.1.3.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.3">𝒩</ci><list id="S3.E3.m1.3.3.1.1.2.2.1.1.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1"><ci id="S3.E3.m1.1.1.cmml" xref="S3.E3.m1.1.1">𝒂</ci><cn id="S3.E3.m1.2.2.cmml" type="integer" xref="S3.E3.m1.2.2">0</cn><apply id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1">subscript</csymbol><ci id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.2">𝚲</ci><ci id="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.3.cmml" xref="S3.E3.m1.3.3.1.1.2.2.1.1.1.1.3">𝑎</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.3c">\displaystyle\bm{\phi}_{n}=\bm{T}\bm{\phi}_{n-1}+\bm{G}\bm{a},\quad\bm{a}|\bm{% \Lambda}_{a}\sim\mathcal{N}(\bm{a};\bm{0},\bm{\Lambda}_{a}).</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.3d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_italic_T bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT + bold_italic_G bold_italic_a , bold_italic_a | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∼ caligraphic_N ( bold_italic_a ; bold_0 , bold_Λ start_POSTSUBSCRIPT italic_a 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="S3.p1.6">Here <math alttext="\bm{T}" class="ltx_Math" display="inline" id="S3.p1.5.m1.1"><semantics id="S3.p1.5.m1.1a"><mi id="S3.p1.5.m1.1.1" xref="S3.p1.5.m1.1.1.cmml">𝑻</mi><annotation-xml encoding="MathML-Content" id="S3.p1.5.m1.1b"><ci id="S3.p1.5.m1.1.1.cmml" xref="S3.p1.5.m1.1.1">𝑻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m1.1c">\bm{T}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m1.1d">bold_italic_T</annotation></semantics></math> denotes the kinematic matrix in Cartesian coordinates, while <math alttext="\bm{G}" class="ltx_Math" display="inline" id="S3.p1.6.m2.1"><semantics id="S3.p1.6.m2.1a"><mi id="S3.p1.6.m2.1.1" xref="S3.p1.6.m2.1.1.cmml">𝑮</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m2.1b"><ci id="S3.p1.6.m2.1.1.cmml" xref="S3.p1.6.m2.1.1">𝑮</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m2.1c">\bm{G}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m2.1d">bold_italic_G</annotation></semantics></math> denotes the process noise matrix, where</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S3.E4"><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{T}=\begin{bmatrix}1&0&\Delta t&0\\ 0&1&0&\Delta t\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\!\!,\ \bm{G}=\begin{bmatrix}\frac{\Delta t^{2}}{2}&0&0&0% \\ 0&\frac{\Delta t^{2}}{2}&0&0\\ 0&0&\Delta t&0\\ 0&0&0&\Delta t\end{bmatrix}\!\!." class="ltx_Math" display="inline" id="S3.E4.m1.3"><semantics id="S3.E4.m1.3a"><mrow id="S3.E4.m1.3.3.1"><mrow id="S3.E4.m1.3.3.1.1.2" xref="S3.E4.m1.3.3.1.1.3.cmml"><mrow id="S3.E4.m1.3.3.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.cmml"><mi id="S3.E4.m1.3.3.1.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.1.2.cmml">𝑻</mi><mo id="S3.E4.m1.3.3.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.1.cmml">=</mo><mrow id="S3.E4.m1.1.1a.3" xref="S3.E4.m1.1.1a.2.cmml"><mo id="S3.E4.m1.1.1a.3.1" xref="S3.E4.m1.1.1a.2.1.cmml">[</mo><mtable columnspacing="5pt" id="S3.E4.m1.1.1.1.1" rowspacing="0pt" xref="S3.E4.m1.1.1.1.1.cmml"><mtr id="S3.E4.m1.1.1.1.1a" xref="S3.E4.m1.1.1.1.1.cmml"><mtd id="S3.E4.m1.1.1.1.1b" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.1.1.1.cmml">1</mn></mtd><mtd id="S3.E4.m1.1.1.1.1c" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.1.2.1" xref="S3.E4.m1.1.1.1.1.1.2.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1d" xref="S3.E4.m1.1.1.1.1.cmml"><mrow id="S3.E4.m1.1.1.1.1.1.3.1" xref="S3.E4.m1.1.1.1.1.1.3.1.cmml"><mi id="S3.E4.m1.1.1.1.1.1.3.1.2" mathvariant="normal" xref="S3.E4.m1.1.1.1.1.1.3.1.2.cmml">Δ</mi><mo id="S3.E4.m1.1.1.1.1.1.3.1.1" xref="S3.E4.m1.1.1.1.1.1.3.1.1.cmml">⁢</mo><mi id="S3.E4.m1.1.1.1.1.1.3.1.3" xref="S3.E4.m1.1.1.1.1.1.3.1.3.cmml">t</mi></mrow></mtd><mtd id="S3.E4.m1.1.1.1.1e" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.1.4.1" xref="S3.E4.m1.1.1.1.1.1.4.1.cmml">0</mn></mtd></mtr><mtr id="S3.E4.m1.1.1.1.1f" xref="S3.E4.m1.1.1.1.1.cmml"><mtd id="S3.E4.m1.1.1.1.1g" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.2.1.1" xref="S3.E4.m1.1.1.1.1.2.1.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1h" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.2.2.1" xref="S3.E4.m1.1.1.1.1.2.2.1.cmml">1</mn></mtd><mtd id="S3.E4.m1.1.1.1.1i" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.2.3.1" xref="S3.E4.m1.1.1.1.1.2.3.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1j" xref="S3.E4.m1.1.1.1.1.cmml"><mrow id="S3.E4.m1.1.1.1.1.2.4.1" xref="S3.E4.m1.1.1.1.1.2.4.1.cmml"><mi id="S3.E4.m1.1.1.1.1.2.4.1.2" mathvariant="normal" xref="S3.E4.m1.1.1.1.1.2.4.1.2.cmml">Δ</mi><mo id="S3.E4.m1.1.1.1.1.2.4.1.1" xref="S3.E4.m1.1.1.1.1.2.4.1.1.cmml">⁢</mo><mi id="S3.E4.m1.1.1.1.1.2.4.1.3" xref="S3.E4.m1.1.1.1.1.2.4.1.3.cmml">t</mi></mrow></mtd></mtr><mtr id="S3.E4.m1.1.1.1.1k" xref="S3.E4.m1.1.1.1.1.cmml"><mtd id="S3.E4.m1.1.1.1.1l" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.3.1.1" xref="S3.E4.m1.1.1.1.1.3.1.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1m" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.3.2.1" xref="S3.E4.m1.1.1.1.1.3.2.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1n" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.3.3.1" xref="S3.E4.m1.1.1.1.1.3.3.1.cmml">1</mn></mtd><mtd id="S3.E4.m1.1.1.1.1o" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.3.4.1" xref="S3.E4.m1.1.1.1.1.3.4.1.cmml">0</mn></mtd></mtr><mtr id="S3.E4.m1.1.1.1.1p" xref="S3.E4.m1.1.1.1.1.cmml"><mtd id="S3.E4.m1.1.1.1.1q" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.4.1.1" xref="S3.E4.m1.1.1.1.1.4.1.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1r" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.4.2.1" xref="S3.E4.m1.1.1.1.1.4.2.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1s" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.4.3.1" xref="S3.E4.m1.1.1.1.1.4.3.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.1.1.1.1t" xref="S3.E4.m1.1.1.1.1.cmml"><mn id="S3.E4.m1.1.1.1.1.4.4.1" xref="S3.E4.m1.1.1.1.1.4.4.1.cmml">1</mn></mtd></mtr></mtable><mo id="S3.E4.m1.1.1a.3.2" xref="S3.E4.m1.1.1a.2.1.cmml">]</mo></mrow></mrow><mo id="S3.E4.m1.3.3.1.1.2.3" rspace="0.667em" xref="S3.E4.m1.3.3.1.1.3a.cmml">,</mo><mrow id="S3.E4.m1.3.3.1.1.2.2" xref="S3.E4.m1.3.3.1.1.2.2.cmml"><mi id="S3.E4.m1.3.3.1.1.2.2.2" xref="S3.E4.m1.3.3.1.1.2.2.2.cmml">𝑮</mi><mo id="S3.E4.m1.3.3.1.1.2.2.1" xref="S3.E4.m1.3.3.1.1.2.2.1.cmml">=</mo><mrow id="S3.E4.m1.2.2a.3" xref="S3.E4.m1.2.2a.2.cmml"><mo id="S3.E4.m1.2.2a.3.1" xref="S3.E4.m1.2.2a.2.1.cmml">[</mo><mtable columnspacing="5pt" id="S3.E4.m1.2.2.1.1" rowspacing="0pt" xref="S3.E4.m1.2.2.1.1.cmml"><mtr id="S3.E4.m1.2.2.1.1a" xref="S3.E4.m1.2.2.1.1.cmml"><mtd id="S3.E4.m1.2.2.1.1b" xref="S3.E4.m1.2.2.1.1.cmml"><mfrac id="S3.E4.m1.2.2.1.1.1.1.1" xref="S3.E4.m1.2.2.1.1.1.1.1.cmml"><mrow id="S3.E4.m1.2.2.1.1.1.1.1.2" xref="S3.E4.m1.2.2.1.1.1.1.1.2.cmml"><mi id="S3.E4.m1.2.2.1.1.1.1.1.2.2" mathvariant="normal" xref="S3.E4.m1.2.2.1.1.1.1.1.2.2.cmml">Δ</mi><mo id="S3.E4.m1.2.2.1.1.1.1.1.2.1" xref="S3.E4.m1.2.2.1.1.1.1.1.2.1.cmml">⁢</mo><msup id="S3.E4.m1.2.2.1.1.1.1.1.2.3" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3.cmml"><mi id="S3.E4.m1.2.2.1.1.1.1.1.2.3.2" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3.2.cmml">t</mi><mn id="S3.E4.m1.2.2.1.1.1.1.1.2.3.3" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3.3.cmml">2</mn></msup></mrow><mn id="S3.E4.m1.2.2.1.1.1.1.1.3" xref="S3.E4.m1.2.2.1.1.1.1.1.3.cmml">2</mn></mfrac></mtd><mtd id="S3.E4.m1.2.2.1.1c" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.1.2.1" xref="S3.E4.m1.2.2.1.1.1.2.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1d" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.1.3.1" xref="S3.E4.m1.2.2.1.1.1.3.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1e" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.1.4.1" xref="S3.E4.m1.2.2.1.1.1.4.1.cmml">0</mn></mtd></mtr><mtr id="S3.E4.m1.2.2.1.1f" xref="S3.E4.m1.2.2.1.1.cmml"><mtd id="S3.E4.m1.2.2.1.1g" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.2.1.1" xref="S3.E4.m1.2.2.1.1.2.1.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1h" xref="S3.E4.m1.2.2.1.1.cmml"><mfrac id="S3.E4.m1.2.2.1.1.2.2.1" xref="S3.E4.m1.2.2.1.1.2.2.1.cmml"><mrow id="S3.E4.m1.2.2.1.1.2.2.1.2" xref="S3.E4.m1.2.2.1.1.2.2.1.2.cmml"><mi id="S3.E4.m1.2.2.1.1.2.2.1.2.2" mathvariant="normal" xref="S3.E4.m1.2.2.1.1.2.2.1.2.2.cmml">Δ</mi><mo id="S3.E4.m1.2.2.1.1.2.2.1.2.1" xref="S3.E4.m1.2.2.1.1.2.2.1.2.1.cmml">⁢</mo><msup id="S3.E4.m1.2.2.1.1.2.2.1.2.3" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3.cmml"><mi id="S3.E4.m1.2.2.1.1.2.2.1.2.3.2" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3.2.cmml">t</mi><mn id="S3.E4.m1.2.2.1.1.2.2.1.2.3.3" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3.3.cmml">2</mn></msup></mrow><mn id="S3.E4.m1.2.2.1.1.2.2.1.3" xref="S3.E4.m1.2.2.1.1.2.2.1.3.cmml">2</mn></mfrac></mtd><mtd id="S3.E4.m1.2.2.1.1i" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.2.3.1" xref="S3.E4.m1.2.2.1.1.2.3.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1j" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.2.4.1" xref="S3.E4.m1.2.2.1.1.2.4.1.cmml">0</mn></mtd></mtr><mtr id="S3.E4.m1.2.2.1.1k" xref="S3.E4.m1.2.2.1.1.cmml"><mtd id="S3.E4.m1.2.2.1.1l" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.3.1.1" xref="S3.E4.m1.2.2.1.1.3.1.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1m" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.3.2.1" xref="S3.E4.m1.2.2.1.1.3.2.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1n" xref="S3.E4.m1.2.2.1.1.cmml"><mrow id="S3.E4.m1.2.2.1.1.3.3.1" xref="S3.E4.m1.2.2.1.1.3.3.1.cmml"><mi id="S3.E4.m1.2.2.1.1.3.3.1.2" mathvariant="normal" xref="S3.E4.m1.2.2.1.1.3.3.1.2.cmml">Δ</mi><mo id="S3.E4.m1.2.2.1.1.3.3.1.1" xref="S3.E4.m1.2.2.1.1.3.3.1.1.cmml">⁢</mo><mi id="S3.E4.m1.2.2.1.1.3.3.1.3" xref="S3.E4.m1.2.2.1.1.3.3.1.3.cmml">t</mi></mrow></mtd><mtd id="S3.E4.m1.2.2.1.1o" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.3.4.1" xref="S3.E4.m1.2.2.1.1.3.4.1.cmml">0</mn></mtd></mtr><mtr id="S3.E4.m1.2.2.1.1p" xref="S3.E4.m1.2.2.1.1.cmml"><mtd id="S3.E4.m1.2.2.1.1q" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.4.1.1" xref="S3.E4.m1.2.2.1.1.4.1.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1r" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.4.2.1" xref="S3.E4.m1.2.2.1.1.4.2.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1s" xref="S3.E4.m1.2.2.1.1.cmml"><mn id="S3.E4.m1.2.2.1.1.4.3.1" xref="S3.E4.m1.2.2.1.1.4.3.1.cmml">0</mn></mtd><mtd id="S3.E4.m1.2.2.1.1t" xref="S3.E4.m1.2.2.1.1.cmml"><mrow id="S3.E4.m1.2.2.1.1.4.4.1" xref="S3.E4.m1.2.2.1.1.4.4.1.cmml"><mi id="S3.E4.m1.2.2.1.1.4.4.1.2" mathvariant="normal" xref="S3.E4.m1.2.2.1.1.4.4.1.2.cmml">Δ</mi><mo id="S3.E4.m1.2.2.1.1.4.4.1.1" xref="S3.E4.m1.2.2.1.1.4.4.1.1.cmml">⁢</mo><mi id="S3.E4.m1.2.2.1.1.4.4.1.3" xref="S3.E4.m1.2.2.1.1.4.4.1.3.cmml">t</mi></mrow></mtd></mtr></mtable><mo id="S3.E4.m1.2.2a.3.2" xref="S3.E4.m1.2.2a.2.1.cmml">]</mo></mrow></mrow></mrow><mo id="S3.E4.m1.3.3.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E4.m1.3b"><apply id="S3.E4.m1.3.3.1.1.3.cmml" xref="S3.E4.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.E4.m1.3.3.1.1.3a.cmml" xref="S3.E4.m1.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E4.m1.3.3.1.1.1.1.cmml" xref="S3.E4.m1.3.3.1.1.1.1"><eq id="S3.E4.m1.3.3.1.1.1.1.1.cmml" xref="S3.E4.m1.3.3.1.1.1.1.1"></eq><ci id="S3.E4.m1.3.3.1.1.1.1.2.cmml" xref="S3.E4.m1.3.3.1.1.1.1.2">𝑻</ci><apply id="S3.E4.m1.1.1a.2.cmml" xref="S3.E4.m1.1.1a.3"><csymbol cd="latexml" id="S3.E4.m1.1.1a.2.1.cmml" xref="S3.E4.m1.1.1a.3.1">matrix</csymbol><matrix id="S3.E4.m1.1.1.1.1.cmml" xref="S3.E4.m1.1.1.1.1"><matrixrow id="S3.E4.m1.1.1.1.1a.cmml" xref="S3.E4.m1.1.1.1.1"><cn id="S3.E4.m1.1.1.1.1.1.1.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.1.1.1">1</cn><cn id="S3.E4.m1.1.1.1.1.1.2.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.1.2.1">0</cn><apply id="S3.E4.m1.1.1.1.1.1.3.1.cmml" xref="S3.E4.m1.1.1.1.1.1.3.1"><times id="S3.E4.m1.1.1.1.1.1.3.1.1.cmml" xref="S3.E4.m1.1.1.1.1.1.3.1.1"></times><ci id="S3.E4.m1.1.1.1.1.1.3.1.2.cmml" xref="S3.E4.m1.1.1.1.1.1.3.1.2">Δ</ci><ci id="S3.E4.m1.1.1.1.1.1.3.1.3.cmml" xref="S3.E4.m1.1.1.1.1.1.3.1.3">𝑡</ci></apply><cn id="S3.E4.m1.1.1.1.1.1.4.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.1.4.1">0</cn></matrixrow><matrixrow id="S3.E4.m1.1.1.1.1b.cmml" xref="S3.E4.m1.1.1.1.1"><cn id="S3.E4.m1.1.1.1.1.2.1.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.2.1.1">0</cn><cn id="S3.E4.m1.1.1.1.1.2.2.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.2.2.1">1</cn><cn id="S3.E4.m1.1.1.1.1.2.3.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.2.3.1">0</cn><apply id="S3.E4.m1.1.1.1.1.2.4.1.cmml" xref="S3.E4.m1.1.1.1.1.2.4.1"><times id="S3.E4.m1.1.1.1.1.2.4.1.1.cmml" xref="S3.E4.m1.1.1.1.1.2.4.1.1"></times><ci id="S3.E4.m1.1.1.1.1.2.4.1.2.cmml" xref="S3.E4.m1.1.1.1.1.2.4.1.2">Δ</ci><ci id="S3.E4.m1.1.1.1.1.2.4.1.3.cmml" xref="S3.E4.m1.1.1.1.1.2.4.1.3">𝑡</ci></apply></matrixrow><matrixrow id="S3.E4.m1.1.1.1.1c.cmml" xref="S3.E4.m1.1.1.1.1"><cn id="S3.E4.m1.1.1.1.1.3.1.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.3.1.1">0</cn><cn id="S3.E4.m1.1.1.1.1.3.2.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.3.2.1">0</cn><cn id="S3.E4.m1.1.1.1.1.3.3.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.3.3.1">1</cn><cn id="S3.E4.m1.1.1.1.1.3.4.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.3.4.1">0</cn></matrixrow><matrixrow id="S3.E4.m1.1.1.1.1d.cmml" xref="S3.E4.m1.1.1.1.1"><cn id="S3.E4.m1.1.1.1.1.4.1.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.4.1.1">0</cn><cn id="S3.E4.m1.1.1.1.1.4.2.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.4.2.1">0</cn><cn id="S3.E4.m1.1.1.1.1.4.3.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.4.3.1">0</cn><cn id="S3.E4.m1.1.1.1.1.4.4.1.cmml" type="integer" xref="S3.E4.m1.1.1.1.1.4.4.1">1</cn></matrixrow></matrix></apply></apply><apply id="S3.E4.m1.3.3.1.1.2.2.cmml" xref="S3.E4.m1.3.3.1.1.2.2"><eq id="S3.E4.m1.3.3.1.1.2.2.1.cmml" xref="S3.E4.m1.3.3.1.1.2.2.1"></eq><ci id="S3.E4.m1.3.3.1.1.2.2.2.cmml" xref="S3.E4.m1.3.3.1.1.2.2.2">𝑮</ci><apply id="S3.E4.m1.2.2a.2.cmml" xref="S3.E4.m1.2.2a.3"><csymbol cd="latexml" id="S3.E4.m1.2.2a.2.1.cmml" xref="S3.E4.m1.2.2a.3.1">matrix</csymbol><matrix id="S3.E4.m1.2.2.1.1.cmml" xref="S3.E4.m1.2.2.1.1"><matrixrow id="S3.E4.m1.2.2.1.1a.cmml" xref="S3.E4.m1.2.2.1.1"><apply id="S3.E4.m1.2.2.1.1.1.1.1.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1"><divide id="S3.E4.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1"></divide><apply id="S3.E4.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1.2"><times id="S3.E4.m1.2.2.1.1.1.1.1.2.1.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1.2.1"></times><ci id="S3.E4.m1.2.2.1.1.1.1.1.2.2.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1.2.2">Δ</ci><apply id="S3.E4.m1.2.2.1.1.1.1.1.2.3.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.E4.m1.2.2.1.1.1.1.1.2.3.1.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3">superscript</csymbol><ci id="S3.E4.m1.2.2.1.1.1.1.1.2.3.2.cmml" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3.2">𝑡</ci><cn id="S3.E4.m1.2.2.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.1.1.1.2.3.3">2</cn></apply></apply><cn id="S3.E4.m1.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.1.1.1.3">2</cn></apply><cn id="S3.E4.m1.2.2.1.1.1.2.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.1.2.1">0</cn><cn id="S3.E4.m1.2.2.1.1.1.3.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.1.3.1">0</cn><cn id="S3.E4.m1.2.2.1.1.1.4.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.1.4.1">0</cn></matrixrow><matrixrow id="S3.E4.m1.2.2.1.1b.cmml" xref="S3.E4.m1.2.2.1.1"><cn id="S3.E4.m1.2.2.1.1.2.1.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.2.1.1">0</cn><apply id="S3.E4.m1.2.2.1.1.2.2.1.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1"><divide id="S3.E4.m1.2.2.1.1.2.2.1.1.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1"></divide><apply id="S3.E4.m1.2.2.1.1.2.2.1.2.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1.2"><times id="S3.E4.m1.2.2.1.1.2.2.1.2.1.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1.2.1"></times><ci id="S3.E4.m1.2.2.1.1.2.2.1.2.2.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1.2.2">Δ</ci><apply id="S3.E4.m1.2.2.1.1.2.2.1.2.3.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3"><csymbol cd="ambiguous" id="S3.E4.m1.2.2.1.1.2.2.1.2.3.1.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3">superscript</csymbol><ci id="S3.E4.m1.2.2.1.1.2.2.1.2.3.2.cmml" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3.2">𝑡</ci><cn id="S3.E4.m1.2.2.1.1.2.2.1.2.3.3.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.2.2.1.2.3.3">2</cn></apply></apply><cn id="S3.E4.m1.2.2.1.1.2.2.1.3.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.2.2.1.3">2</cn></apply><cn id="S3.E4.m1.2.2.1.1.2.3.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.2.3.1">0</cn><cn id="S3.E4.m1.2.2.1.1.2.4.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.2.4.1">0</cn></matrixrow><matrixrow id="S3.E4.m1.2.2.1.1c.cmml" xref="S3.E4.m1.2.2.1.1"><cn id="S3.E4.m1.2.2.1.1.3.1.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.3.1.1">0</cn><cn id="S3.E4.m1.2.2.1.1.3.2.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.3.2.1">0</cn><apply id="S3.E4.m1.2.2.1.1.3.3.1.cmml" xref="S3.E4.m1.2.2.1.1.3.3.1"><times id="S3.E4.m1.2.2.1.1.3.3.1.1.cmml" xref="S3.E4.m1.2.2.1.1.3.3.1.1"></times><ci id="S3.E4.m1.2.2.1.1.3.3.1.2.cmml" xref="S3.E4.m1.2.2.1.1.3.3.1.2">Δ</ci><ci id="S3.E4.m1.2.2.1.1.3.3.1.3.cmml" xref="S3.E4.m1.2.2.1.1.3.3.1.3">𝑡</ci></apply><cn id="S3.E4.m1.2.2.1.1.3.4.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.3.4.1">0</cn></matrixrow><matrixrow id="S3.E4.m1.2.2.1.1d.cmml" xref="S3.E4.m1.2.2.1.1"><cn id="S3.E4.m1.2.2.1.1.4.1.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.4.1.1">0</cn><cn id="S3.E4.m1.2.2.1.1.4.2.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.4.2.1">0</cn><cn id="S3.E4.m1.2.2.1.1.4.3.1.cmml" type="integer" xref="S3.E4.m1.2.2.1.1.4.3.1">0</cn><apply id="S3.E4.m1.2.2.1.1.4.4.1.cmml" xref="S3.E4.m1.2.2.1.1.4.4.1"><times id="S3.E4.m1.2.2.1.1.4.4.1.1.cmml" xref="S3.E4.m1.2.2.1.1.4.4.1.1"></times><ci id="S3.E4.m1.2.2.1.1.4.4.1.2.cmml" xref="S3.E4.m1.2.2.1.1.4.4.1.2">Δ</ci><ci id="S3.E4.m1.2.2.1.1.4.4.1.3.cmml" xref="S3.E4.m1.2.2.1.1.4.4.1.3">𝑡</ci></apply></matrixrow></matrix></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m1.3c">\displaystyle\bm{T}=\begin{bmatrix}1&0&\Delta t&0\\ 0&1&0&\Delta t\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\!\!,\ \bm{G}=\begin{bmatrix}\frac{\Delta t^{2}}{2}&0&0&0% \\ 0&\frac{\Delta t^{2}}{2}&0&0\\ 0&0&\Delta t&0\\ 0&0&0&\Delta t\end{bmatrix}\!\!.</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.3d">bold_italic_T = [ start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL roman_Δ italic_t end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL roman_Δ italic_t end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARG ] , bold_italic_G = [ start_ARG start_ROW start_CELL divide start_ARG roman_Δ italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL divide start_ARG roman_Δ italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL roman_Δ italic_t end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL roman_Δ italic_t 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">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p1.10">The joint distribution thus reads,</p> <table class="ltx_equation ltx_eqn_table" id="S3.E5"> <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="p\!\left(\left\{\bm{Z}^{(k)}_{0},\dots,\bm{Z}^{(k)}_{n}\right\},\bm{\phi}_{0},% \dots,\bm{\phi}_{n},\bm{\Lambda}_{a}\right)=p(\bm{\phi}_{0}|\bm{\Lambda}_{a})p% (\bm{\Lambda}_{a})\\ \times\prod_{n=1}^{N}\left(\prod_{k=1}^{N_{\text{radar}}}p\left(\bm{Z}^{(k)}_{% n}|\bm{\phi}_{n}\right)\!\right)p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{% a})." class="ltx_Math" display="block" id="S3.E5.m1.76"><semantics id="S3.E5.m1.76a"><mtable displaystyle="true" id="S3.E5.m1.76.76.7" rowspacing="0pt"><mtr id="S3.E5.m1.76.76.7a"><mtd class="ltx_align_left" columnalign="left" id="S3.E5.m1.76.76.7b"><mrow id="S3.E5.m1.76.76.7.75.45.45"><mrow id="S3.E5.m1.74.74.5.73.43.43.43"><mpadded width="0.333em"><mi id="S3.E5.m1.1.1.1.1.1.1" xref="S3.E5.m1.1.1.1.1.1.1.cmml">p</mi></mpadded><mo id="S3.E5.m1.74.74.5.73.43.43.43.5" xref="S3.E5.m1.70.70.1.1.1.cmml">⁢</mo><mrow id="S3.E5.m1.74.74.5.73.43.43.43.4.4"><mo id="S3.E5.m1.2.2.2.2.2.2" xref="S3.E5.m1.70.70.1.1.1.cmml">(</mo><mrow id="S3.E5.m1.71.71.2.70.40.40.40.1.1.1"><mo id="S3.E5.m1.3.3.3.3.3.3" xref="S3.E5.m1.70.70.1.1.1.cmml">{</mo><msubsup id="S3.E5.m1.71.71.2.70.40.40.40.1.1.1.1.1"><mi id="S3.E5.m1.4.4.4.4.4.4" xref="S3.E5.m1.4.4.4.4.4.4.cmml">𝒁</mi><mn id="S3.E5.m1.6.6.6.6.6.6.1" xref="S3.E5.m1.6.6.6.6.6.6.1.cmml">0</mn><mrow id="S3.E5.m1.5.5.5.5.5.5.1.3"><mo id="S3.E5.m1.5.5.5.5.5.5.1.3.1" stretchy="false">(</mo><mi id="S3.E5.m1.5.5.5.5.5.5.1.1" xref="S3.E5.m1.5.5.5.5.5.5.1.1.cmml">k</mi><mo id="S3.E5.m1.5.5.5.5.5.5.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E5.m1.7.7.7.7.7.7" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><mi id="S3.E5.m1.8.8.8.8.8.8" mathvariant="normal" xref="S3.E5.m1.8.8.8.8.8.8.cmml">…</mi><mo id="S3.E5.m1.9.9.9.9.9.9" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><msubsup id="S3.E5.m1.71.71.2.70.40.40.40.1.1.1.2.2"><mi id="S3.E5.m1.10.10.10.10.10.10" xref="S3.E5.m1.10.10.10.10.10.10.cmml">𝒁</mi><mi id="S3.E5.m1.12.12.12.12.12.12.1" xref="S3.E5.m1.12.12.12.12.12.12.1.cmml">n</mi><mrow id="S3.E5.m1.11.11.11.11.11.11.1.3"><mo id="S3.E5.m1.11.11.11.11.11.11.1.3.1" stretchy="false">(</mo><mi id="S3.E5.m1.11.11.11.11.11.11.1.1" xref="S3.E5.m1.11.11.11.11.11.11.1.1.cmml">k</mi><mo id="S3.E5.m1.11.11.11.11.11.11.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E5.m1.13.13.13.13.13.13" xref="S3.E5.m1.70.70.1.1.1.cmml">}</mo></mrow><mo id="S3.E5.m1.14.14.14.14.14.14" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><msub id="S3.E5.m1.72.72.3.71.41.41.41.2.2.2"><mi class="ltx_mathvariant_bold-italic" id="S3.E5.m1.15.15.15.15.15.15" mathvariant="bold-italic" xref="S3.E5.m1.15.15.15.15.15.15.cmml">ϕ</mi><mn id="S3.E5.m1.16.16.16.16.16.16.1" xref="S3.E5.m1.16.16.16.16.16.16.1.cmml">0</mn></msub><mo id="S3.E5.m1.17.17.17.17.17.17" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><mi id="S3.E5.m1.18.18.18.18.18.18" mathvariant="normal" xref="S3.E5.m1.18.18.18.18.18.18.cmml">…</mi><mo id="S3.E5.m1.19.19.19.19.19.19" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><msub id="S3.E5.m1.73.73.4.72.42.42.42.3.3.3"><mi class="ltx_mathvariant_bold-italic" id="S3.E5.m1.20.20.20.20.20.20" mathvariant="bold-italic" xref="S3.E5.m1.20.20.20.20.20.20.cmml">ϕ</mi><mi id="S3.E5.m1.21.21.21.21.21.21.1" xref="S3.E5.m1.21.21.21.21.21.21.1.cmml">n</mi></msub><mo id="S3.E5.m1.22.22.22.22.22.22" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><msub id="S3.E5.m1.74.74.5.73.43.43.43.4.4.4"><mi id="S3.E5.m1.23.23.23.23.23.23" xref="S3.E5.m1.23.23.23.23.23.23.cmml">𝚲</mi><mi id="S3.E5.m1.24.24.24.24.24.24.1" xref="S3.E5.m1.24.24.24.24.24.24.1.cmml">a</mi></msub><mo id="S3.E5.m1.25.25.25.25.25.25" xref="S3.E5.m1.70.70.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E5.m1.26.26.26.26.26.26" xref="S3.E5.m1.26.26.26.26.26.26.cmml">=</mo><mrow id="S3.E5.m1.76.76.7.75.45.45.45"><mi id="S3.E5.m1.27.27.27.27.27.27" xref="S3.E5.m1.27.27.27.27.27.27.cmml">p</mi><mo id="S3.E5.m1.76.76.7.75.45.45.45.3" xref="S3.E5.m1.70.70.1.1.1.cmml">⁢</mo><mrow id="S3.E5.m1.75.75.6.74.44.44.44.1.1"><mo id="S3.E5.m1.28.28.28.28.28.28" stretchy="false" xref="S3.E5.m1.70.70.1.1.1.cmml">(</mo><mrow id="S3.E5.m1.75.75.6.74.44.44.44.1.1.1"><msub id="S3.E5.m1.75.75.6.74.44.44.44.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S3.E5.m1.29.29.29.29.29.29" mathvariant="bold-italic" xref="S3.E5.m1.29.29.29.29.29.29.cmml">ϕ</mi><mn id="S3.E5.m1.30.30.30.30.30.30.1" xref="S3.E5.m1.30.30.30.30.30.30.1.cmml">0</mn></msub><mo fence="false" id="S3.E5.m1.31.31.31.31.31.31" xref="S3.E5.m1.31.31.31.31.31.31.cmml">|</mo><msub id="S3.E5.m1.75.75.6.74.44.44.44.1.1.1.2"><mi id="S3.E5.m1.32.32.32.32.32.32" xref="S3.E5.m1.32.32.32.32.32.32.cmml">𝚲</mi><mi id="S3.E5.m1.33.33.33.33.33.33.1" xref="S3.E5.m1.33.33.33.33.33.33.1.cmml">a</mi></msub></mrow><mo id="S3.E5.m1.34.34.34.34.34.34" stretchy="false" xref="S3.E5.m1.70.70.1.1.1.cmml">)</mo></mrow><mo id="S3.E5.m1.76.76.7.75.45.45.45.3a" xref="S3.E5.m1.70.70.1.1.1.cmml">⁢</mo><mi id="S3.E5.m1.35.35.35.35.35.35" xref="S3.E5.m1.35.35.35.35.35.35.cmml">p</mi><mo id="S3.E5.m1.76.76.7.75.45.45.45.3b" xref="S3.E5.m1.70.70.1.1.1.cmml">⁢</mo><mrow id="S3.E5.m1.76.76.7.75.45.45.45.2.1"><mo id="S3.E5.m1.36.36.36.36.36.36" stretchy="false" xref="S3.E5.m1.70.70.1.1.1.cmml">(</mo><msub id="S3.E5.m1.76.76.7.75.45.45.45.2.1.1"><mi id="S3.E5.m1.37.37.37.37.37.37" xref="S3.E5.m1.37.37.37.37.37.37.cmml">𝚲</mi><mi id="S3.E5.m1.38.38.38.38.38.38.1" xref="S3.E5.m1.38.38.38.38.38.38.1.cmml">a</mi></msub><mo id="S3.E5.m1.39.39.39.39.39.39" stretchy="false" xref="S3.E5.m1.70.70.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E5.m1.76.76.7c"><mtd class="ltx_align_right" columnalign="right" id="S3.E5.m1.76.76.7d"><mrow id="S3.E5.m1.69.69.69.30.30"><mo id="S3.E5.m1.40.40.40.1.1.1" rspace="0.055em" xref="S3.E5.m1.40.40.40.1.1.1.cmml">×</mo><munderover id="S3.E5.m1.69.69.69.30.30.31"><mo id="S3.E5.m1.41.41.41.2.2.2" movablelimits="false" rspace="0em" xref="S3.E5.m1.41.41.41.2.2.2.cmml">∏</mo><mrow id="S3.E5.m1.42.42.42.3.3.3.1" xref="S3.E5.m1.42.42.42.3.3.3.1.cmml"><mi id="S3.E5.m1.42.42.42.3.3.3.1.2" xref="S3.E5.m1.42.42.42.3.3.3.1.2.cmml">n</mi><mo id="S3.E5.m1.42.42.42.3.3.3.1.1" xref="S3.E5.m1.42.42.42.3.3.3.1.1.cmml">=</mo><mn id="S3.E5.m1.42.42.42.3.3.3.1.3" xref="S3.E5.m1.42.42.42.3.3.3.1.3.cmml">1</mn></mrow><mi id="S3.E5.m1.43.43.43.4.4.4.1" xref="S3.E5.m1.43.43.43.4.4.4.1.cmml">N</mi></munderover><mrow id="S3.E5.m1.69.69.69.30.30.32"><mo id="S3.E5.m1.44.44.44.5.5.5" xref="S3.E5.m1.70.70.1.1.1.cmml">(</mo><munderover id="S3.E5.m1.69.69.69.30.30.32.1"><mo id="S3.E5.m1.45.45.45.6.6.6" lspace="0em" movablelimits="false" xref="S3.E5.m1.45.45.45.6.6.6.cmml">∏</mo><mrow id="S3.E5.m1.46.46.46.7.7.7.1" xref="S3.E5.m1.46.46.46.7.7.7.1.cmml"><mi id="S3.E5.m1.46.46.46.7.7.7.1.2" xref="S3.E5.m1.46.46.46.7.7.7.1.2.cmml">k</mi><mo id="S3.E5.m1.46.46.46.7.7.7.1.1" xref="S3.E5.m1.46.46.46.7.7.7.1.1.cmml">=</mo><mn id="S3.E5.m1.46.46.46.7.7.7.1.3" xref="S3.E5.m1.46.46.46.7.7.7.1.3.cmml">1</mn></mrow><msub id="S3.E5.m1.47.47.47.8.8.8.1" xref="S3.E5.m1.47.47.47.8.8.8.1.cmml"><mi id="S3.E5.m1.47.47.47.8.8.8.1.2" xref="S3.E5.m1.47.47.47.8.8.8.1.2.cmml">N</mi><mtext id="S3.E5.m1.47.47.47.8.8.8.1.3" xref="S3.E5.m1.47.47.47.8.8.8.1.3a.cmml">radar</mtext></msub></munderover><mi id="S3.E5.m1.48.48.48.9.9.9" xref="S3.E5.m1.48.48.48.9.9.9.cmml">p</mi><mrow id="S3.E5.m1.69.69.69.30.30.32.2"><mo id="S3.E5.m1.49.49.49.10.10.10" xref="S3.E5.m1.70.70.1.1.1.cmml">(</mo><msubsup id="S3.E5.m1.69.69.69.30.30.32.2.1"><mi id="S3.E5.m1.50.50.50.11.11.11" xref="S3.E5.m1.50.50.50.11.11.11.cmml">𝒁</mi><mi id="S3.E5.m1.52.52.52.13.13.13.1" xref="S3.E5.m1.52.52.52.13.13.13.1.cmml">n</mi><mrow id="S3.E5.m1.51.51.51.12.12.12.1.3"><mo id="S3.E5.m1.51.51.51.12.12.12.1.3.1" stretchy="false">(</mo><mi id="S3.E5.m1.51.51.51.12.12.12.1.1" xref="S3.E5.m1.51.51.51.12.12.12.1.1.cmml">k</mi><mo id="S3.E5.m1.51.51.51.12.12.12.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo fence="false" id="S3.E5.m1.53.53.53.14.14.14" rspace="0.167em" stretchy="false" xref="S3.E5.m1.53.53.53.14.14.14.cmml">|</mo><msub id="S3.E5.m1.69.69.69.30.30.32.2.2"><mi class="ltx_mathvariant_bold-italic" id="S3.E5.m1.54.54.54.15.15.15" mathvariant="bold-italic" xref="S3.E5.m1.54.54.54.15.15.15.cmml">ϕ</mi><mi id="S3.E5.m1.55.55.55.16.16.16.1" xref="S3.E5.m1.55.55.55.16.16.16.1.cmml">n</mi></msub><mpadded width="0.288em"><mo id="S3.E5.m1.56.56.56.17.17.17" xref="S3.E5.m1.70.70.1.1.1.cmml">)</mo></mpadded></mrow><mo id="S3.E5.m1.57.57.57.18.18.18" xref="S3.E5.m1.70.70.1.1.1.cmml">)</mo></mrow><mi id="S3.E5.m1.58.58.58.19.19.19" xref="S3.E5.m1.58.58.58.19.19.19.cmml">p</mi><mrow id="S3.E5.m1.69.69.69.30.30.33"><mo id="S3.E5.m1.59.59.59.20.20.20" stretchy="false" xref="S3.E5.m1.70.70.1.1.1.cmml">(</mo><msub id="S3.E5.m1.69.69.69.30.30.33.1"><mi class="ltx_mathvariant_bold-italic" id="S3.E5.m1.60.60.60.21.21.21" mathvariant="bold-italic" xref="S3.E5.m1.60.60.60.21.21.21.cmml">ϕ</mi><mi id="S3.E5.m1.61.61.61.22.22.22.1" xref="S3.E5.m1.61.61.61.22.22.22.1.cmml">n</mi></msub><mo fence="false" id="S3.E5.m1.62.62.62.23.23.23" rspace="0.167em" stretchy="false" xref="S3.E5.m1.62.62.62.23.23.23.cmml">|</mo><msub id="S3.E5.m1.69.69.69.30.30.33.2"><mi class="ltx_mathvariant_bold-italic" id="S3.E5.m1.63.63.63.24.24.24" mathvariant="bold-italic" xref="S3.E5.m1.63.63.63.24.24.24.cmml">ϕ</mi><mrow id="S3.E5.m1.64.64.64.25.25.25.1" xref="S3.E5.m1.64.64.64.25.25.25.1.cmml"><mi id="S3.E5.m1.64.64.64.25.25.25.1.2" xref="S3.E5.m1.64.64.64.25.25.25.1.2.cmml">n</mi><mo id="S3.E5.m1.64.64.64.25.25.25.1.1" xref="S3.E5.m1.64.64.64.25.25.25.1.1.cmml">−</mo><mn id="S3.E5.m1.64.64.64.25.25.25.1.3" xref="S3.E5.m1.64.64.64.25.25.25.1.3.cmml">1</mn></mrow></msub><mo id="S3.E5.m1.65.65.65.26.26.26" xref="S3.E5.m1.70.70.1.1.1.cmml">,</mo><msub id="S3.E5.m1.69.69.69.30.30.33.3"><mi id="S3.E5.m1.66.66.66.27.27.27" xref="S3.E5.m1.66.66.66.27.27.27.cmml">𝚲</mi><mi id="S3.E5.m1.67.67.67.28.28.28.1" xref="S3.E5.m1.67.67.67.28.28.28.1.cmml">a</mi></msub><mo id="S3.E5.m1.68.68.68.29.29.29" stretchy="false" xref="S3.E5.m1.70.70.1.1.1.cmml">)</mo></mrow><mo id="S3.E5.m1.69.69.69.30.30.30" lspace="0em" xref="S3.E5.m1.70.70.1.1.1.cmml">.</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S3.E5.m1.76b"><apply id="S3.E5.m1.70.70.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><eq id="S3.E5.m1.26.26.26.26.26.26.cmml" xref="S3.E5.m1.26.26.26.26.26.26"></eq><apply id="S3.E5.m1.70.70.1.1.1.4.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><times id="S3.E5.m1.70.70.1.1.1.4.5.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"></times><ci id="S3.E5.m1.1.1.1.1.1.1.cmml" xref="S3.E5.m1.1.1.1.1.1.1">𝑝</ci><vector id="S3.E5.m1.70.70.1.1.1.4.4.5.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><set id="S3.E5.m1.70.70.1.1.1.1.1.1.1.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><apply id="S3.E5.m1.70.70.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><apply id="S3.E5.m1.70.70.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">superscript</csymbol><ci id="S3.E5.m1.4.4.4.4.4.4.cmml" xref="S3.E5.m1.4.4.4.4.4.4">𝒁</ci><ci id="S3.E5.m1.5.5.5.5.5.5.1.1.cmml" xref="S3.E5.m1.5.5.5.5.5.5.1.1">𝑘</ci></apply><cn id="S3.E5.m1.6.6.6.6.6.6.1.cmml" type="integer" xref="S3.E5.m1.6.6.6.6.6.6.1">0</cn></apply><ci id="S3.E5.m1.8.8.8.8.8.8.cmml" xref="S3.E5.m1.8.8.8.8.8.8">…</ci><apply id="S3.E5.m1.70.70.1.1.1.1.1.1.1.2.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><apply id="S3.E5.m1.70.70.1.1.1.1.1.1.1.2.2.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">superscript</csymbol><ci id="S3.E5.m1.10.10.10.10.10.10.cmml" xref="S3.E5.m1.10.10.10.10.10.10">𝒁</ci><ci id="S3.E5.m1.11.11.11.11.11.11.1.1.cmml" xref="S3.E5.m1.11.11.11.11.11.11.1.1">𝑘</ci></apply><ci id="S3.E5.m1.12.12.12.12.12.12.1.cmml" xref="S3.E5.m1.12.12.12.12.12.12.1">𝑛</ci></apply></set><apply id="S3.E5.m1.70.70.1.1.1.2.2.2.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.2.2.2.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.15.15.15.15.15.15.cmml" xref="S3.E5.m1.15.15.15.15.15.15">bold-italic-ϕ</ci><cn id="S3.E5.m1.16.16.16.16.16.16.1.cmml" type="integer" xref="S3.E5.m1.16.16.16.16.16.16.1">0</cn></apply><ci id="S3.E5.m1.18.18.18.18.18.18.cmml" xref="S3.E5.m1.18.18.18.18.18.18">…</ci><apply id="S3.E5.m1.70.70.1.1.1.3.3.3.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.3.3.3.3.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.20.20.20.20.20.20.cmml" xref="S3.E5.m1.20.20.20.20.20.20">bold-italic-ϕ</ci><ci id="S3.E5.m1.21.21.21.21.21.21.1.cmml" xref="S3.E5.m1.21.21.21.21.21.21.1">𝑛</ci></apply><apply id="S3.E5.m1.70.70.1.1.1.4.4.4.4.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.4.4.4.4.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.23.23.23.23.23.23.cmml" xref="S3.E5.m1.23.23.23.23.23.23">𝚲</ci><ci id="S3.E5.m1.24.24.24.24.24.24.1.cmml" xref="S3.E5.m1.24.24.24.24.24.24.1">𝑎</ci></apply></vector></apply><apply id="S3.E5.m1.70.70.1.1.1.8.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><times id="S3.E5.m1.40.40.40.1.1.1.cmml" xref="S3.E5.m1.40.40.40.1.1.1"></times><apply id="S3.E5.m1.70.70.1.1.1.6.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><times id="S3.E5.m1.70.70.1.1.1.6.2.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"></times><ci id="S3.E5.m1.27.27.27.27.27.27.cmml" xref="S3.E5.m1.27.27.27.27.27.27">𝑝</ci><apply id="S3.E5.m1.70.70.1.1.1.5.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="latexml" id="S3.E5.m1.31.31.31.31.31.31.cmml" xref="S3.E5.m1.31.31.31.31.31.31">conditional</csymbol><apply id="S3.E5.m1.70.70.1.1.1.5.1.1.1.1.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.5.1.1.1.1.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.29.29.29.29.29.29.cmml" xref="S3.E5.m1.29.29.29.29.29.29">bold-italic-ϕ</ci><cn id="S3.E5.m1.30.30.30.30.30.30.1.cmml" type="integer" xref="S3.E5.m1.30.30.30.30.30.30.1">0</cn></apply><apply id="S3.E5.m1.70.70.1.1.1.5.1.1.1.1.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.5.1.1.1.1.3.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.32.32.32.32.32.32.cmml" xref="S3.E5.m1.32.32.32.32.32.32">𝚲</ci><ci id="S3.E5.m1.33.33.33.33.33.33.1.cmml" xref="S3.E5.m1.33.33.33.33.33.33.1">𝑎</ci></apply></apply><ci id="S3.E5.m1.35.35.35.35.35.35.cmml" xref="S3.E5.m1.35.35.35.35.35.35">𝑝</ci><apply id="S3.E5.m1.70.70.1.1.1.6.2.2.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.6.2.2.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.37.37.37.37.37.37.cmml" xref="S3.E5.m1.37.37.37.37.37.37">𝚲</ci><ci id="S3.E5.m1.38.38.38.38.38.38.1.cmml" xref="S3.E5.m1.38.38.38.38.38.38.1">𝑎</ci></apply></apply><apply id="S3.E5.m1.70.70.1.1.1.8.4.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><apply id="S3.E5.m1.70.70.1.1.1.8.4.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.8.4.3.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">superscript</csymbol><apply id="S3.E5.m1.70.70.1.1.1.8.4.3.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.8.4.3.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><csymbol cd="latexml" id="S3.E5.m1.41.41.41.2.2.2.cmml" xref="S3.E5.m1.41.41.41.2.2.2">product</csymbol><apply id="S3.E5.m1.42.42.42.3.3.3.1.cmml" xref="S3.E5.m1.42.42.42.3.3.3.1"><eq id="S3.E5.m1.42.42.42.3.3.3.1.1.cmml" xref="S3.E5.m1.42.42.42.3.3.3.1.1"></eq><ci id="S3.E5.m1.42.42.42.3.3.3.1.2.cmml" xref="S3.E5.m1.42.42.42.3.3.3.1.2">𝑛</ci><cn id="S3.E5.m1.42.42.42.3.3.3.1.3.cmml" type="integer" xref="S3.E5.m1.42.42.42.3.3.3.1.3">1</cn></apply></apply><ci id="S3.E5.m1.43.43.43.4.4.4.1.cmml" xref="S3.E5.m1.43.43.43.4.4.4.1">𝑁</ci></apply><apply id="S3.E5.m1.70.70.1.1.1.8.4.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><times id="S3.E5.m1.70.70.1.1.1.8.4.2.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"></times><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">superscript</csymbol><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.2.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.2.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><csymbol cd="latexml" id="S3.E5.m1.45.45.45.6.6.6.cmml" xref="S3.E5.m1.45.45.45.6.6.6">product</csymbol><apply id="S3.E5.m1.46.46.46.7.7.7.1.cmml" xref="S3.E5.m1.46.46.46.7.7.7.1"><eq id="S3.E5.m1.46.46.46.7.7.7.1.1.cmml" xref="S3.E5.m1.46.46.46.7.7.7.1.1"></eq><ci id="S3.E5.m1.46.46.46.7.7.7.1.2.cmml" xref="S3.E5.m1.46.46.46.7.7.7.1.2">𝑘</ci><cn id="S3.E5.m1.46.46.46.7.7.7.1.3.cmml" type="integer" xref="S3.E5.m1.46.46.46.7.7.7.1.3">1</cn></apply></apply><apply id="S3.E5.m1.47.47.47.8.8.8.1.cmml" xref="S3.E5.m1.47.47.47.8.8.8.1"><csymbol cd="ambiguous" id="S3.E5.m1.47.47.47.8.8.8.1.1.cmml" xref="S3.E5.m1.47.47.47.8.8.8.1">subscript</csymbol><ci id="S3.E5.m1.47.47.47.8.8.8.1.2.cmml" xref="S3.E5.m1.47.47.47.8.8.8.1.2">𝑁</ci><ci id="S3.E5.m1.47.47.47.8.8.8.1.3a.cmml" xref="S3.E5.m1.47.47.47.8.8.8.1.3"><mtext id="S3.E5.m1.47.47.47.8.8.8.1.3.cmml" mathsize="50%" xref="S3.E5.m1.47.47.47.8.8.8.1.3">radar</mtext></ci></apply></apply><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><times id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"></times><ci id="S3.E5.m1.48.48.48.9.9.9.cmml" xref="S3.E5.m1.48.48.48.9.9.9">𝑝</ci><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><ci id="S3.E5.m1.53.53.53.14.14.14.cmml" xref="S3.E5.m1.53.53.53.14.14.14">|</ci><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">superscript</csymbol><ci id="S3.E5.m1.50.50.50.11.11.11.cmml" xref="S3.E5.m1.50.50.50.11.11.11">𝒁</ci><ci id="S3.E5.m1.51.51.51.12.12.12.1.1.cmml" xref="S3.E5.m1.51.51.51.12.12.12.1.1">𝑘</ci></apply><ci id="S3.E5.m1.52.52.52.13.13.13.1.cmml" xref="S3.E5.m1.52.52.52.13.13.13.1">𝑛</ci></apply><apply id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.7.3.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.54.54.54.15.15.15.cmml" xref="S3.E5.m1.54.54.54.15.15.15">bold-italic-ϕ</ci><ci id="S3.E5.m1.55.55.55.16.16.16.1.cmml" xref="S3.E5.m1.55.55.55.16.16.16.1">𝑛</ci></apply></apply></apply></apply><ci id="S3.E5.m1.58.58.58.19.19.19.cmml" xref="S3.E5.m1.58.58.58.19.19.19">𝑝</ci><apply id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><ci id="S3.E5.m1.62.62.62.23.23.23.cmml" xref="S3.E5.m1.62.62.62.23.23.23">|</ci><apply id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.4.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.4.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.60.60.60.21.21.21.cmml" xref="S3.E5.m1.60.60.60.21.21.21">bold-italic-ϕ</ci><ci id="S3.E5.m1.61.61.61.22.22.22.1.cmml" xref="S3.E5.m1.61.61.61.22.22.22.1">𝑛</ci></apply><list id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.2.3.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><apply id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.1.1.1.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.63.63.63.24.24.24.cmml" xref="S3.E5.m1.63.63.63.24.24.24">bold-italic-ϕ</ci><apply id="S3.E5.m1.64.64.64.25.25.25.1.cmml" xref="S3.E5.m1.64.64.64.25.25.25.1"><minus id="S3.E5.m1.64.64.64.25.25.25.1.1.cmml" xref="S3.E5.m1.64.64.64.25.25.25.1.1"></minus><ci id="S3.E5.m1.64.64.64.25.25.25.1.2.cmml" xref="S3.E5.m1.64.64.64.25.25.25.1.2">𝑛</ci><cn id="S3.E5.m1.64.64.64.25.25.25.1.3.cmml" type="integer" xref="S3.E5.m1.64.64.64.25.25.25.1.3">1</cn></apply></apply><apply id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.2.2.2.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5"><csymbol cd="ambiguous" id="S3.E5.m1.70.70.1.1.1.8.4.2.2.1.1.2.2.2.1.cmml" xref="S3.E5.m1.74.74.5.73.43.43.43.5">subscript</csymbol><ci id="S3.E5.m1.66.66.66.27.27.27.cmml" xref="S3.E5.m1.66.66.66.27.27.27">𝚲</ci><ci id="S3.E5.m1.67.67.67.28.28.28.1.cmml" xref="S3.E5.m1.67.67.67.28.28.28.1">𝑎</ci></apply></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.76c">p\!\left(\left\{\bm{Z}^{(k)}_{0},\dots,\bm{Z}^{(k)}_{n}\right\},\bm{\phi}_{0},% \dots,\bm{\phi}_{n},\bm{\Lambda}_{a}\right)=p(\bm{\phi}_{0}|\bm{\Lambda}_{a})p% (\bm{\Lambda}_{a})\\ \times\prod_{n=1}^{N}\left(\prod_{k=1}^{N_{\text{radar}}}p\left(\bm{Z}^{(k)}_{% n}|\bm{\phi}_{n}\right)\!\right)p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{% a}).</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.76d">start_ROW start_CELL italic_p ( { bold_italic_Z start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , bold_italic_Z start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , bold_italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) = italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) italic_p ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL × ∏ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( ∏ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p ( bold_italic_Z start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) . 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">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p1.9">Note that the prior <math alttext="p(\bm{\phi}_{0}|\bm{\Lambda}_{a})" class="ltx_Math" display="inline" id="S3.p1.7.m1.1"><semantics id="S3.p1.7.m1.1a"><mrow id="S3.p1.7.m1.1.1" xref="S3.p1.7.m1.1.1.cmml"><mi id="S3.p1.7.m1.1.1.3" xref="S3.p1.7.m1.1.1.3.cmml">p</mi><mo id="S3.p1.7.m1.1.1.2" xref="S3.p1.7.m1.1.1.2.cmml">⁢</mo><mrow id="S3.p1.7.m1.1.1.1.1" xref="S3.p1.7.m1.1.1.1.1.1.cmml"><mo id="S3.p1.7.m1.1.1.1.1.2" stretchy="false" xref="S3.p1.7.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p1.7.m1.1.1.1.1.1" xref="S3.p1.7.m1.1.1.1.1.1.cmml"><msub id="S3.p1.7.m1.1.1.1.1.1.2" xref="S3.p1.7.m1.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p1.7.m1.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S3.p1.7.m1.1.1.1.1.1.2.2.cmml">ϕ</mi><mn id="S3.p1.7.m1.1.1.1.1.1.2.3" xref="S3.p1.7.m1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo fence="false" id="S3.p1.7.m1.1.1.1.1.1.1" xref="S3.p1.7.m1.1.1.1.1.1.1.cmml">|</mo><msub id="S3.p1.7.m1.1.1.1.1.1.3" xref="S3.p1.7.m1.1.1.1.1.1.3.cmml"><mi id="S3.p1.7.m1.1.1.1.1.1.3.2" xref="S3.p1.7.m1.1.1.1.1.1.3.2.cmml">𝚲</mi><mi id="S3.p1.7.m1.1.1.1.1.1.3.3" xref="S3.p1.7.m1.1.1.1.1.1.3.3.cmml">a</mi></msub></mrow><mo id="S3.p1.7.m1.1.1.1.1.3" stretchy="false" xref="S3.p1.7.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m1.1b"><apply id="S3.p1.7.m1.1.1.cmml" xref="S3.p1.7.m1.1.1"><times id="S3.p1.7.m1.1.1.2.cmml" xref="S3.p1.7.m1.1.1.2"></times><ci id="S3.p1.7.m1.1.1.3.cmml" xref="S3.p1.7.m1.1.1.3">𝑝</ci><apply id="S3.p1.7.m1.1.1.1.1.1.cmml" xref="S3.p1.7.m1.1.1.1.1"><csymbol cd="latexml" id="S3.p1.7.m1.1.1.1.1.1.1.cmml" xref="S3.p1.7.m1.1.1.1.1.1.1">conditional</csymbol><apply id="S3.p1.7.m1.1.1.1.1.1.2.cmml" xref="S3.p1.7.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p1.7.m1.1.1.1.1.1.2.1.cmml" xref="S3.p1.7.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.p1.7.m1.1.1.1.1.1.2.2.cmml" xref="S3.p1.7.m1.1.1.1.1.1.2.2">bold-italic-ϕ</ci><cn id="S3.p1.7.m1.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.p1.7.m1.1.1.1.1.1.2.3">0</cn></apply><apply id="S3.p1.7.m1.1.1.1.1.1.3.cmml" xref="S3.p1.7.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p1.7.m1.1.1.1.1.1.3.1.cmml" xref="S3.p1.7.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.p1.7.m1.1.1.1.1.1.3.2.cmml" xref="S3.p1.7.m1.1.1.1.1.1.3.2">𝚲</ci><ci id="S3.p1.7.m1.1.1.1.1.1.3.3.cmml" xref="S3.p1.7.m1.1.1.1.1.1.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m1.1c">p(\bm{\phi}_{0}|\bm{\Lambda}_{a})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m1.1d">italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT )</annotation></semantics></math> is complicated by the kinematic process (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S3.E3" title="In III Bayesian Network ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">3</span></a>) as any stationary solution will have the variance of <math alttext="p(\bm{\phi}_{0}|\bm{\Lambda}_{a})" class="ltx_Math" display="inline" id="S3.p1.8.m2.1"><semantics id="S3.p1.8.m2.1a"><mrow id="S3.p1.8.m2.1.1" xref="S3.p1.8.m2.1.1.cmml"><mi id="S3.p1.8.m2.1.1.3" xref="S3.p1.8.m2.1.1.3.cmml">p</mi><mo id="S3.p1.8.m2.1.1.2" xref="S3.p1.8.m2.1.1.2.cmml">⁢</mo><mrow id="S3.p1.8.m2.1.1.1.1" xref="S3.p1.8.m2.1.1.1.1.1.cmml"><mo id="S3.p1.8.m2.1.1.1.1.2" stretchy="false" xref="S3.p1.8.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.p1.8.m2.1.1.1.1.1" xref="S3.p1.8.m2.1.1.1.1.1.cmml"><msub id="S3.p1.8.m2.1.1.1.1.1.2" xref="S3.p1.8.m2.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p1.8.m2.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S3.p1.8.m2.1.1.1.1.1.2.2.cmml">ϕ</mi><mn id="S3.p1.8.m2.1.1.1.1.1.2.3" xref="S3.p1.8.m2.1.1.1.1.1.2.3.cmml">0</mn></msub><mo fence="false" id="S3.p1.8.m2.1.1.1.1.1.1" xref="S3.p1.8.m2.1.1.1.1.1.1.cmml">|</mo><msub id="S3.p1.8.m2.1.1.1.1.1.3" xref="S3.p1.8.m2.1.1.1.1.1.3.cmml"><mi id="S3.p1.8.m2.1.1.1.1.1.3.2" xref="S3.p1.8.m2.1.1.1.1.1.3.2.cmml">𝚲</mi><mi id="S3.p1.8.m2.1.1.1.1.1.3.3" xref="S3.p1.8.m2.1.1.1.1.1.3.3.cmml">a</mi></msub></mrow><mo id="S3.p1.8.m2.1.1.1.1.3" stretchy="false" xref="S3.p1.8.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m2.1b"><apply id="S3.p1.8.m2.1.1.cmml" xref="S3.p1.8.m2.1.1"><times id="S3.p1.8.m2.1.1.2.cmml" xref="S3.p1.8.m2.1.1.2"></times><ci id="S3.p1.8.m2.1.1.3.cmml" xref="S3.p1.8.m2.1.1.3">𝑝</ci><apply id="S3.p1.8.m2.1.1.1.1.1.cmml" xref="S3.p1.8.m2.1.1.1.1"><csymbol cd="latexml" id="S3.p1.8.m2.1.1.1.1.1.1.cmml" xref="S3.p1.8.m2.1.1.1.1.1.1">conditional</csymbol><apply id="S3.p1.8.m2.1.1.1.1.1.2.cmml" xref="S3.p1.8.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p1.8.m2.1.1.1.1.1.2.1.cmml" xref="S3.p1.8.m2.1.1.1.1.1.2">subscript</csymbol><ci id="S3.p1.8.m2.1.1.1.1.1.2.2.cmml" xref="S3.p1.8.m2.1.1.1.1.1.2.2">bold-italic-ϕ</ci><cn id="S3.p1.8.m2.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.p1.8.m2.1.1.1.1.1.2.3">0</cn></apply><apply id="S3.p1.8.m2.1.1.1.1.1.3.cmml" xref="S3.p1.8.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p1.8.m2.1.1.1.1.1.3.1.cmml" xref="S3.p1.8.m2.1.1.1.1.1.3">subscript</csymbol><ci id="S3.p1.8.m2.1.1.1.1.1.3.2.cmml" xref="S3.p1.8.m2.1.1.1.1.1.3.2">𝚲</ci><ci id="S3.p1.8.m2.1.1.1.1.1.3.3.cmml" xref="S3.p1.8.m2.1.1.1.1.1.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m2.1c">p(\bm{\phi}_{0}|\bm{\Lambda}_{a})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m2.1d">italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT )</annotation></semantics></math> tending to infinity. For this reason we use an improper distribution <math alttext="p(\bm{\phi}_{0}|\bm{\Lambda}_{a})=1" class="ltx_Math" display="inline" id="S3.p1.9.m3.1"><semantics id="S3.p1.9.m3.1a"><mrow id="S3.p1.9.m3.1.1" xref="S3.p1.9.m3.1.1.cmml"><mrow id="S3.p1.9.m3.1.1.1" xref="S3.p1.9.m3.1.1.1.cmml"><mi id="S3.p1.9.m3.1.1.1.3" xref="S3.p1.9.m3.1.1.1.3.cmml">p</mi><mo id="S3.p1.9.m3.1.1.1.2" xref="S3.p1.9.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.p1.9.m3.1.1.1.1.1" xref="S3.p1.9.m3.1.1.1.1.1.1.cmml"><mo id="S3.p1.9.m3.1.1.1.1.1.2" stretchy="false" xref="S3.p1.9.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p1.9.m3.1.1.1.1.1.1" xref="S3.p1.9.m3.1.1.1.1.1.1.cmml"><msub id="S3.p1.9.m3.1.1.1.1.1.1.2" xref="S3.p1.9.m3.1.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.p1.9.m3.1.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S3.p1.9.m3.1.1.1.1.1.1.2.2.cmml">ϕ</mi><mn id="S3.p1.9.m3.1.1.1.1.1.1.2.3" xref="S3.p1.9.m3.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo fence="false" id="S3.p1.9.m3.1.1.1.1.1.1.1" xref="S3.p1.9.m3.1.1.1.1.1.1.1.cmml">|</mo><msub id="S3.p1.9.m3.1.1.1.1.1.1.3" xref="S3.p1.9.m3.1.1.1.1.1.1.3.cmml"><mi id="S3.p1.9.m3.1.1.1.1.1.1.3.2" xref="S3.p1.9.m3.1.1.1.1.1.1.3.2.cmml">𝚲</mi><mi id="S3.p1.9.m3.1.1.1.1.1.1.3.3" xref="S3.p1.9.m3.1.1.1.1.1.1.3.3.cmml">a</mi></msub></mrow><mo id="S3.p1.9.m3.1.1.1.1.1.3" stretchy="false" xref="S3.p1.9.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p1.9.m3.1.1.2" xref="S3.p1.9.m3.1.1.2.cmml">=</mo><mn id="S3.p1.9.m3.1.1.3" xref="S3.p1.9.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.9.m3.1b"><apply id="S3.p1.9.m3.1.1.cmml" xref="S3.p1.9.m3.1.1"><eq id="S3.p1.9.m3.1.1.2.cmml" xref="S3.p1.9.m3.1.1.2"></eq><apply id="S3.p1.9.m3.1.1.1.cmml" xref="S3.p1.9.m3.1.1.1"><times id="S3.p1.9.m3.1.1.1.2.cmml" xref="S3.p1.9.m3.1.1.1.2"></times><ci id="S3.p1.9.m3.1.1.1.3.cmml" xref="S3.p1.9.m3.1.1.1.3">𝑝</ci><apply id="S3.p1.9.m3.1.1.1.1.1.1.cmml" xref="S3.p1.9.m3.1.1.1.1.1"><csymbol cd="latexml" id="S3.p1.9.m3.1.1.1.1.1.1.1.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.1">conditional</csymbol><apply id="S3.p1.9.m3.1.1.1.1.1.1.2.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p1.9.m3.1.1.1.1.1.1.2.1.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.p1.9.m3.1.1.1.1.1.1.2.2.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.2.2">bold-italic-ϕ</ci><cn id="S3.p1.9.m3.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.p1.9.m3.1.1.1.1.1.1.2.3">0</cn></apply><apply id="S3.p1.9.m3.1.1.1.1.1.1.3.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p1.9.m3.1.1.1.1.1.1.3.1.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.p1.9.m3.1.1.1.1.1.1.3.2.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.3.2">𝚲</ci><ci id="S3.p1.9.m3.1.1.1.1.1.1.3.3.cmml" xref="S3.p1.9.m3.1.1.1.1.1.1.3.3">𝑎</ci></apply></apply></apply><cn id="S3.p1.9.m3.1.1.3.cmml" type="integer" xref="S3.p1.9.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m3.1c">p(\bm{\phi}_{0}|\bm{\Lambda}_{a})=1</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m3.1d">italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) = 1</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="170" id="S3.F2.g1" src="x2.png" width="345"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Variational framework where each node represents a stochastic variable. The shaded nodes represent observed variables.</figcaption> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Variational Message Passing</span> </h2> <div class="ltx_para ltx_noindent" id="S4.p1"> <p class="ltx_p" id="S4.p1.2">To estimate <math alttext="\{\bm{\phi}_{n}\}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mrow id="S4.p1.1.m1.1.1.1" xref="S4.p1.1.m1.1.1.2.cmml"><mo id="S4.p1.1.m1.1.1.1.2" stretchy="false" xref="S4.p1.1.m1.1.1.2.cmml">{</mo><msub id="S4.p1.1.m1.1.1.1.1" xref="S4.p1.1.m1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p1.1.m1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.p1.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.p1.1.m1.1.1.1.1.3" xref="S4.p1.1.m1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.p1.1.m1.1.1.1.3" stretchy="false" xref="S4.p1.1.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><set id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.1"><apply id="S4.p1.1.m1.1.1.1.1.cmml" xref="S4.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.1.m1.1.1.1.1.1.cmml" xref="S4.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.p1.1.m1.1.1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.p1.1.m1.1.1.1.1.3.cmml" xref="S4.p1.1.m1.1.1.1.1.3">𝑛</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\{\bm{\phi}_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">{ bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math> we seek the posterior, <math alttext="p(\{\bm{\phi}_{n}\}|\{\bm{Z}\},\bm{\Lambda}_{a})" class="ltx_Math" display="inline" id="S4.p1.2.m2.2"><semantics id="S4.p1.2.m2.2a"><mrow id="S4.p1.2.m2.2.2" xref="S4.p1.2.m2.2.2.cmml"><mi id="S4.p1.2.m2.2.2.3" xref="S4.p1.2.m2.2.2.3.cmml">p</mi><mo id="S4.p1.2.m2.2.2.2" xref="S4.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S4.p1.2.m2.2.2.1.1" xref="S4.p1.2.m2.2.2.1.1.1.cmml"><mo id="S4.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S4.p1.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S4.p1.2.m2.2.2.1.1.1" xref="S4.p1.2.m2.2.2.1.1.1.cmml"><mrow id="S4.p1.2.m2.2.2.1.1.1.1.1" xref="S4.p1.2.m2.2.2.1.1.1.1.2.cmml"><mo id="S4.p1.2.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.p1.2.m2.2.2.1.1.1.1.2.cmml">{</mo><msub id="S4.p1.2.m2.2.2.1.1.1.1.1.1" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p1.2.m2.2.2.1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.p1.2.m2.2.2.1.1.1.1.1.1.3" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.p1.2.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.p1.2.m2.2.2.1.1.1.1.2.cmml">}</mo></mrow><mo fence="false" id="S4.p1.2.m2.2.2.1.1.1.4" xref="S4.p1.2.m2.2.2.1.1.1.4.cmml">|</mo><mrow id="S4.p1.2.m2.2.2.1.1.1.3.2" xref="S4.p1.2.m2.2.2.1.1.1.3.3.cmml"><mrow id="S4.p1.2.m2.2.2.1.1.1.2.1.1.2" xref="S4.p1.2.m2.2.2.1.1.1.2.1.1.1.cmml"><mo id="S4.p1.2.m2.2.2.1.1.1.2.1.1.2.1" stretchy="false" xref="S4.p1.2.m2.2.2.1.1.1.2.1.1.1.cmml">{</mo><mi id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml">𝒁</mi><mo id="S4.p1.2.m2.2.2.1.1.1.2.1.1.2.2" stretchy="false" xref="S4.p1.2.m2.2.2.1.1.1.2.1.1.1.cmml">}</mo></mrow><mo id="S4.p1.2.m2.2.2.1.1.1.3.2.3" xref="S4.p1.2.m2.2.2.1.1.1.3.3.cmml">,</mo><msub id="S4.p1.2.m2.2.2.1.1.1.3.2.2" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2.cmml"><mi id="S4.p1.2.m2.2.2.1.1.1.3.2.2.2" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2.2.cmml">𝚲</mi><mi id="S4.p1.2.m2.2.2.1.1.1.3.2.2.3" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2.3.cmml">a</mi></msub></mrow></mrow><mo id="S4.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S4.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.2b"><apply id="S4.p1.2.m2.2.2.cmml" xref="S4.p1.2.m2.2.2"><times id="S4.p1.2.m2.2.2.2.cmml" xref="S4.p1.2.m2.2.2.2"></times><ci id="S4.p1.2.m2.2.2.3.cmml" xref="S4.p1.2.m2.2.2.3">𝑝</ci><apply id="S4.p1.2.m2.2.2.1.1.1.cmml" xref="S4.p1.2.m2.2.2.1.1"><csymbol cd="latexml" id="S4.p1.2.m2.2.2.1.1.1.4.cmml" xref="S4.p1.2.m2.2.2.1.1.1.4">conditional</csymbol><set id="S4.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S4.p1.2.m2.2.2.1.1.1.1.1"><apply id="S4.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml" xref="S4.p1.2.m2.2.2.1.1.1.1.1.1.3">𝑛</ci></apply></set><list id="S4.p1.2.m2.2.2.1.1.1.3.3.cmml" xref="S4.p1.2.m2.2.2.1.1.1.3.2"><set id="S4.p1.2.m2.2.2.1.1.1.2.1.1.1.cmml" xref="S4.p1.2.m2.2.2.1.1.1.2.1.1.2"><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">𝒁</ci></set><apply id="S4.p1.2.m2.2.2.1.1.1.3.2.2.cmml" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.p1.2.m2.2.2.1.1.1.3.2.2.1.cmml" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2">subscript</csymbol><ci id="S4.p1.2.m2.2.2.1.1.1.3.2.2.2.cmml" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2.2">𝚲</ci><ci id="S4.p1.2.m2.2.2.1.1.1.3.2.2.3.cmml" xref="S4.p1.2.m2.2.2.1.1.1.3.2.2.3">𝑎</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.2c">p(\{\bm{\phi}_{n}\}|\{\bm{Z}\},\bm{\Lambda}_{a})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.2d">italic_p ( { bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } | { bold_italic_Z } , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT )</annotation></semantics></math>, however this is intractable. In lieu, we employ the mean field approach and approximate it by a surrogate function,</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="q(\bm{\phi}_{0},\ldots,\bm{\phi}_{N},\bm{\Lambda}_{a})=q(\bm{\Lambda}_{a})% \prod_{n=0}^{N}q(\bm{\phi}_{n})," class="ltx_Math" display="block" id="S4.E6.m1.2"><semantics id="S4.E6.m1.2a"><mrow id="S4.E6.m1.2.2.1" xref="S4.E6.m1.2.2.1.1.cmml"><mrow id="S4.E6.m1.2.2.1.1" xref="S4.E6.m1.2.2.1.1.cmml"><mrow id="S4.E6.m1.2.2.1.1.3" xref="S4.E6.m1.2.2.1.1.3.cmml"><mi id="S4.E6.m1.2.2.1.1.3.5" xref="S4.E6.m1.2.2.1.1.3.5.cmml">q</mi><mo id="S4.E6.m1.2.2.1.1.3.4" xref="S4.E6.m1.2.2.1.1.3.4.cmml">⁢</mo><mrow id="S4.E6.m1.2.2.1.1.3.3.3" xref="S4.E6.m1.2.2.1.1.3.3.4.cmml"><mo id="S4.E6.m1.2.2.1.1.3.3.3.4" stretchy="false" xref="S4.E6.m1.2.2.1.1.3.3.4.cmml">(</mo><msub id="S4.E6.m1.2.2.1.1.1.1.1.1" xref="S4.E6.m1.2.2.1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E6.m1.2.2.1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.E6.m1.2.2.1.1.1.1.1.1.2.cmml">ϕ</mi><mn id="S4.E6.m1.2.2.1.1.1.1.1.1.3" xref="S4.E6.m1.2.2.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.E6.m1.2.2.1.1.3.3.3.5" xref="S4.E6.m1.2.2.1.1.3.3.4.cmml">,</mo><mi id="S4.E6.m1.1.1" mathvariant="normal" xref="S4.E6.m1.1.1.cmml">…</mi><mo id="S4.E6.m1.2.2.1.1.3.3.3.6" xref="S4.E6.m1.2.2.1.1.3.3.4.cmml">,</mo><msub id="S4.E6.m1.2.2.1.1.2.2.2.2" xref="S4.E6.m1.2.2.1.1.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E6.m1.2.2.1.1.2.2.2.2.2" mathvariant="bold-italic" xref="S4.E6.m1.2.2.1.1.2.2.2.2.2.cmml">ϕ</mi><mi id="S4.E6.m1.2.2.1.1.2.2.2.2.3" xref="S4.E6.m1.2.2.1.1.2.2.2.2.3.cmml">N</mi></msub><mo id="S4.E6.m1.2.2.1.1.3.3.3.7" xref="S4.E6.m1.2.2.1.1.3.3.4.cmml">,</mo><msub id="S4.E6.m1.2.2.1.1.3.3.3.3" xref="S4.E6.m1.2.2.1.1.3.3.3.3.cmml"><mi id="S4.E6.m1.2.2.1.1.3.3.3.3.2" xref="S4.E6.m1.2.2.1.1.3.3.3.3.2.cmml">𝚲</mi><mi id="S4.E6.m1.2.2.1.1.3.3.3.3.3" xref="S4.E6.m1.2.2.1.1.3.3.3.3.3.cmml">a</mi></msub><mo id="S4.E6.m1.2.2.1.1.3.3.3.8" stretchy="false" xref="S4.E6.m1.2.2.1.1.3.3.4.cmml">)</mo></mrow></mrow><mo id="S4.E6.m1.2.2.1.1.6" xref="S4.E6.m1.2.2.1.1.6.cmml">=</mo><mrow id="S4.E6.m1.2.2.1.1.5" xref="S4.E6.m1.2.2.1.1.5.cmml"><mi id="S4.E6.m1.2.2.1.1.5.4" xref="S4.E6.m1.2.2.1.1.5.4.cmml">q</mi><mo id="S4.E6.m1.2.2.1.1.5.3" xref="S4.E6.m1.2.2.1.1.5.3.cmml">⁢</mo><mrow id="S4.E6.m1.2.2.1.1.4.1.1" xref="S4.E6.m1.2.2.1.1.4.1.1.1.cmml"><mo id="S4.E6.m1.2.2.1.1.4.1.1.2" stretchy="false" xref="S4.E6.m1.2.2.1.1.4.1.1.1.cmml">(</mo><msub id="S4.E6.m1.2.2.1.1.4.1.1.1" xref="S4.E6.m1.2.2.1.1.4.1.1.1.cmml"><mi id="S4.E6.m1.2.2.1.1.4.1.1.1.2" xref="S4.E6.m1.2.2.1.1.4.1.1.1.2.cmml">𝚲</mi><mi id="S4.E6.m1.2.2.1.1.4.1.1.1.3" xref="S4.E6.m1.2.2.1.1.4.1.1.1.3.cmml">a</mi></msub><mo id="S4.E6.m1.2.2.1.1.4.1.1.3" stretchy="false" xref="S4.E6.m1.2.2.1.1.4.1.1.1.cmml">)</mo></mrow><mo id="S4.E6.m1.2.2.1.1.5.3a" xref="S4.E6.m1.2.2.1.1.5.3.cmml">⁢</mo><mrow id="S4.E6.m1.2.2.1.1.5.2" xref="S4.E6.m1.2.2.1.1.5.2.cmml"><munderover id="S4.E6.m1.2.2.1.1.5.2.2" xref="S4.E6.m1.2.2.1.1.5.2.2.cmml"><mo id="S4.E6.m1.2.2.1.1.5.2.2.2.2" movablelimits="false" xref="S4.E6.m1.2.2.1.1.5.2.2.2.2.cmml">∏</mo><mrow id="S4.E6.m1.2.2.1.1.5.2.2.2.3" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.cmml"><mi id="S4.E6.m1.2.2.1.1.5.2.2.2.3.2" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.2.cmml">n</mi><mo id="S4.E6.m1.2.2.1.1.5.2.2.2.3.1" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.1.cmml">=</mo><mn id="S4.E6.m1.2.2.1.1.5.2.2.2.3.3" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.3.cmml">0</mn></mrow><mi id="S4.E6.m1.2.2.1.1.5.2.2.3" xref="S4.E6.m1.2.2.1.1.5.2.2.3.cmml">N</mi></munderover><mrow id="S4.E6.m1.2.2.1.1.5.2.1" xref="S4.E6.m1.2.2.1.1.5.2.1.cmml"><mi id="S4.E6.m1.2.2.1.1.5.2.1.3" xref="S4.E6.m1.2.2.1.1.5.2.1.3.cmml">q</mi><mo id="S4.E6.m1.2.2.1.1.5.2.1.2" xref="S4.E6.m1.2.2.1.1.5.2.1.2.cmml">⁢</mo><mrow id="S4.E6.m1.2.2.1.1.5.2.1.1.1" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.cmml"><mo id="S4.E6.m1.2.2.1.1.5.2.1.1.1.2" stretchy="false" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.cmml">(</mo><msub id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.2" mathvariant="bold-italic" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.3" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.E6.m1.2.2.1.1.5.2.1.1.1.3" stretchy="false" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="S4.E6.m1.2.2.1.2" xref="S4.E6.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.2b"><apply id="S4.E6.m1.2.2.1.1.cmml" xref="S4.E6.m1.2.2.1"><eq id="S4.E6.m1.2.2.1.1.6.cmml" xref="S4.E6.m1.2.2.1.1.6"></eq><apply id="S4.E6.m1.2.2.1.1.3.cmml" xref="S4.E6.m1.2.2.1.1.3"><times id="S4.E6.m1.2.2.1.1.3.4.cmml" xref="S4.E6.m1.2.2.1.1.3.4"></times><ci id="S4.E6.m1.2.2.1.1.3.5.cmml" xref="S4.E6.m1.2.2.1.1.3.5">𝑞</ci><vector id="S4.E6.m1.2.2.1.1.3.3.4.cmml" xref="S4.E6.m1.2.2.1.1.3.3.3"><apply id="S4.E6.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E6.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E6.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E6.m1.2.2.1.1.1.1.1.1.2">bold-italic-ϕ</ci><cn id="S4.E6.m1.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E6.m1.2.2.1.1.1.1.1.1.3">0</cn></apply><ci id="S4.E6.m1.1.1.cmml" xref="S4.E6.m1.1.1">…</ci><apply id="S4.E6.m1.2.2.1.1.2.2.2.2.cmml" xref="S4.E6.m1.2.2.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.2.2.2.2.1.cmml" xref="S4.E6.m1.2.2.1.1.2.2.2.2">subscript</csymbol><ci id="S4.E6.m1.2.2.1.1.2.2.2.2.2.cmml" xref="S4.E6.m1.2.2.1.1.2.2.2.2.2">bold-italic-ϕ</ci><ci id="S4.E6.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S4.E6.m1.2.2.1.1.2.2.2.2.3">𝑁</ci></apply><apply id="S4.E6.m1.2.2.1.1.3.3.3.3.cmml" xref="S4.E6.m1.2.2.1.1.3.3.3.3"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.3.3.3.3.1.cmml" xref="S4.E6.m1.2.2.1.1.3.3.3.3">subscript</csymbol><ci id="S4.E6.m1.2.2.1.1.3.3.3.3.2.cmml" xref="S4.E6.m1.2.2.1.1.3.3.3.3.2">𝚲</ci><ci id="S4.E6.m1.2.2.1.1.3.3.3.3.3.cmml" xref="S4.E6.m1.2.2.1.1.3.3.3.3.3">𝑎</ci></apply></vector></apply><apply id="S4.E6.m1.2.2.1.1.5.cmml" xref="S4.E6.m1.2.2.1.1.5"><times id="S4.E6.m1.2.2.1.1.5.3.cmml" xref="S4.E6.m1.2.2.1.1.5.3"></times><ci id="S4.E6.m1.2.2.1.1.5.4.cmml" xref="S4.E6.m1.2.2.1.1.5.4">𝑞</ci><apply id="S4.E6.m1.2.2.1.1.4.1.1.1.cmml" xref="S4.E6.m1.2.2.1.1.4.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.4.1.1.1.1.cmml" xref="S4.E6.m1.2.2.1.1.4.1.1">subscript</csymbol><ci id="S4.E6.m1.2.2.1.1.4.1.1.1.2.cmml" xref="S4.E6.m1.2.2.1.1.4.1.1.1.2">𝚲</ci><ci id="S4.E6.m1.2.2.1.1.4.1.1.1.3.cmml" xref="S4.E6.m1.2.2.1.1.4.1.1.1.3">𝑎</ci></apply><apply id="S4.E6.m1.2.2.1.1.5.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2"><apply id="S4.E6.m1.2.2.1.1.5.2.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.5.2.2.1.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2">superscript</csymbol><apply id="S4.E6.m1.2.2.1.1.5.2.2.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.5.2.2.2.1.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2">subscript</csymbol><csymbol cd="latexml" id="S4.E6.m1.2.2.1.1.5.2.2.2.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2.2.2">product</csymbol><apply id="S4.E6.m1.2.2.1.1.5.2.2.2.3.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3"><eq id="S4.E6.m1.2.2.1.1.5.2.2.2.3.1.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.1"></eq><ci id="S4.E6.m1.2.2.1.1.5.2.2.2.3.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.2">𝑛</ci><cn id="S4.E6.m1.2.2.1.1.5.2.2.2.3.3.cmml" type="integer" xref="S4.E6.m1.2.2.1.1.5.2.2.2.3.3">0</cn></apply></apply><ci id="S4.E6.m1.2.2.1.1.5.2.2.3.cmml" xref="S4.E6.m1.2.2.1.1.5.2.2.3">𝑁</ci></apply><apply id="S4.E6.m1.2.2.1.1.5.2.1.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1"><times id="S4.E6.m1.2.2.1.1.5.2.1.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1.2"></times><ci id="S4.E6.m1.2.2.1.1.5.2.1.3.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1.3">𝑞</ci><apply id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.1.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1">subscript</csymbol><ci id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.2.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.3.cmml" xref="S4.E6.m1.2.2.1.1.5.2.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.2c">q(\bm{\phi}_{0},\ldots,\bm{\phi}_{N},\bm{\Lambda}_{a})=q(\bm{\Lambda}_{a})% \prod_{n=0}^{N}q(\bm{\phi}_{n}),</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.2d">italic_q ( bold_italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) = italic_q ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_q ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n 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">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p1.4">obtained by minimizing the Kullback-Leibler (KL) divergence w.r.t. the true posterior. Following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#bib.bib22" title="">22</a>]</cite>, the surrogates <math alttext="q(\bm{\phi}_{n})" class="ltx_Math" display="inline" id="S4.p1.3.m1.1"><semantics id="S4.p1.3.m1.1a"><mrow id="S4.p1.3.m1.1.1" xref="S4.p1.3.m1.1.1.cmml"><mi id="S4.p1.3.m1.1.1.3" xref="S4.p1.3.m1.1.1.3.cmml">q</mi><mo id="S4.p1.3.m1.1.1.2" xref="S4.p1.3.m1.1.1.2.cmml">⁢</mo><mrow id="S4.p1.3.m1.1.1.1.1" xref="S4.p1.3.m1.1.1.1.1.1.cmml"><mo id="S4.p1.3.m1.1.1.1.1.2" stretchy="false" xref="S4.p1.3.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.p1.3.m1.1.1.1.1.1" xref="S4.p1.3.m1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p1.3.m1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.p1.3.m1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.p1.3.m1.1.1.1.1.1.3" xref="S4.p1.3.m1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.p1.3.m1.1.1.1.1.3" stretchy="false" xref="S4.p1.3.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m1.1b"><apply id="S4.p1.3.m1.1.1.cmml" xref="S4.p1.3.m1.1.1"><times id="S4.p1.3.m1.1.1.2.cmml" xref="S4.p1.3.m1.1.1.2"></times><ci id="S4.p1.3.m1.1.1.3.cmml" xref="S4.p1.3.m1.1.1.3">𝑞</ci><apply id="S4.p1.3.m1.1.1.1.1.1.cmml" xref="S4.p1.3.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.3.m1.1.1.1.1.1.1.cmml" xref="S4.p1.3.m1.1.1.1.1">subscript</csymbol><ci id="S4.p1.3.m1.1.1.1.1.1.2.cmml" xref="S4.p1.3.m1.1.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.p1.3.m1.1.1.1.1.1.3.cmml" xref="S4.p1.3.m1.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m1.1c">q(\bm{\phi}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m1.1d">italic_q ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="q(\bm{\Lambda}_{a})" class="ltx_Math" display="inline" id="S4.p1.4.m2.1"><semantics id="S4.p1.4.m2.1a"><mrow id="S4.p1.4.m2.1.1" xref="S4.p1.4.m2.1.1.cmml"><mi id="S4.p1.4.m2.1.1.3" xref="S4.p1.4.m2.1.1.3.cmml">q</mi><mo id="S4.p1.4.m2.1.1.2" xref="S4.p1.4.m2.1.1.2.cmml">⁢</mo><mrow id="S4.p1.4.m2.1.1.1.1" xref="S4.p1.4.m2.1.1.1.1.1.cmml"><mo id="S4.p1.4.m2.1.1.1.1.2" stretchy="false" xref="S4.p1.4.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.p1.4.m2.1.1.1.1.1" xref="S4.p1.4.m2.1.1.1.1.1.cmml"><mi id="S4.p1.4.m2.1.1.1.1.1.2" xref="S4.p1.4.m2.1.1.1.1.1.2.cmml">𝚲</mi><mi id="S4.p1.4.m2.1.1.1.1.1.3" xref="S4.p1.4.m2.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.p1.4.m2.1.1.1.1.3" stretchy="false" xref="S4.p1.4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.4.m2.1b"><apply id="S4.p1.4.m2.1.1.cmml" xref="S4.p1.4.m2.1.1"><times id="S4.p1.4.m2.1.1.2.cmml" xref="S4.p1.4.m2.1.1.2"></times><ci id="S4.p1.4.m2.1.1.3.cmml" xref="S4.p1.4.m2.1.1.3">𝑞</ci><apply id="S4.p1.4.m2.1.1.1.1.1.cmml" xref="S4.p1.4.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.4.m2.1.1.1.1.1.1.cmml" xref="S4.p1.4.m2.1.1.1.1">subscript</csymbol><ci id="S4.p1.4.m2.1.1.1.1.1.2.cmml" xref="S4.p1.4.m2.1.1.1.1.1.2">𝚲</ci><ci id="S4.p1.4.m2.1.1.1.1.1.3.cmml" xref="S4.p1.4.m2.1.1.1.1.1.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m2.1c">q(\bm{\Lambda}_{a})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m2.1d">italic_q ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT )</annotation></semantics></math> can be expressed as</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="\ln(q(\bm{\phi}_{n}))=\sum_{k=1}^{N_{\text{radars}}}\left(\ln(p(\bm{Z}_{n}^{(k% )})|\bm{\phi}_{n})\right)\\ +\mathbb{E}_{\backslash\bm{\phi}_{n}}[\ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{% \Lambda}_{a}))]+\mathbb{E}_{\backslash\bm{\phi}_{n}}[\ln(p(\bm{\phi}_{n+1}|\bm% {\phi}_{n},\bm{\Lambda}_{a}))]\\ +\text{const.}" class="ltx_Math" display="block" id="S4.E7.m1.74"><semantics id="S4.E7.m1.74a"><mtable displaystyle="true" id="S4.E7.m1.74.74.8" rowspacing="0pt" xref="S4.E7.m1.70.70.4.cmml"><mtr id="S4.E7.m1.74.74.8a" xref="S4.E7.m1.70.70.4.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E7.m1.74.74.8b" xref="S4.E7.m1.70.70.4.cmml"><mrow id="S4.E7.m1.72.72.6.68.28.28" xref="S4.E7.m1.70.70.4.cmml"><mrow id="S4.E7.m1.71.71.5.67.27.27.27.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.cmml">ln</mi><mo id="S4.E7.m1.71.71.5.67.27.27.27.1a" xref="S4.E7.m1.70.70.4.cmml">⁡</mo><mrow id="S4.E7.m1.71.71.5.67.27.27.27.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.2.2.2.2.2.2" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.71.71.5.67.27.27.27.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.3.3.3.3.3.3" xref="S4.E7.m1.3.3.3.3.3.3.cmml">q</mi><mo id="S4.E7.m1.71.71.5.67.27.27.27.1.1.1.2" xref="S4.E7.m1.70.70.4.cmml">⁢</mo><mrow id="S4.E7.m1.71.71.5.67.27.27.27.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.4.4.4.4.4.4" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><msub id="S4.E7.m1.71.71.5.67.27.27.27.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.5.5.5.5.5.5" mathvariant="bold-italic" xref="S4.E7.m1.5.5.5.5.5.5.cmml">ϕ</mi><mi id="S4.E7.m1.6.6.6.6.6.6.1" xref="S4.E7.m1.6.6.6.6.6.6.1.cmml">n</mi></msub><mo id="S4.E7.m1.7.7.7.7.7.7" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.8.8.8.8.8.8" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.9.9.9.9.9.9" rspace="0.111em" xref="S4.E7.m1.9.9.9.9.9.9.cmml">=</mo><mrow id="S4.E7.m1.72.72.6.68.28.28.28" xref="S4.E7.m1.70.70.4.cmml"><munderover id="S4.E7.m1.72.72.6.68.28.28.28.2" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.10.10.10.10.10.10" movablelimits="false" rspace="0em" xref="S4.E7.m1.10.10.10.10.10.10.cmml">∑</mo><mrow id="S4.E7.m1.11.11.11.11.11.11.1" xref="S4.E7.m1.11.11.11.11.11.11.1.cmml"><mi id="S4.E7.m1.11.11.11.11.11.11.1.2" xref="S4.E7.m1.11.11.11.11.11.11.1.2.cmml">k</mi><mo id="S4.E7.m1.11.11.11.11.11.11.1.1" xref="S4.E7.m1.11.11.11.11.11.11.1.1.cmml">=</mo><mn id="S4.E7.m1.11.11.11.11.11.11.1.3" xref="S4.E7.m1.11.11.11.11.11.11.1.3.cmml">1</mn></mrow><msub id="S4.E7.m1.12.12.12.12.12.12.1" xref="S4.E7.m1.12.12.12.12.12.12.1.cmml"><mi id="S4.E7.m1.12.12.12.12.12.12.1.2" xref="S4.E7.m1.12.12.12.12.12.12.1.2.cmml">N</mi><mtext id="S4.E7.m1.12.12.12.12.12.12.1.3" xref="S4.E7.m1.12.12.12.12.12.12.1.3a.cmml">radars</mtext></msub></munderover><mrow id="S4.E7.m1.72.72.6.68.28.28.28.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.13.13.13.13.13.13" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.14.14.14.14.14.14" xref="S4.E7.m1.14.14.14.14.14.14.cmml">ln</mi><mo id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1a" xref="S4.E7.m1.70.70.4.cmml">⁡</mo><mrow id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.15.15.15.15.15.15" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mrow id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.16.16.16.16.16.16" xref="S4.E7.m1.16.16.16.16.16.16.cmml">p</mi><mo id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1.1.1.2" xref="S4.E7.m1.70.70.4.cmml">⁢</mo><mrow id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.17.17.17.17.17.17" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><msubsup id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.18.18.18.18.18.18" xref="S4.E7.m1.18.18.18.18.18.18.cmml">𝒁</mi><mi id="S4.E7.m1.19.19.19.19.19.19.1" xref="S4.E7.m1.19.19.19.19.19.19.1.cmml">n</mi><mrow id="S4.E7.m1.20.20.20.20.20.20.1.3" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.20.20.20.20.20.20.1.3.1" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mi id="S4.E7.m1.20.20.20.20.20.20.1.1" xref="S4.E7.m1.20.20.20.20.20.20.1.1.cmml">k</mi><mo id="S4.E7.m1.20.20.20.20.20.20.1.3.2" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></msubsup><mo id="S4.E7.m1.21.21.21.21.21.21" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo fence="false" id="S4.E7.m1.22.22.22.22.22.22" xref="S4.E7.m1.22.22.22.22.22.22.cmml">|</mo><msub id="S4.E7.m1.72.72.6.68.28.28.28.1.1.1.1.1.1.2" xref="S4.E7.m1.70.70.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.23.23.23.23.23.23" mathvariant="bold-italic" xref="S4.E7.m1.23.23.23.23.23.23.cmml">ϕ</mi><mi id="S4.E7.m1.24.24.24.24.24.24.1" xref="S4.E7.m1.24.24.24.24.24.24.1.cmml">n</mi></msub></mrow><mo id="S4.E7.m1.25.25.25.25.25.25" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.26.26.26.26.26.26" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E7.m1.74.74.8c" xref="S4.E7.m1.70.70.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E7.m1.74.74.8d" xref="S4.E7.m1.70.70.4.cmml"><mrow id="S4.E7.m1.74.74.8.70.40.40" xref="S4.E7.m1.70.70.4.cmml"><mrow id="S4.E7.m1.73.73.7.69.39.39.39" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.73.73.7.69.39.39.39a" xref="S4.E7.m1.70.70.4.cmml">+</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1" xref="S4.E7.m1.70.70.4.cmml"><msub id="S4.E7.m1.73.73.7.69.39.39.39.1.3" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.28.28.28.2.2.2" xref="S4.E7.m1.28.28.28.2.2.2.cmml">𝔼</mi><mrow id="S4.E7.m1.29.29.29.3.3.3.1" xref="S4.E7.m1.29.29.29.3.3.3.1.cmml"><mi id="S4.E7.m1.29.29.29.3.3.3.1.2" xref="S4.E7.m1.29.29.29.3.3.3.1.2.cmml"></mi><mo id="S4.E7.m1.29.29.29.3.3.3.1.1" lspace="0.222em" rspace="0.222em" xref="S4.E7.m1.29.29.29.3.3.3.1.1.cmml">\</mo><msub id="S4.E7.m1.29.29.29.3.3.3.1.3" xref="S4.E7.m1.29.29.29.3.3.3.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.29.29.29.3.3.3.1.3.2" mathvariant="bold-italic" xref="S4.E7.m1.29.29.29.3.3.3.1.3.2.cmml">ϕ</mi><mi id="S4.E7.m1.29.29.29.3.3.3.1.3.3" xref="S4.E7.m1.29.29.29.3.3.3.1.3.3.cmml">n</mi></msub></mrow></msub><mo id="S4.E7.m1.73.73.7.69.39.39.39.1.2" xref="S4.E7.m1.70.70.4.cmml">⁢</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.30.30.30.4.4.4" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">[</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.31.31.31.5.5.5" xref="S4.E7.m1.31.31.31.5.5.5.cmml">ln</mi><mo id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1a" xref="S4.E7.m1.70.70.4.cmml">⁡</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.32.32.32.6.6.6" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.33.33.33.7.7.7" xref="S4.E7.m1.33.33.33.7.7.7.cmml">p</mi><mo id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.2" xref="S4.E7.m1.70.70.4.cmml">⁢</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.34.34.34.8.8.8" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><msub id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.1.1.1.3" xref="S4.E7.m1.70.70.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.35.35.35.9.9.9" mathvariant="bold-italic" xref="S4.E7.m1.35.35.35.9.9.9.cmml">ϕ</mi><mi id="S4.E7.m1.36.36.36.10.10.10.1" xref="S4.E7.m1.36.36.36.10.10.10.1.cmml">n</mi></msub><mo fence="false" id="S4.E7.m1.37.37.37.11.11.11" xref="S4.E7.m1.37.37.37.11.11.11.cmml">|</mo><mrow id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E7.m1.70.70.4.cmml"><msub id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.38.38.38.12.12.12" mathvariant="bold-italic" xref="S4.E7.m1.38.38.38.12.12.12.cmml">ϕ</mi><mrow id="S4.E7.m1.39.39.39.13.13.13.1" xref="S4.E7.m1.39.39.39.13.13.13.1.cmml"><mi id="S4.E7.m1.39.39.39.13.13.13.1.2" xref="S4.E7.m1.39.39.39.13.13.13.1.2.cmml">n</mi><mo id="S4.E7.m1.39.39.39.13.13.13.1.1" xref="S4.E7.m1.39.39.39.13.13.13.1.1.cmml">−</mo><mn id="S4.E7.m1.39.39.39.13.13.13.1.3" xref="S4.E7.m1.39.39.39.13.13.13.1.3.cmml">1</mn></mrow></msub><mo id="S4.E7.m1.40.40.40.14.14.14" xref="S4.E7.m1.70.70.4.cmml">,</mo><msub id="S4.E7.m1.73.73.7.69.39.39.39.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.41.41.41.15.15.15" xref="S4.E7.m1.41.41.41.15.15.15.cmml">𝚲</mi><mi id="S4.E7.m1.42.42.42.16.16.16.1" xref="S4.E7.m1.42.42.42.16.16.16.1.cmml">a</mi></msub></mrow></mrow><mo id="S4.E7.m1.43.43.43.17.17.17" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.44.44.44.18.18.18" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.45.45.45.19.19.19" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">]</mo></mrow></mrow></mrow><mo id="S4.E7.m1.46.46.46.20.20.20" xref="S4.E7.m1.70.70.4.cmml">+</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40" xref="S4.E7.m1.70.70.4.cmml"><msub id="S4.E7.m1.74.74.8.70.40.40.40.3" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.47.47.47.21.21.21" xref="S4.E7.m1.47.47.47.21.21.21.cmml">𝔼</mi><mrow id="S4.E7.m1.48.48.48.22.22.22.1" xref="S4.E7.m1.48.48.48.22.22.22.1.cmml"><mi id="S4.E7.m1.48.48.48.22.22.22.1.2" xref="S4.E7.m1.48.48.48.22.22.22.1.2.cmml"></mi><mo id="S4.E7.m1.48.48.48.22.22.22.1.1" lspace="0.222em" rspace="0.222em" xref="S4.E7.m1.48.48.48.22.22.22.1.1.cmml">\</mo><msub id="S4.E7.m1.48.48.48.22.22.22.1.3" xref="S4.E7.m1.48.48.48.22.22.22.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.48.48.48.22.22.22.1.3.2" mathvariant="bold-italic" xref="S4.E7.m1.48.48.48.22.22.22.1.3.2.cmml">ϕ</mi><mi id="S4.E7.m1.48.48.48.22.22.22.1.3.3" xref="S4.E7.m1.48.48.48.22.22.22.1.3.3.cmml">n</mi></msub></mrow></msub><mo id="S4.E7.m1.74.74.8.70.40.40.40.2" xref="S4.E7.m1.70.70.4.cmml">⁢</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.49.49.49.23.23.23" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">[</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.50.50.50.24.24.24" xref="S4.E7.m1.50.50.50.24.24.24.cmml">ln</mi><mo id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1a" xref="S4.E7.m1.70.70.4.cmml">⁡</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.51.51.51.25.25.25" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.52.52.52.26.26.26" xref="S4.E7.m1.52.52.52.26.26.26.cmml">p</mi><mo id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.2" xref="S4.E7.m1.70.70.4.cmml">⁢</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.53.53.53.27.27.27" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">(</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><msub id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.1.1.1.3" xref="S4.E7.m1.70.70.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.54.54.54.28.28.28" mathvariant="bold-italic" xref="S4.E7.m1.54.54.54.28.28.28.cmml">ϕ</mi><mrow id="S4.E7.m1.55.55.55.29.29.29.1" xref="S4.E7.m1.55.55.55.29.29.29.1.cmml"><mi id="S4.E7.m1.55.55.55.29.29.29.1.2" xref="S4.E7.m1.55.55.55.29.29.29.1.2.cmml">n</mi><mo id="S4.E7.m1.55.55.55.29.29.29.1.1" xref="S4.E7.m1.55.55.55.29.29.29.1.1.cmml">+</mo><mn id="S4.E7.m1.55.55.55.29.29.29.1.3" xref="S4.E7.m1.55.55.55.29.29.29.1.3.cmml">1</mn></mrow></msub><mo fence="false" id="S4.E7.m1.56.56.56.30.30.30" xref="S4.E7.m1.56.56.56.30.30.30.cmml">|</mo><mrow id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E7.m1.70.70.4.cmml"><msub id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.70.70.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E7.m1.57.57.57.31.31.31" mathvariant="bold-italic" xref="S4.E7.m1.57.57.57.31.31.31.cmml">ϕ</mi><mi id="S4.E7.m1.58.58.58.32.32.32.1" xref="S4.E7.m1.58.58.58.32.32.32.1.cmml">n</mi></msub><mo id="S4.E7.m1.59.59.59.33.33.33" xref="S4.E7.m1.70.70.4.cmml">,</mo><msub id="S4.E7.m1.74.74.8.70.40.40.40.1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E7.m1.70.70.4.cmml"><mi id="S4.E7.m1.60.60.60.34.34.34" xref="S4.E7.m1.60.60.60.34.34.34.cmml">𝚲</mi><mi id="S4.E7.m1.61.61.61.35.35.35.1" xref="S4.E7.m1.61.61.61.35.35.35.1.cmml">a</mi></msub></mrow></mrow><mo id="S4.E7.m1.62.62.62.36.36.36" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.63.63.63.37.37.37" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.64.64.64.38.38.38" stretchy="false" xref="S4.E7.m1.70.70.4.cmml">]</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E7.m1.74.74.8e" xref="S4.E7.m1.70.70.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E7.m1.74.74.8f" xref="S4.E7.m1.70.70.4.cmml"><mrow id="S4.E7.m1.66.66.66.2.2" xref="S4.E7.m1.70.70.4.cmml"><mo id="S4.E7.m1.66.66.66.2.2a" xref="S4.E7.m1.70.70.4.cmml">+</mo><mtext id="S4.E7.m1.66.66.66.2.2.2" xref="S4.E7.m1.66.66.66.2.2.2a.cmml">const.</mtext></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E7.m1.74b"><apply id="S4.E7.m1.70.70.4.cmml" xref="S4.E7.m1.74.74.8"><eq id="S4.E7.m1.9.9.9.9.9.9.cmml" xref="S4.E7.m1.9.9.9.9.9.9"></eq><apply id="S4.E7.m1.67.67.1.1.2.cmml" xref="S4.E7.m1.74.74.8"><ln id="S4.E7.m1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.1.1"></ln><apply id="S4.E7.m1.67.67.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><times id="S4.E7.m1.67.67.1.1.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"></times><ci id="S4.E7.m1.3.3.3.3.3.3.cmml" xref="S4.E7.m1.3.3.3.3.3.3">𝑞</ci><apply id="S4.E7.m1.67.67.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.67.67.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.5.5.5.5.5.5.cmml" xref="S4.E7.m1.5.5.5.5.5.5">bold-italic-ϕ</ci><ci id="S4.E7.m1.6.6.6.6.6.6.1.cmml" xref="S4.E7.m1.6.6.6.6.6.6.1">𝑛</ci></apply></apply></apply><apply id="S4.E7.m1.70.70.4.4.cmml" xref="S4.E7.m1.74.74.8"><plus id="S4.E7.m1.27.27.27.1.1.1.cmml" xref="S4.E7.m1.74.74.8"></plus><apply id="S4.E7.m1.68.68.2.2.1.cmml" xref="S4.E7.m1.74.74.8"><apply id="S4.E7.m1.68.68.2.2.1.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.68.68.2.2.1.2.1.cmml" xref="S4.E7.m1.74.74.8">superscript</csymbol><apply id="S4.E7.m1.68.68.2.2.1.2.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.68.68.2.2.1.2.2.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><sum id="S4.E7.m1.10.10.10.10.10.10.cmml" xref="S4.E7.m1.10.10.10.10.10.10"></sum><apply id="S4.E7.m1.11.11.11.11.11.11.1.cmml" xref="S4.E7.m1.11.11.11.11.11.11.1"><eq id="S4.E7.m1.11.11.11.11.11.11.1.1.cmml" xref="S4.E7.m1.11.11.11.11.11.11.1.1"></eq><ci id="S4.E7.m1.11.11.11.11.11.11.1.2.cmml" xref="S4.E7.m1.11.11.11.11.11.11.1.2">𝑘</ci><cn id="S4.E7.m1.11.11.11.11.11.11.1.3.cmml" type="integer" xref="S4.E7.m1.11.11.11.11.11.11.1.3">1</cn></apply></apply><apply id="S4.E7.m1.12.12.12.12.12.12.1.cmml" xref="S4.E7.m1.12.12.12.12.12.12.1"><csymbol cd="ambiguous" id="S4.E7.m1.12.12.12.12.12.12.1.1.cmml" xref="S4.E7.m1.12.12.12.12.12.12.1">subscript</csymbol><ci id="S4.E7.m1.12.12.12.12.12.12.1.2.cmml" xref="S4.E7.m1.12.12.12.12.12.12.1.2">𝑁</ci><ci id="S4.E7.m1.12.12.12.12.12.12.1.3a.cmml" xref="S4.E7.m1.12.12.12.12.12.12.1.3"><mtext id="S4.E7.m1.12.12.12.12.12.12.1.3.cmml" mathsize="50%" xref="S4.E7.m1.12.12.12.12.12.12.1.3">radars</mtext></ci></apply></apply><apply id="S4.E7.m1.68.68.2.2.1.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"><ln id="S4.E7.m1.14.14.14.14.14.14.cmml" xref="S4.E7.m1.14.14.14.14.14.14"></ln><apply id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="latexml" id="S4.E7.m1.22.22.22.22.22.22.cmml" xref="S4.E7.m1.22.22.22.22.22.22">conditional</csymbol><apply id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><times id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"></times><ci id="S4.E7.m1.16.16.16.16.16.16.cmml" xref="S4.E7.m1.16.16.16.16.16.16">𝑝</ci><apply id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8">superscript</csymbol><apply id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.18.18.18.18.18.18.cmml" xref="S4.E7.m1.18.18.18.18.18.18">𝒁</ci><ci id="S4.E7.m1.19.19.19.19.19.19.1.cmml" xref="S4.E7.m1.19.19.19.19.19.19.1">𝑛</ci></apply><ci id="S4.E7.m1.20.20.20.20.20.20.1.1.cmml" xref="S4.E7.m1.20.20.20.20.20.20.1.1">𝑘</ci></apply></apply><apply id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.68.68.2.2.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.23.23.23.23.23.23.cmml" xref="S4.E7.m1.23.23.23.23.23.23">bold-italic-ϕ</ci><ci id="S4.E7.m1.24.24.24.24.24.24.1.cmml" xref="S4.E7.m1.24.24.24.24.24.24.1">𝑛</ci></apply></apply></apply></apply><apply id="S4.E7.m1.69.69.3.3.2.cmml" xref="S4.E7.m1.74.74.8"><times id="S4.E7.m1.69.69.3.3.2.2.cmml" xref="S4.E7.m1.74.74.8"></times><apply id="S4.E7.m1.69.69.3.3.2.3.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.69.69.3.3.2.3.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.28.28.28.2.2.2.cmml" xref="S4.E7.m1.28.28.28.2.2.2">𝔼</ci><apply id="S4.E7.m1.29.29.29.3.3.3.1.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1"><ci id="S4.E7.m1.29.29.29.3.3.3.1.1.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1.1">\</ci><csymbol cd="latexml" id="S4.E7.m1.29.29.29.3.3.3.1.2.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1.2">absent</csymbol><apply id="S4.E7.m1.29.29.29.3.3.3.1.3.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1.3"><csymbol cd="ambiguous" id="S4.E7.m1.29.29.29.3.3.3.1.3.1.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1.3">subscript</csymbol><ci id="S4.E7.m1.29.29.29.3.3.3.1.3.2.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1.3.2">bold-italic-ϕ</ci><ci id="S4.E7.m1.29.29.29.3.3.3.1.3.3.cmml" xref="S4.E7.m1.29.29.29.3.3.3.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E7.m1.69.69.3.3.2.1.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="latexml" id="S4.E7.m1.69.69.3.3.2.1.2.1.cmml" xref="S4.E7.m1.74.74.8">delimited-[]</csymbol><apply id="S4.E7.m1.69.69.3.3.2.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"><ln id="S4.E7.m1.31.31.31.5.5.5.cmml" xref="S4.E7.m1.31.31.31.5.5.5"></ln><apply id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><times id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"></times><ci id="S4.E7.m1.33.33.33.7.7.7.cmml" xref="S4.E7.m1.33.33.33.7.7.7">𝑝</ci><apply id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="latexml" id="S4.E7.m1.37.37.37.11.11.11.cmml" xref="S4.E7.m1.37.37.37.11.11.11">conditional</csymbol><apply id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.35.35.35.9.9.9.cmml" xref="S4.E7.m1.35.35.35.9.9.9">bold-italic-ϕ</ci><ci id="S4.E7.m1.36.36.36.10.10.10.1.cmml" xref="S4.E7.m1.36.36.36.10.10.10.1">𝑛</ci></apply><list id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E7.m1.74.74.8"><apply id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.38.38.38.12.12.12.cmml" xref="S4.E7.m1.38.38.38.12.12.12">bold-italic-ϕ</ci><apply id="S4.E7.m1.39.39.39.13.13.13.1.cmml" xref="S4.E7.m1.39.39.39.13.13.13.1"><minus id="S4.E7.m1.39.39.39.13.13.13.1.1.cmml" xref="S4.E7.m1.39.39.39.13.13.13.1.1"></minus><ci id="S4.E7.m1.39.39.39.13.13.13.1.2.cmml" xref="S4.E7.m1.39.39.39.13.13.13.1.2">𝑛</ci><cn id="S4.E7.m1.39.39.39.13.13.13.1.3.cmml" type="integer" xref="S4.E7.m1.39.39.39.13.13.13.1.3">1</cn></apply></apply><apply id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.69.69.3.3.2.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.41.41.41.15.15.15.cmml" xref="S4.E7.m1.41.41.41.15.15.15">𝚲</ci><ci id="S4.E7.m1.42.42.42.16.16.16.1.cmml" xref="S4.E7.m1.42.42.42.16.16.16.1">𝑎</ci></apply></list></apply></apply></apply></apply></apply><apply id="S4.E7.m1.70.70.4.4.3.cmml" xref="S4.E7.m1.74.74.8"><times id="S4.E7.m1.70.70.4.4.3.2.cmml" xref="S4.E7.m1.74.74.8"></times><apply id="S4.E7.m1.70.70.4.4.3.3.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.70.70.4.4.3.3.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.47.47.47.21.21.21.cmml" xref="S4.E7.m1.47.47.47.21.21.21">𝔼</ci><apply id="S4.E7.m1.48.48.48.22.22.22.1.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1"><ci id="S4.E7.m1.48.48.48.22.22.22.1.1.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1.1">\</ci><csymbol cd="latexml" id="S4.E7.m1.48.48.48.22.22.22.1.2.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1.2">absent</csymbol><apply id="S4.E7.m1.48.48.48.22.22.22.1.3.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1.3"><csymbol cd="ambiguous" id="S4.E7.m1.48.48.48.22.22.22.1.3.1.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1.3">subscript</csymbol><ci id="S4.E7.m1.48.48.48.22.22.22.1.3.2.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1.3.2">bold-italic-ϕ</ci><ci id="S4.E7.m1.48.48.48.22.22.22.1.3.3.cmml" xref="S4.E7.m1.48.48.48.22.22.22.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E7.m1.70.70.4.4.3.1.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="latexml" id="S4.E7.m1.70.70.4.4.3.1.2.1.cmml" xref="S4.E7.m1.74.74.8">delimited-[]</csymbol><apply id="S4.E7.m1.70.70.4.4.3.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"><ln id="S4.E7.m1.50.50.50.24.24.24.cmml" xref="S4.E7.m1.50.50.50.24.24.24"></ln><apply id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><times id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.74.74.8"></times><ci id="S4.E7.m1.52.52.52.26.26.26.cmml" xref="S4.E7.m1.52.52.52.26.26.26">𝑝</ci><apply id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="latexml" id="S4.E7.m1.56.56.56.30.30.30.cmml" xref="S4.E7.m1.56.56.56.30.30.30">conditional</csymbol><apply id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.54.54.54.28.28.28.cmml" xref="S4.E7.m1.54.54.54.28.28.28">bold-italic-ϕ</ci><apply id="S4.E7.m1.55.55.55.29.29.29.1.cmml" xref="S4.E7.m1.55.55.55.29.29.29.1"><plus id="S4.E7.m1.55.55.55.29.29.29.1.1.cmml" xref="S4.E7.m1.55.55.55.29.29.29.1.1"></plus><ci id="S4.E7.m1.55.55.55.29.29.29.1.2.cmml" xref="S4.E7.m1.55.55.55.29.29.29.1.2">𝑛</ci><cn id="S4.E7.m1.55.55.55.29.29.29.1.3.cmml" type="integer" xref="S4.E7.m1.55.55.55.29.29.29.1.3">1</cn></apply></apply><list id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E7.m1.74.74.8"><apply id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.57.57.57.31.31.31.cmml" xref="S4.E7.m1.57.57.57.31.31.31">bold-italic-ϕ</ci><ci id="S4.E7.m1.58.58.58.32.32.32.1.cmml" xref="S4.E7.m1.58.58.58.32.32.32.1">𝑛</ci></apply><apply id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E7.m1.74.74.8"><csymbol cd="ambiguous" id="S4.E7.m1.70.70.4.4.3.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E7.m1.74.74.8">subscript</csymbol><ci id="S4.E7.m1.60.60.60.34.34.34.cmml" xref="S4.E7.m1.60.60.60.34.34.34">𝚲</ci><ci id="S4.E7.m1.61.61.61.35.35.35.1.cmml" xref="S4.E7.m1.61.61.61.35.35.35.1">𝑎</ci></apply></list></apply></apply></apply></apply></apply><ci id="S4.E7.m1.66.66.66.2.2.2a.cmml" xref="S4.E7.m1.66.66.66.2.2.2"><mtext id="S4.E7.m1.66.66.66.2.2.2.cmml" xref="S4.E7.m1.66.66.66.2.2.2">const.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.74c">\ln(q(\bm{\phi}_{n}))=\sum_{k=1}^{N_{\text{radars}}}\left(\ln(p(\bm{Z}_{n}^{(k% )})|\bm{\phi}_{n})\right)\\ +\mathbb{E}_{\backslash\bm{\phi}_{n}}[\ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{% \Lambda}_{a}))]+\mathbb{E}_{\backslash\bm{\phi}_{n}}[\ln(p(\bm{\phi}_{n+1}|\bm% {\phi}_{n},\bm{\Lambda}_{a}))]\\ +\text{const.}</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.74d">start_ROW start_CELL roman_ln ( italic_q ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) = ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT radars end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( roman_ln ( italic_p ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) end_CELL end_ROW start_ROW start_CELL + blackboard_E start_POSTSUBSCRIPT \ bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ roman_ln ( italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) ] + blackboard_E start_POSTSUBSCRIPT \ bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ roman_ln ( italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) ] end_CELL end_ROW start_ROW start_CELL + const. 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">(7)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <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="\ln q(\bm{\Lambda}_{a})=\sum_{n=1}^{N}\mathbb{E}_{\backslash\bm{\Lambda}_{a}}[% \ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}))]\\ +\ln(p(\bm{\Lambda}_{a}))+\text{const.}" class="ltx_Math" display="block" id="S4.E8.m1.45"><semantics id="S4.E8.m1.45a"><mtable displaystyle="true" id="S4.E8.m1.45.45.6" rowspacing="0pt" xref="S4.E8.m1.42.42.3.cmml"><mtr id="S4.E8.m1.45.45.6a" xref="S4.E8.m1.42.42.3.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E8.m1.45.45.6b" xref="S4.E8.m1.42.42.3.cmml"><mrow id="S4.E8.m1.44.44.5.41.30.30" xref="S4.E8.m1.42.42.3.cmml"><mrow id="S4.E8.m1.43.43.4.40.29.29.29" xref="S4.E8.m1.42.42.3.cmml"><mrow id="S4.E8.m1.43.43.4.40.29.29.29.3" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.1.1.1.1.1.1" xref="S4.E8.m1.1.1.1.1.1.1.cmml">ln</mi><mo id="S4.E8.m1.43.43.4.40.29.29.29.3a" lspace="0.167em" xref="S4.E8.m1.42.42.3.cmml">⁡</mo><mi id="S4.E8.m1.2.2.2.2.2.2" xref="S4.E8.m1.2.2.2.2.2.2.cmml">q</mi></mrow><mo id="S4.E8.m1.43.43.4.40.29.29.29.2" xref="S4.E8.m1.42.42.3.cmml">⁢</mo><mrow id="S4.E8.m1.43.43.4.40.29.29.29.1.1" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.3.3.3.3.3.3" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">(</mo><msub id="S4.E8.m1.43.43.4.40.29.29.29.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.4.4.4.4.4.4" xref="S4.E8.m1.4.4.4.4.4.4.cmml">𝚲</mi><mi id="S4.E8.m1.5.5.5.5.5.5.1" xref="S4.E8.m1.5.5.5.5.5.5.1.cmml">a</mi></msub><mo id="S4.E8.m1.6.6.6.6.6.6" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">)</mo></mrow></mrow><mo id="S4.E8.m1.7.7.7.7.7.7" rspace="0.111em" xref="S4.E8.m1.7.7.7.7.7.7.cmml">=</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30" xref="S4.E8.m1.42.42.3.cmml"><munderover id="S4.E8.m1.44.44.5.41.30.30.30.2" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.8.8.8.8.8.8" movablelimits="false" xref="S4.E8.m1.8.8.8.8.8.8.cmml">∑</mo><mrow id="S4.E8.m1.9.9.9.9.9.9.1" xref="S4.E8.m1.9.9.9.9.9.9.1.cmml"><mi id="S4.E8.m1.9.9.9.9.9.9.1.2" xref="S4.E8.m1.9.9.9.9.9.9.1.2.cmml">n</mi><mo id="S4.E8.m1.9.9.9.9.9.9.1.1" xref="S4.E8.m1.9.9.9.9.9.9.1.1.cmml">=</mo><mn id="S4.E8.m1.9.9.9.9.9.9.1.3" xref="S4.E8.m1.9.9.9.9.9.9.1.3.cmml">1</mn></mrow><mi id="S4.E8.m1.10.10.10.10.10.10.1" xref="S4.E8.m1.10.10.10.10.10.10.1.cmml">N</mi></munderover><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1" xref="S4.E8.m1.42.42.3.cmml"><msub id="S4.E8.m1.44.44.5.41.30.30.30.1.3" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.11.11.11.11.11.11" xref="S4.E8.m1.11.11.11.11.11.11.cmml">𝔼</mi><mrow id="S4.E8.m1.12.12.12.12.12.12.1" xref="S4.E8.m1.12.12.12.12.12.12.1.cmml"><mi id="S4.E8.m1.12.12.12.12.12.12.1.2" xref="S4.E8.m1.12.12.12.12.12.12.1.2.cmml"></mi><mo id="S4.E8.m1.12.12.12.12.12.12.1.1" lspace="0.222em" rspace="0.222em" xref="S4.E8.m1.12.12.12.12.12.12.1.1.cmml">\</mo><msub id="S4.E8.m1.12.12.12.12.12.12.1.3" xref="S4.E8.m1.12.12.12.12.12.12.1.3.cmml"><mi id="S4.E8.m1.12.12.12.12.12.12.1.3.2" xref="S4.E8.m1.12.12.12.12.12.12.1.3.2.cmml">𝚲</mi><mi id="S4.E8.m1.12.12.12.12.12.12.1.3.3" xref="S4.E8.m1.12.12.12.12.12.12.1.3.3.cmml">a</mi></msub></mrow></msub><mo id="S4.E8.m1.44.44.5.41.30.30.30.1.2" xref="S4.E8.m1.42.42.3.cmml">⁢</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.13.13.13.13.13.13" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">[</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.14.14.14.14.14.14" xref="S4.E8.m1.14.14.14.14.14.14.cmml">ln</mi><mo id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1a" xref="S4.E8.m1.42.42.3.cmml">⁡</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.15.15.15.15.15.15" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">(</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.16.16.16.16.16.16" xref="S4.E8.m1.16.16.16.16.16.16.cmml">p</mi><mo id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.2" xref="S4.E8.m1.42.42.3.cmml">⁢</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.17.17.17.17.17.17" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">(</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><msub id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.1.1.1.3" xref="S4.E8.m1.42.42.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E8.m1.18.18.18.18.18.18" mathvariant="bold-italic" xref="S4.E8.m1.18.18.18.18.18.18.cmml">ϕ</mi><mi id="S4.E8.m1.19.19.19.19.19.19.1" xref="S4.E8.m1.19.19.19.19.19.19.1.cmml">n</mi></msub><mo fence="false" id="S4.E8.m1.20.20.20.20.20.20" xref="S4.E8.m1.20.20.20.20.20.20.cmml">|</mo><mrow id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E8.m1.42.42.3.cmml"><msub id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E8.m1.21.21.21.21.21.21" mathvariant="bold-italic" xref="S4.E8.m1.21.21.21.21.21.21.cmml">ϕ</mi><mrow id="S4.E8.m1.22.22.22.22.22.22.1" xref="S4.E8.m1.22.22.22.22.22.22.1.cmml"><mi id="S4.E8.m1.22.22.22.22.22.22.1.2" xref="S4.E8.m1.22.22.22.22.22.22.1.2.cmml">n</mi><mo id="S4.E8.m1.22.22.22.22.22.22.1.1" xref="S4.E8.m1.22.22.22.22.22.22.1.1.cmml">−</mo><mn id="S4.E8.m1.22.22.22.22.22.22.1.3" xref="S4.E8.m1.22.22.22.22.22.22.1.3.cmml">1</mn></mrow></msub><mo id="S4.E8.m1.23.23.23.23.23.23" xref="S4.E8.m1.42.42.3.cmml">,</mo><msub id="S4.E8.m1.44.44.5.41.30.30.30.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.24.24.24.24.24.24" xref="S4.E8.m1.24.24.24.24.24.24.cmml">𝚲</mi><mi id="S4.E8.m1.25.25.25.25.25.25.1" xref="S4.E8.m1.25.25.25.25.25.25.1.cmml">a</mi></msub></mrow></mrow><mo id="S4.E8.m1.26.26.26.26.26.26" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">)</mo></mrow></mrow><mo id="S4.E8.m1.27.27.27.27.27.27" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">)</mo></mrow></mrow><mo id="S4.E8.m1.28.28.28.28.28.28" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">]</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E8.m1.45.45.6c" xref="S4.E8.m1.42.42.3.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E8.m1.45.45.6d" xref="S4.E8.m1.42.42.3.cmml"><mrow id="S4.E8.m1.45.45.6.42.12.12" xref="S4.E8.m1.42.42.3.cmml"><mrow id="S4.E8.m1.45.45.6.42.12.12.12" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.45.45.6.42.12.12.12a" rspace="0.167em" xref="S4.E8.m1.42.42.3.cmml">+</mo><mrow id="S4.E8.m1.45.45.6.42.12.12.12.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.30.30.30.2.2.2" xref="S4.E8.m1.30.30.30.2.2.2.cmml">ln</mi><mo id="S4.E8.m1.45.45.6.42.12.12.12.1.1a" xref="S4.E8.m1.42.42.3.cmml">⁡</mo><mrow id="S4.E8.m1.45.45.6.42.12.12.12.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.31.31.31.3.3.3" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">(</mo><mrow id="S4.E8.m1.45.45.6.42.12.12.12.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.32.32.32.4.4.4" xref="S4.E8.m1.32.32.32.4.4.4.cmml">p</mi><mo id="S4.E8.m1.45.45.6.42.12.12.12.1.1.1.1.2" xref="S4.E8.m1.42.42.3.cmml">⁢</mo><mrow id="S4.E8.m1.45.45.6.42.12.12.12.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mo id="S4.E8.m1.33.33.33.5.5.5" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">(</mo><msub id="S4.E8.m1.45.45.6.42.12.12.12.1.1.1.1.1.1.1" xref="S4.E8.m1.42.42.3.cmml"><mi id="S4.E8.m1.34.34.34.6.6.6" xref="S4.E8.m1.34.34.34.6.6.6.cmml">𝚲</mi><mi id="S4.E8.m1.35.35.35.7.7.7.1" xref="S4.E8.m1.35.35.35.7.7.7.1.cmml">a</mi></msub><mo id="S4.E8.m1.36.36.36.8.8.8" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">)</mo></mrow></mrow><mo id="S4.E8.m1.37.37.37.9.9.9" stretchy="false" xref="S4.E8.m1.42.42.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E8.m1.38.38.38.10.10.10" xref="S4.E8.m1.42.42.3.cmml">+</mo><mtext id="S4.E8.m1.39.39.39.11.11.11" xref="S4.E8.m1.39.39.39.11.11.11a.cmml">const.</mtext></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E8.m1.45b"><apply id="S4.E8.m1.42.42.3.cmml" xref="S4.E8.m1.45.45.6"><eq id="S4.E8.m1.7.7.7.7.7.7.cmml" xref="S4.E8.m1.7.7.7.7.7.7"></eq><apply id="S4.E8.m1.40.40.1.1.cmml" xref="S4.E8.m1.45.45.6"><times id="S4.E8.m1.40.40.1.1.2.cmml" xref="S4.E8.m1.45.45.6"></times><apply id="S4.E8.m1.40.40.1.1.3.cmml" xref="S4.E8.m1.45.45.6"><ln id="S4.E8.m1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1"></ln><ci id="S4.E8.m1.2.2.2.2.2.2.cmml" xref="S4.E8.m1.2.2.2.2.2.2">𝑞</ci></apply><apply id="S4.E8.m1.40.40.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.40.40.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><ci id="S4.E8.m1.4.4.4.4.4.4.cmml" xref="S4.E8.m1.4.4.4.4.4.4">𝚲</ci><ci id="S4.E8.m1.5.5.5.5.5.5.1.cmml" xref="S4.E8.m1.5.5.5.5.5.5.1">𝑎</ci></apply></apply><apply id="S4.E8.m1.42.42.3.3.cmml" xref="S4.E8.m1.45.45.6"><plus id="S4.E8.m1.29.29.29.1.1.1.cmml" xref="S4.E8.m1.45.45.6"></plus><apply id="S4.E8.m1.41.41.2.2.1.cmml" xref="S4.E8.m1.45.45.6"><apply id="S4.E8.m1.41.41.2.2.1.2.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.41.41.2.2.1.2.1.cmml" xref="S4.E8.m1.45.45.6">superscript</csymbol><apply id="S4.E8.m1.41.41.2.2.1.2.2.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.41.41.2.2.1.2.2.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><sum id="S4.E8.m1.8.8.8.8.8.8.cmml" xref="S4.E8.m1.8.8.8.8.8.8"></sum><apply id="S4.E8.m1.9.9.9.9.9.9.1.cmml" xref="S4.E8.m1.9.9.9.9.9.9.1"><eq id="S4.E8.m1.9.9.9.9.9.9.1.1.cmml" xref="S4.E8.m1.9.9.9.9.9.9.1.1"></eq><ci id="S4.E8.m1.9.9.9.9.9.9.1.2.cmml" xref="S4.E8.m1.9.9.9.9.9.9.1.2">𝑛</ci><cn id="S4.E8.m1.9.9.9.9.9.9.1.3.cmml" type="integer" xref="S4.E8.m1.9.9.9.9.9.9.1.3">1</cn></apply></apply><ci id="S4.E8.m1.10.10.10.10.10.10.1.cmml" xref="S4.E8.m1.10.10.10.10.10.10.1">𝑁</ci></apply><apply id="S4.E8.m1.41.41.2.2.1.1.cmml" xref="S4.E8.m1.45.45.6"><times id="S4.E8.m1.41.41.2.2.1.1.2.cmml" xref="S4.E8.m1.45.45.6"></times><apply id="S4.E8.m1.41.41.2.2.1.1.3.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.41.41.2.2.1.1.3.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><ci id="S4.E8.m1.11.11.11.11.11.11.cmml" xref="S4.E8.m1.11.11.11.11.11.11">𝔼</ci><apply id="S4.E8.m1.12.12.12.12.12.12.1.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1"><ci id="S4.E8.m1.12.12.12.12.12.12.1.1.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1.1">\</ci><csymbol cd="latexml" id="S4.E8.m1.12.12.12.12.12.12.1.2.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1.2">absent</csymbol><apply id="S4.E8.m1.12.12.12.12.12.12.1.3.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1.3"><csymbol cd="ambiguous" id="S4.E8.m1.12.12.12.12.12.12.1.3.1.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1.3">subscript</csymbol><ci id="S4.E8.m1.12.12.12.12.12.12.1.3.2.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1.3.2">𝚲</ci><ci id="S4.E8.m1.12.12.12.12.12.12.1.3.3.cmml" xref="S4.E8.m1.12.12.12.12.12.12.1.3.3">𝑎</ci></apply></apply></apply><apply id="S4.E8.m1.41.41.2.2.1.1.1.2.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="latexml" id="S4.E8.m1.41.41.2.2.1.1.1.2.1.cmml" xref="S4.E8.m1.45.45.6">delimited-[]</csymbol><apply id="S4.E8.m1.41.41.2.2.1.1.1.1.1.2.cmml" xref="S4.E8.m1.45.45.6"><ln id="S4.E8.m1.14.14.14.14.14.14.cmml" xref="S4.E8.m1.14.14.14.14.14.14"></ln><apply id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6"><times id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E8.m1.45.45.6"></times><ci id="S4.E8.m1.16.16.16.16.16.16.cmml" xref="S4.E8.m1.16.16.16.16.16.16">𝑝</ci><apply id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="latexml" id="S4.E8.m1.20.20.20.20.20.20.cmml" xref="S4.E8.m1.20.20.20.20.20.20">conditional</csymbol><apply id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><ci id="S4.E8.m1.18.18.18.18.18.18.cmml" xref="S4.E8.m1.18.18.18.18.18.18">bold-italic-ϕ</ci><ci id="S4.E8.m1.19.19.19.19.19.19.1.cmml" xref="S4.E8.m1.19.19.19.19.19.19.1">𝑛</ci></apply><list id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E8.m1.45.45.6"><apply id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><ci id="S4.E8.m1.21.21.21.21.21.21.cmml" xref="S4.E8.m1.21.21.21.21.21.21">bold-italic-ϕ</ci><apply id="S4.E8.m1.22.22.22.22.22.22.1.cmml" xref="S4.E8.m1.22.22.22.22.22.22.1"><minus id="S4.E8.m1.22.22.22.22.22.22.1.1.cmml" xref="S4.E8.m1.22.22.22.22.22.22.1.1"></minus><ci id="S4.E8.m1.22.22.22.22.22.22.1.2.cmml" xref="S4.E8.m1.22.22.22.22.22.22.1.2">𝑛</ci><cn id="S4.E8.m1.22.22.22.22.22.22.1.3.cmml" type="integer" xref="S4.E8.m1.22.22.22.22.22.22.1.3">1</cn></apply></apply><apply id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.41.41.2.2.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><ci id="S4.E8.m1.24.24.24.24.24.24.cmml" xref="S4.E8.m1.24.24.24.24.24.24">𝚲</ci><ci id="S4.E8.m1.25.25.25.25.25.25.1.cmml" xref="S4.E8.m1.25.25.25.25.25.25.1">𝑎</ci></apply></list></apply></apply></apply></apply></apply></apply><apply id="S4.E8.m1.42.42.3.3.2.2.cmml" xref="S4.E8.m1.45.45.6"><ln id="S4.E8.m1.30.30.30.2.2.2.cmml" xref="S4.E8.m1.30.30.30.2.2.2"></ln><apply id="S4.E8.m1.42.42.3.3.2.1.1.1.cmml" xref="S4.E8.m1.45.45.6"><times id="S4.E8.m1.42.42.3.3.2.1.1.1.2.cmml" xref="S4.E8.m1.45.45.6"></times><ci id="S4.E8.m1.32.32.32.4.4.4.cmml" xref="S4.E8.m1.32.32.32.4.4.4">𝑝</ci><apply id="S4.E8.m1.42.42.3.3.2.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6"><csymbol cd="ambiguous" id="S4.E8.m1.42.42.3.3.2.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.45.45.6">subscript</csymbol><ci id="S4.E8.m1.34.34.34.6.6.6.cmml" xref="S4.E8.m1.34.34.34.6.6.6">𝚲</ci><ci id="S4.E8.m1.35.35.35.7.7.7.1.cmml" xref="S4.E8.m1.35.35.35.7.7.7.1">𝑎</ci></apply></apply></apply><ci id="S4.E8.m1.39.39.39.11.11.11a.cmml" xref="S4.E8.m1.39.39.39.11.11.11"><mtext id="S4.E8.m1.39.39.39.11.11.11.cmml" xref="S4.E8.m1.39.39.39.11.11.11">const.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.45c">\ln q(\bm{\Lambda}_{a})=\sum_{n=1}^{N}\mathbb{E}_{\backslash\bm{\Lambda}_{a}}[% \ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}))]\\ +\ln(p(\bm{\Lambda}_{a}))+\text{const.}</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.45d">start_ROW start_CELL roman_ln italic_q ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT blackboard_E start_POSTSUBSCRIPT \ bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ roman_ln ( italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) ] end_CELL end_ROW start_ROW start_CELL + roman_ln ( italic_p ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) + const. 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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p1.5">The surrogates are calculated recursively using a message passing scheme. Each term in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E7" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">7</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E8" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">8</span></a>) can be viewed as a message passed from the nodes connected to the node under consideration, i.e., the nodes in the neighborhood denoted <math alttext="\mathcal{N}_{\text{node}}" class="ltx_Math" display="inline" id="S4.p1.5.m1.1"><semantics id="S4.p1.5.m1.1a"><msub id="S4.p1.5.m1.1.1" xref="S4.p1.5.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p1.5.m1.1.1.2" xref="S4.p1.5.m1.1.1.2.cmml">𝒩</mi><mtext id="S4.p1.5.m1.1.1.3" xref="S4.p1.5.m1.1.1.3a.cmml">node</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.p1.5.m1.1b"><apply id="S4.p1.5.m1.1.1.cmml" xref="S4.p1.5.m1.1.1"><csymbol cd="ambiguous" id="S4.p1.5.m1.1.1.1.cmml" xref="S4.p1.5.m1.1.1">subscript</csymbol><ci id="S4.p1.5.m1.1.1.2.cmml" xref="S4.p1.5.m1.1.1.2">𝒩</ci><ci id="S4.p1.5.m1.1.1.3a.cmml" xref="S4.p1.5.m1.1.1.3"><mtext id="S4.p1.5.m1.1.1.3.cmml" mathsize="70%" xref="S4.p1.5.m1.1.1.3">node</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m1.1c">\mathcal{N}_{\text{node}}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m1.1d">caligraphic_N start_POSTSUBSCRIPT node end_POSTSUBSCRIPT</annotation></semantics></math>. The messages are in the following calculated one by one in the order they appear in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E7" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">7</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E8" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">8</span></a>).</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">The message <math alttext="\epsilon^{(\bm{Z}^{(k)}_{n}\to\bm{\phi}_{n})}" class="ltx_Math" display="inline" id="S4.p2.1.m1.2"><semantics id="S4.p2.1.m1.2a"><msup id="S4.p2.1.m1.2.3" xref="S4.p2.1.m1.2.3.cmml"><mi id="S4.p2.1.m1.2.3.2" xref="S4.p2.1.m1.2.3.2.cmml">ϵ</mi><mrow id="S4.p2.1.m1.2.2.2.2" xref="S4.p2.1.m1.2.2.2.2.1.cmml"><mo id="S4.p2.1.m1.2.2.2.2.2" stretchy="false" xref="S4.p2.1.m1.2.2.2.2.1.cmml">(</mo><mrow id="S4.p2.1.m1.2.2.2.2.1" xref="S4.p2.1.m1.2.2.2.2.1.cmml"><msubsup id="S4.p2.1.m1.2.2.2.2.1.2" xref="S4.p2.1.m1.2.2.2.2.1.2.cmml"><mi id="S4.p2.1.m1.2.2.2.2.1.2.2.2" xref="S4.p2.1.m1.2.2.2.2.1.2.2.2.cmml">𝒁</mi><mi id="S4.p2.1.m1.2.2.2.2.1.2.3" xref="S4.p2.1.m1.2.2.2.2.1.2.3.cmml">n</mi><mrow id="S4.p2.1.m1.1.1.1.1.1.3" xref="S4.p2.1.m1.2.2.2.2.1.2.cmml"><mo id="S4.p2.1.m1.1.1.1.1.1.3.1" stretchy="false" xref="S4.p2.1.m1.2.2.2.2.1.2.cmml">(</mo><mi id="S4.p2.1.m1.1.1.1.1.1.1" xref="S4.p2.1.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S4.p2.1.m1.1.1.1.1.1.3.2" stretchy="false" xref="S4.p2.1.m1.2.2.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="S4.p2.1.m1.2.2.2.2.1.1" stretchy="false" xref="S4.p2.1.m1.2.2.2.2.1.1.cmml">→</mo><msub id="S4.p2.1.m1.2.2.2.2.1.3" xref="S4.p2.1.m1.2.2.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.1.m1.2.2.2.2.1.3.2" mathvariant="bold-italic" xref="S4.p2.1.m1.2.2.2.2.1.3.2.cmml">ϕ</mi><mi id="S4.p2.1.m1.2.2.2.2.1.3.3" xref="S4.p2.1.m1.2.2.2.2.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.p2.1.m1.2.2.2.2.3" stretchy="false" xref="S4.p2.1.m1.2.2.2.2.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.2b"><apply id="S4.p2.1.m1.2.3.cmml" xref="S4.p2.1.m1.2.3"><csymbol cd="ambiguous" id="S4.p2.1.m1.2.3.1.cmml" xref="S4.p2.1.m1.2.3">superscript</csymbol><ci id="S4.p2.1.m1.2.3.2.cmml" xref="S4.p2.1.m1.2.3.2">italic-ϵ</ci><apply id="S4.p2.1.m1.2.2.2.2.1.cmml" xref="S4.p2.1.m1.2.2.2.2"><ci id="S4.p2.1.m1.2.2.2.2.1.1.cmml" xref="S4.p2.1.m1.2.2.2.2.1.1">→</ci><apply id="S4.p2.1.m1.2.2.2.2.1.2.cmml" xref="S4.p2.1.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.p2.1.m1.2.2.2.2.1.2.1.cmml" xref="S4.p2.1.m1.2.2.2.2.1.2">subscript</csymbol><apply id="S4.p2.1.m1.2.2.2.2.1.2.2.cmml" xref="S4.p2.1.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.p2.1.m1.2.2.2.2.1.2.2.1.cmml" xref="S4.p2.1.m1.2.2.2.2.1.2">superscript</csymbol><ci id="S4.p2.1.m1.2.2.2.2.1.2.2.2.cmml" xref="S4.p2.1.m1.2.2.2.2.1.2.2.2">𝒁</ci><ci id="S4.p2.1.m1.1.1.1.1.1.1.cmml" xref="S4.p2.1.m1.1.1.1.1.1.1">𝑘</ci></apply><ci id="S4.p2.1.m1.2.2.2.2.1.2.3.cmml" xref="S4.p2.1.m1.2.2.2.2.1.2.3">𝑛</ci></apply><apply id="S4.p2.1.m1.2.2.2.2.1.3.cmml" xref="S4.p2.1.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.p2.1.m1.2.2.2.2.1.3.1.cmml" xref="S4.p2.1.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S4.p2.1.m1.2.2.2.2.1.3.2.cmml" xref="S4.p2.1.m1.2.2.2.2.1.3.2">bold-italic-ϕ</ci><ci id="S4.p2.1.m1.2.2.2.2.1.3.3.cmml" xref="S4.p2.1.m1.2.2.2.2.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.2c">\epsilon^{(\bm{Z}^{(k)}_{n}\to\bm{\phi}_{n})}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.2d">italic_ϵ start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math> can be expanded as,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E9"> <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="\ln\!\left(p\!\left(\bm{Z}^{(k)}_{n}|\bm{\phi}_{n}\right)\!\right)\!\propto\\ -\!\left\langle\!\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}|\bm{% \Lambda}_{Z}|\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\!\right\rangle." class="ltx_Math" display="block" id="S4.E9.m1.46"><semantics id="S4.E9.m1.46a"><mtable displaystyle="true" id="S4.E9.m1.46.46.3" rowspacing="0pt"><mtr id="S4.E9.m1.46.46.3a"><mtd class="ltx_align_left" columnalign="left" id="S4.E9.m1.46.46.3b"><mrow id="S4.E9.m1.45.45.2.44.14.14"><mrow id="S4.E9.m1.45.45.2.44.14.14.14.1"><mpadded width="0.748em"><mi id="S4.E9.m1.1.1.1.1.1.1" xref="S4.E9.m1.1.1.1.1.1.1.cmml">ln</mi></mpadded><mo id="S4.E9.m1.45.45.2.44.14.14.14.1a" xref="S4.E9.m1.44.44.1.1.1.cmml">⁡</mo><mrow id="S4.E9.m1.45.45.2.44.14.14.14.1.1"><mo id="S4.E9.m1.2.2.2.2.2.2" xref="S4.E9.m1.44.44.1.1.1.cmml">(</mo><mrow id="S4.E9.m1.45.45.2.44.14.14.14.1.1.1"><mpadded width="0.333em"><mi id="S4.E9.m1.3.3.3.3.3.3" xref="S4.E9.m1.3.3.3.3.3.3.cmml">p</mi></mpadded><mo id="S4.E9.m1.45.45.2.44.14.14.14.1.1.1.2" xref="S4.E9.m1.44.44.1.1.1.cmml">⁢</mo><mrow id="S4.E9.m1.45.45.2.44.14.14.14.1.1.1.1.1"><mo id="S4.E9.m1.4.4.4.4.4.4" xref="S4.E9.m1.44.44.1.1.1.cmml">(</mo><mrow id="S4.E9.m1.45.45.2.44.14.14.14.1.1.1.1.1.1"><msubsup id="S4.E9.m1.45.45.2.44.14.14.14.1.1.1.1.1.1.1"><mi id="S4.E9.m1.5.5.5.5.5.5" xref="S4.E9.m1.5.5.5.5.5.5.cmml">𝒁</mi><mi id="S4.E9.m1.7.7.7.7.7.7.1" xref="S4.E9.m1.7.7.7.7.7.7.1.cmml">n</mi><mrow id="S4.E9.m1.6.6.6.6.6.6.1.3"><mo id="S4.E9.m1.6.6.6.6.6.6.1.3.1" stretchy="false">(</mo><mi id="S4.E9.m1.6.6.6.6.6.6.1.1" xref="S4.E9.m1.6.6.6.6.6.6.1.1.cmml">k</mi><mo id="S4.E9.m1.6.6.6.6.6.6.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo fence="false" id="S4.E9.m1.8.8.8.8.8.8" xref="S4.E9.m1.8.8.8.8.8.8.cmml">|</mo><msub id="S4.E9.m1.45.45.2.44.14.14.14.1.1.1.1.1.1.2"><mi class="ltx_mathvariant_bold-italic" id="S4.E9.m1.9.9.9.9.9.9" mathvariant="bold-italic" xref="S4.E9.m1.9.9.9.9.9.9.cmml">ϕ</mi><mi id="S4.E9.m1.10.10.10.10.10.10.1" xref="S4.E9.m1.10.10.10.10.10.10.1.cmml">n</mi></msub></mrow><mpadded width="0.288em"><mo id="S4.E9.m1.11.11.11.11.11.11" xref="S4.E9.m1.44.44.1.1.1.cmml">)</mo></mpadded></mrow></mrow><mo id="S4.E9.m1.12.12.12.12.12.12" xref="S4.E9.m1.44.44.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E9.m1.13.13.13.13.13.13" lspace="0.108em" xref="S4.E9.m1.13.13.13.13.13.13.cmml">∝</mo><mi id="S4.E9.m1.45.45.2.44.14.14.15" xref="S4.E9.m1.44.44.1.1.1.cmml"></mi></mrow></mtd></mtr><mtr id="S4.E9.m1.46.46.3c"><mtd class="ltx_align_right" columnalign="right" id="S4.E9.m1.46.46.3d"><mrow id="S4.E9.m1.46.46.3.45.31.31.31"><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1"><mo id="S4.E9.m1.46.46.3.45.31.31.31.1a" xref="S4.E9.m1.44.44.1.1.1.cmml">−</mo><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3"><mpadded width="0.219em"><mo id="S4.E9.m1.15.15.15.2.2.2" xref="S4.E9.m1.44.44.1.1.1.cmml">⟨</mo></mpadded><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1"><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1.1"><msubsup id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1.1.3"><mi id="S4.E9.m1.16.16.16.3.3.3" xref="S4.E9.m1.16.16.16.3.3.3.cmml">𝑺</mi><mi id="S4.E9.m1.17.17.17.4.4.4.1" xref="S4.E9.m1.17.17.17.4.4.4.1.cmml">n</mi><mrow id="S4.E9.m1.18.18.18.5.5.5.1.3"><mo id="S4.E9.m1.18.18.18.5.5.5.1.3.1" stretchy="false">(</mo><mi id="S4.E9.m1.18.18.18.5.5.5.1.1" xref="S4.E9.m1.18.18.18.5.5.5.1.1.cmml">k</mi><mo id="S4.E9.m1.18.18.18.5.5.5.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1.1.2" xref="S4.E9.m1.44.44.1.1.1.cmml">⁢</mo><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1.1.1.1"><mo id="S4.E9.m1.19.19.19.6.6.6" stretchy="false" xref="S4.E9.m1.44.44.1.1.1.cmml">(</mo><msub id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E9.m1.20.20.20.7.7.7" mathvariant="bold-italic" xref="S4.E9.m1.20.20.20.7.7.7.cmml">ϕ</mi><mi id="S4.E9.m1.21.21.21.8.8.8.1" xref="S4.E9.m1.21.21.21.8.8.8.1.cmml">n</mi></msub><mo id="S4.E9.m1.22.22.22.9.9.9" stretchy="false" xref="S4.E9.m1.44.44.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E9.m1.23.23.23.10.10.10" xref="S4.E9.m1.23.23.23.10.10.10.cmml">−</mo><msubsup id="S4.E9.m1.46.46.3.45.31.31.31.1.1.1.1.2"><mi id="S4.E9.m1.24.24.24.11.11.11" xref="S4.E9.m1.24.24.24.11.11.11.cmml">𝒁</mi><mi id="S4.E9.m1.25.25.25.12.12.12.1" xref="S4.E9.m1.25.25.25.12.12.12.1.cmml">n</mi><mrow id="S4.E9.m1.26.26.26.13.13.13.1.3"><mo id="S4.E9.m1.26.26.26.13.13.13.1.3.1" stretchy="false">(</mo><mi id="S4.E9.m1.26.26.26.13.13.13.1.1" xref="S4.E9.m1.26.26.26.13.13.13.1.1.cmml">k</mi><mo id="S4.E9.m1.26.26.26.13.13.13.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E9.m1.27.27.27.14.14.14" stretchy="false" xref="S4.E9.m1.44.44.1.1.1.cmml">|</mo><msub id="S4.E9.m1.46.46.3.45.31.31.31.1.2.2.2"><mi id="S4.E9.m1.28.28.28.15.15.15" xref="S4.E9.m1.28.28.28.15.15.15.cmml">𝚲</mi><mi id="S4.E9.m1.29.29.29.16.16.16.1" xref="S4.E9.m1.29.29.29.16.16.16.1.cmml">Z</mi></msub><mo id="S4.E9.m1.30.30.30.17.17.17" stretchy="false" xref="S4.E9.m1.44.44.1.1.1.cmml">|</mo><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3"><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3.1"><msubsup id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3.1.3"><mi id="S4.E9.m1.31.31.31.18.18.18" xref="S4.E9.m1.31.31.31.18.18.18.cmml">𝑺</mi><mi id="S4.E9.m1.32.32.32.19.19.19.1" xref="S4.E9.m1.32.32.32.19.19.19.1.cmml">n</mi><mrow id="S4.E9.m1.33.33.33.20.20.20.1.3"><mo id="S4.E9.m1.33.33.33.20.20.20.1.3.1" stretchy="false">(</mo><mi id="S4.E9.m1.33.33.33.20.20.20.1.1" xref="S4.E9.m1.33.33.33.20.20.20.1.1.cmml">k</mi><mo id="S4.E9.m1.33.33.33.20.20.20.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3.1.2" xref="S4.E9.m1.44.44.1.1.1.cmml">⁢</mo><mrow id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3.1.1.1"><mo id="S4.E9.m1.34.34.34.21.21.21" stretchy="false" xref="S4.E9.m1.44.44.1.1.1.cmml">(</mo><msub id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E9.m1.35.35.35.22.22.22" mathvariant="bold-italic" xref="S4.E9.m1.35.35.35.22.22.22.cmml">ϕ</mi><mi id="S4.E9.m1.36.36.36.23.23.23.1" xref="S4.E9.m1.36.36.36.23.23.23.1.cmml">n</mi></msub><mo id="S4.E9.m1.37.37.37.24.24.24" stretchy="false" xref="S4.E9.m1.44.44.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E9.m1.38.38.38.25.25.25" xref="S4.E9.m1.38.38.38.25.25.25.cmml">−</mo><msubsup id="S4.E9.m1.46.46.3.45.31.31.31.1.3.3.3.2"><mi id="S4.E9.m1.39.39.39.26.26.26" xref="S4.E9.m1.39.39.39.26.26.26.cmml">𝒁</mi><mi id="S4.E9.m1.40.40.40.27.27.27.1" xref="S4.E9.m1.40.40.40.27.27.27.1.cmml">n</mi><mrow id="S4.E9.m1.41.41.41.28.28.28.1.3"><mo id="S4.E9.m1.41.41.41.28.28.28.1.3.1" stretchy="false">(</mo><mi id="S4.E9.m1.41.41.41.28.28.28.1.1" xref="S4.E9.m1.41.41.41.28.28.28.1.1.cmml">k</mi><mo id="S4.E9.m1.41.41.41.28.28.28.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E9.m1.42.42.42.29.29.29" xref="S4.E9.m1.44.44.1.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.E9.m1.43.43.43.30.30.30" lspace="0em" xref="S4.E9.m1.44.44.1.1.1.cmml">.</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E9.m1.46b"><apply id="S4.E9.m1.44.44.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="latexml" id="S4.E9.m1.13.13.13.13.13.13.cmml" xref="S4.E9.m1.13.13.13.13.13.13">proportional-to</csymbol><apply id="S4.E9.m1.44.44.1.1.1.1.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><ln id="S4.E9.m1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.1.1.1.1.1.1"></ln><apply id="S4.E9.m1.44.44.1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><times id="S4.E9.m1.44.44.1.1.1.1.1.1.1.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"></times><ci id="S4.E9.m1.3.3.3.3.3.3.cmml" xref="S4.E9.m1.3.3.3.3.3.3">𝑝</ci><apply id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="latexml" id="S4.E9.m1.8.8.8.8.8.8.cmml" xref="S4.E9.m1.8.8.8.8.8.8">conditional</csymbol><apply id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><apply id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">superscript</csymbol><ci id="S4.E9.m1.5.5.5.5.5.5.cmml" xref="S4.E9.m1.5.5.5.5.5.5">𝒁</ci><ci id="S4.E9.m1.6.6.6.6.6.6.1.1.cmml" xref="S4.E9.m1.6.6.6.6.6.6.1.1">𝑘</ci></apply><ci id="S4.E9.m1.7.7.7.7.7.7.1.cmml" xref="S4.E9.m1.7.7.7.7.7.7.1">𝑛</ci></apply><apply id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.9.9.9.9.9.9.cmml" xref="S4.E9.m1.9.9.9.9.9.9">bold-italic-ϕ</ci><ci id="S4.E9.m1.10.10.10.10.10.10.1.cmml" xref="S4.E9.m1.10.10.10.10.10.10.1">𝑛</ci></apply></apply></apply></apply><apply id="S4.E9.m1.44.44.1.1.1.4.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><minus id="S4.E9.m1.14.14.14.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"></minus><apply id="S4.E9.m1.44.44.1.1.1.4.3.4.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="latexml" id="S4.E9.m1.44.44.1.1.1.4.3.4.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">quantum-operator-product</csymbol><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><minus id="S4.E9.m1.23.23.23.10.10.10.cmml" xref="S4.E9.m1.23.23.23.10.10.10"></minus><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><times id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"></times><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.3.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">superscript</csymbol><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.3.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.3.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.16.16.16.3.3.3.cmml" xref="S4.E9.m1.16.16.16.3.3.3">𝑺</ci><ci id="S4.E9.m1.17.17.17.4.4.4.1.cmml" xref="S4.E9.m1.17.17.17.4.4.4.1">𝑛</ci></apply><ci id="S4.E9.m1.18.18.18.5.5.5.1.1.cmml" xref="S4.E9.m1.18.18.18.5.5.5.1.1">𝑘</ci></apply><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.20.20.20.7.7.7.cmml" xref="S4.E9.m1.20.20.20.7.7.7">bold-italic-ϕ</ci><ci id="S4.E9.m1.21.21.21.8.8.8.1.cmml" xref="S4.E9.m1.21.21.21.8.8.8.1">𝑛</ci></apply></apply><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.3.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.2.1.1.1.3.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">superscript</csymbol><apply id="S4.E9.m1.44.44.1.1.1.2.1.1.1.3.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.2.1.1.1.3.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.24.24.24.11.11.11.cmml" xref="S4.E9.m1.24.24.24.11.11.11">𝒁</ci><ci id="S4.E9.m1.25.25.25.12.12.12.1.cmml" xref="S4.E9.m1.25.25.25.12.12.12.1">𝑛</ci></apply><ci id="S4.E9.m1.26.26.26.13.13.13.1.1.cmml" xref="S4.E9.m1.26.26.26.13.13.13.1.1">𝑘</ci></apply></apply><apply id="S4.E9.m1.44.44.1.1.1.3.2.2.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.3.2.2.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.28.28.28.15.15.15.cmml" xref="S4.E9.m1.28.28.28.15.15.15">𝚲</ci><ci id="S4.E9.m1.29.29.29.16.16.16.1.cmml" xref="S4.E9.m1.29.29.29.16.16.16.1">𝑍</ci></apply><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><minus id="S4.E9.m1.38.38.38.25.25.25.cmml" xref="S4.E9.m1.38.38.38.25.25.25"></minus><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><times id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"></times><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.3.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.3.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">superscript</csymbol><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.3.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.3.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.31.31.31.18.18.18.cmml" xref="S4.E9.m1.31.31.31.18.18.18">𝑺</ci><ci id="S4.E9.m1.32.32.32.19.19.19.1.cmml" xref="S4.E9.m1.32.32.32.19.19.19.1">𝑛</ci></apply><ci id="S4.E9.m1.33.33.33.20.20.20.1.1.cmml" xref="S4.E9.m1.33.33.33.20.20.20.1.1">𝑘</ci></apply><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.4.3.3.3.1.1.1.1.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.35.35.35.22.22.22.cmml" xref="S4.E9.m1.35.35.35.22.22.22">bold-italic-ϕ</ci><ci id="S4.E9.m1.36.36.36.23.23.23.1.cmml" xref="S4.E9.m1.36.36.36.23.23.23.1">𝑛</ci></apply></apply><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.3.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.4.3.3.3.3.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">superscript</csymbol><apply id="S4.E9.m1.44.44.1.1.1.4.3.3.3.3.2.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a"><csymbol cd="ambiguous" id="S4.E9.m1.44.44.1.1.1.4.3.3.3.3.2.1.cmml" xref="S4.E9.m1.45.45.2.44.14.14.14.1a">subscript</csymbol><ci id="S4.E9.m1.39.39.39.26.26.26.cmml" xref="S4.E9.m1.39.39.39.26.26.26">𝒁</ci><ci id="S4.E9.m1.40.40.40.27.27.27.1.cmml" xref="S4.E9.m1.40.40.40.27.27.27.1">𝑛</ci></apply><ci id="S4.E9.m1.41.41.41.28.28.28.1.1.cmml" xref="S4.E9.m1.41.41.41.28.28.28.1.1">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9.m1.46c">\ln\!\left(p\!\left(\bm{Z}^{(k)}_{n}|\bm{\phi}_{n}\right)\!\right)\!\propto\\ -\!\left\langle\!\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}|\bm{% \Lambda}_{Z}|\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\!\right\rangle.</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.46d">start_ROW start_CELL roman_ln ( italic_p ( bold_italic_Z start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) ∝ end_CELL end_ROW start_ROW start_CELL - ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) 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">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.3">Here <math alttext="\langle\cdot|\cdot\rangle" class="ltx_math_unparsed" display="inline" id="S4.p2.2.m1.1"><semantics id="S4.p2.2.m1.1a"><mrow id="S4.p2.2.m1.1b"><mo id="S4.p2.2.m1.1.1" stretchy="false">⟨</mo><mo id="S4.p2.2.m1.1.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="S4.p2.2.m1.1.3" stretchy="false">|</mo><mo id="S4.p2.2.m1.1.4" lspace="0em" rspace="0em">⋅</mo><mo id="S4.p2.2.m1.1.5" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S4.p2.2.m1.1c">\langle\cdot|\cdot\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m1.1d">⟨ ⋅ | ⋅ ⟩</annotation></semantics></math> is the bra-ket notation for inner products. The function in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E9" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">9</span></a>) is not recognized as a known distribution in <math alttext="\bm{\phi}_{n}" class="ltx_Math" display="inline" id="S4.p2.3.m2.1"><semantics id="S4.p2.3.m2.1a"><msub id="S4.p2.3.m2.1.1" xref="S4.p2.3.m2.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.3.m2.1.1.2" mathvariant="bold-italic" xref="S4.p2.3.m2.1.1.2.cmml">ϕ</mi><mi id="S4.p2.3.m2.1.1.3" xref="S4.p2.3.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.3.m2.1b"><apply id="S4.p2.3.m2.1.1.cmml" xref="S4.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.3.m2.1.1.1.cmml" xref="S4.p2.3.m2.1.1">subscript</csymbol><ci id="S4.p2.3.m2.1.1.2.cmml" xref="S4.p2.3.m2.1.1.2">bold-italic-ϕ</ci><ci id="S4.p2.3.m2.1.1.3.cmml" xref="S4.p2.3.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m2.1c">\bm{\phi}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m2.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Instead, we approximate this message by a Gaussian,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E10"> <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="\epsilon^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}=\mathcal{N}\!\left(\bm{% \phi};\bar{\bm{\epsilon}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})},\left(% \bar{\bar{\bm{\epsilon}}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}\right)^% {-1}\right)\!," class="ltx_Math" display="block" id="S4.E10.m1.8"><semantics id="S4.E10.m1.8a"><mrow id="S4.E10.m1.8.8.1" xref="S4.E10.m1.8.8.1.1.cmml"><mrow id="S4.E10.m1.8.8.1.1" xref="S4.E10.m1.8.8.1.1.cmml"><msup id="S4.E10.m1.8.8.1.1.4" xref="S4.E10.m1.8.8.1.1.4.cmml"><mi id="S4.E10.m1.8.8.1.1.4.2" xref="S4.E10.m1.8.8.1.1.4.2.cmml">ϵ</mi><mrow id="S4.E10.m1.2.2.2.2" xref="S4.E10.m1.2.2.2.2.1.cmml"><mo id="S4.E10.m1.2.2.2.2.2" stretchy="false" xref="S4.E10.m1.2.2.2.2.1.cmml">(</mo><mrow id="S4.E10.m1.2.2.2.2.1" xref="S4.E10.m1.2.2.2.2.1.cmml"><msubsup id="S4.E10.m1.2.2.2.2.1.2" xref="S4.E10.m1.2.2.2.2.1.2.cmml"><mi id="S4.E10.m1.2.2.2.2.1.2.2.2" xref="S4.E10.m1.2.2.2.2.1.2.2.2.cmml">𝒁</mi><mi id="S4.E10.m1.2.2.2.2.1.2.2.3" xref="S4.E10.m1.2.2.2.2.1.2.2.3.cmml">n</mi><mrow id="S4.E10.m1.1.1.1.1.1.3" xref="S4.E10.m1.2.2.2.2.1.2.cmml"><mo id="S4.E10.m1.1.1.1.1.1.3.1" stretchy="false" xref="S4.E10.m1.2.2.2.2.1.2.cmml">(</mo><mi id="S4.E10.m1.1.1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S4.E10.m1.1.1.1.1.1.3.2" stretchy="false" xref="S4.E10.m1.2.2.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="S4.E10.m1.2.2.2.2.1.1" stretchy="false" xref="S4.E10.m1.2.2.2.2.1.1.cmml">→</mo><msub id="S4.E10.m1.2.2.2.2.1.3" xref="S4.E10.m1.2.2.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E10.m1.2.2.2.2.1.3.2" mathvariant="bold-italic" xref="S4.E10.m1.2.2.2.2.1.3.2.cmml">ϕ</mi><mi id="S4.E10.m1.2.2.2.2.1.3.3" xref="S4.E10.m1.2.2.2.2.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.E10.m1.2.2.2.2.3" stretchy="false" xref="S4.E10.m1.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="S4.E10.m1.8.8.1.1.3" xref="S4.E10.m1.8.8.1.1.3.cmml">=</mo><mrow id="S4.E10.m1.8.8.1.1.2" xref="S4.E10.m1.8.8.1.1.2.cmml"><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E10.m1.8.8.1.1.2.4" xref="S4.E10.m1.8.8.1.1.2.4.cmml">𝒩</mi></mpadded><mo id="S4.E10.m1.8.8.1.1.2.3" xref="S4.E10.m1.8.8.1.1.2.3.cmml">⁢</mo><mrow id="S4.E10.m1.8.8.1.1.2.2.2" xref="S4.E10.m1.8.8.1.1.2.2.3.cmml"><mo id="S4.E10.m1.8.8.1.1.2.2.2.3" xref="S4.E10.m1.8.8.1.1.2.2.3.cmml">(</mo><mi class="ltx_mathvariant_bold-italic" id="S4.E10.m1.7.7" mathvariant="bold-italic" xref="S4.E10.m1.7.7.cmml">ϕ</mi><mo id="S4.E10.m1.8.8.1.1.2.2.2.4" xref="S4.E10.m1.8.8.1.1.2.2.3.cmml">;</mo><msup id="S4.E10.m1.8.8.1.1.1.1.1.1" xref="S4.E10.m1.8.8.1.1.1.1.1.1.cmml"><mover accent="true" id="S4.E10.m1.8.8.1.1.1.1.1.1.2" xref="S4.E10.m1.8.8.1.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E10.m1.8.8.1.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S4.E10.m1.8.8.1.1.1.1.1.1.2.2.cmml">ϵ</mi><mo id="S4.E10.m1.8.8.1.1.1.1.1.1.2.1" xref="S4.E10.m1.8.8.1.1.1.1.1.1.2.1.cmml">¯</mo></mover><mrow id="S4.E10.m1.4.4.2.2" xref="S4.E10.m1.4.4.2.2.1.cmml"><mo id="S4.E10.m1.4.4.2.2.2" stretchy="false" xref="S4.E10.m1.4.4.2.2.1.cmml">(</mo><mrow id="S4.E10.m1.4.4.2.2.1" xref="S4.E10.m1.4.4.2.2.1.cmml"><msubsup id="S4.E10.m1.4.4.2.2.1.2" xref="S4.E10.m1.4.4.2.2.1.2.cmml"><mi id="S4.E10.m1.4.4.2.2.1.2.2.2" xref="S4.E10.m1.4.4.2.2.1.2.2.2.cmml">𝒁</mi><mi id="S4.E10.m1.4.4.2.2.1.2.2.3" xref="S4.E10.m1.4.4.2.2.1.2.2.3.cmml">n</mi><mrow id="S4.E10.m1.3.3.1.1.1.3" xref="S4.E10.m1.4.4.2.2.1.2.cmml"><mo id="S4.E10.m1.3.3.1.1.1.3.1" stretchy="false" xref="S4.E10.m1.4.4.2.2.1.2.cmml">(</mo><mi id="S4.E10.m1.3.3.1.1.1.1" xref="S4.E10.m1.3.3.1.1.1.1.cmml">k</mi><mo id="S4.E10.m1.3.3.1.1.1.3.2" stretchy="false" xref="S4.E10.m1.4.4.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="S4.E10.m1.4.4.2.2.1.1" stretchy="false" xref="S4.E10.m1.4.4.2.2.1.1.cmml">→</mo><msub id="S4.E10.m1.4.4.2.2.1.3" xref="S4.E10.m1.4.4.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E10.m1.4.4.2.2.1.3.2" mathvariant="bold-italic" xref="S4.E10.m1.4.4.2.2.1.3.2.cmml">ϕ</mi><mi id="S4.E10.m1.4.4.2.2.1.3.3" xref="S4.E10.m1.4.4.2.2.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.E10.m1.4.4.2.2.3" stretchy="false" xref="S4.E10.m1.4.4.2.2.1.cmml">)</mo></mrow></msup><mo id="S4.E10.m1.8.8.1.1.2.2.2.5" xref="S4.E10.m1.8.8.1.1.2.2.3.cmml">,</mo><msup id="S4.E10.m1.8.8.1.1.2.2.2.2" xref="S4.E10.m1.8.8.1.1.2.2.2.2.cmml"><mrow id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.cmml"><mo id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.2" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.cmml">(</mo><msup id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.cmml"><mover accent="true" id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.cmml"><mover accent="true" id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.2" mathvariant="bold-italic" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.2.cmml">ϵ</mi><mo id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.1" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.1.cmml">¯</mo></mover><mo id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.1" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.1.cmml">¯</mo></mover><mrow id="S4.E10.m1.6.6.2.2" xref="S4.E10.m1.6.6.2.2.1.cmml"><mo id="S4.E10.m1.6.6.2.2.2" stretchy="false" xref="S4.E10.m1.6.6.2.2.1.cmml">(</mo><mrow id="S4.E10.m1.6.6.2.2.1" xref="S4.E10.m1.6.6.2.2.1.cmml"><msubsup id="S4.E10.m1.6.6.2.2.1.2" xref="S4.E10.m1.6.6.2.2.1.2.cmml"><mi id="S4.E10.m1.6.6.2.2.1.2.2.2" xref="S4.E10.m1.6.6.2.2.1.2.2.2.cmml">𝒁</mi><mi id="S4.E10.m1.6.6.2.2.1.2.2.3" xref="S4.E10.m1.6.6.2.2.1.2.2.3.cmml">n</mi><mrow id="S4.E10.m1.5.5.1.1.1.3" xref="S4.E10.m1.6.6.2.2.1.2.cmml"><mo id="S4.E10.m1.5.5.1.1.1.3.1" stretchy="false" xref="S4.E10.m1.6.6.2.2.1.2.cmml">(</mo><mi id="S4.E10.m1.5.5.1.1.1.1" xref="S4.E10.m1.5.5.1.1.1.1.cmml">k</mi><mo id="S4.E10.m1.5.5.1.1.1.3.2" stretchy="false" xref="S4.E10.m1.6.6.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="S4.E10.m1.6.6.2.2.1.1" stretchy="false" xref="S4.E10.m1.6.6.2.2.1.1.cmml">→</mo><msub id="S4.E10.m1.6.6.2.2.1.3" xref="S4.E10.m1.6.6.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E10.m1.6.6.2.2.1.3.2" mathvariant="bold-italic" xref="S4.E10.m1.6.6.2.2.1.3.2.cmml">ϕ</mi><mi id="S4.E10.m1.6.6.2.2.1.3.3" xref="S4.E10.m1.6.6.2.2.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.E10.m1.6.6.2.2.3" stretchy="false" xref="S4.E10.m1.6.6.2.2.1.cmml">)</mo></mrow></msup><mo id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.3" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S4.E10.m1.8.8.1.1.2.2.2.2.3" xref="S4.E10.m1.8.8.1.1.2.2.2.2.3.cmml"><mo id="S4.E10.m1.8.8.1.1.2.2.2.2.3a" xref="S4.E10.m1.8.8.1.1.2.2.2.2.3.cmml">−</mo><mn id="S4.E10.m1.8.8.1.1.2.2.2.2.3.2" xref="S4.E10.m1.8.8.1.1.2.2.2.2.3.2.cmml">1</mn></mrow></msup><mpadded width="0.288em"><mo id="S4.E10.m1.8.8.1.1.2.2.2.6" xref="S4.E10.m1.8.8.1.1.2.2.3.cmml">)</mo></mpadded></mrow></mrow></mrow><mo id="S4.E10.m1.8.8.1.2" xref="S4.E10.m1.8.8.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E10.m1.8b"><apply id="S4.E10.m1.8.8.1.1.cmml" xref="S4.E10.m1.8.8.1"><eq id="S4.E10.m1.8.8.1.1.3.cmml" xref="S4.E10.m1.8.8.1.1.3"></eq><apply id="S4.E10.m1.8.8.1.1.4.cmml" xref="S4.E10.m1.8.8.1.1.4"><csymbol cd="ambiguous" id="S4.E10.m1.8.8.1.1.4.1.cmml" xref="S4.E10.m1.8.8.1.1.4">superscript</csymbol><ci id="S4.E10.m1.8.8.1.1.4.2.cmml" xref="S4.E10.m1.8.8.1.1.4.2">italic-ϵ</ci><apply id="S4.E10.m1.2.2.2.2.1.cmml" xref="S4.E10.m1.2.2.2.2"><ci id="S4.E10.m1.2.2.2.2.1.1.cmml" xref="S4.E10.m1.2.2.2.2.1.1">→</ci><apply id="S4.E10.m1.2.2.2.2.1.2.cmml" xref="S4.E10.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.2.2.1.2.1.cmml" xref="S4.E10.m1.2.2.2.2.1.2">superscript</csymbol><apply id="S4.E10.m1.2.2.2.2.1.2.2.cmml" xref="S4.E10.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.2.2.1.2.2.1.cmml" xref="S4.E10.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S4.E10.m1.2.2.2.2.1.2.2.2.cmml" xref="S4.E10.m1.2.2.2.2.1.2.2.2">𝒁</ci><ci id="S4.E10.m1.2.2.2.2.1.2.2.3.cmml" xref="S4.E10.m1.2.2.2.2.1.2.2.3">𝑛</ci></apply><ci id="S4.E10.m1.1.1.1.1.1.1.cmml" xref="S4.E10.m1.1.1.1.1.1.1">𝑘</ci></apply><apply id="S4.E10.m1.2.2.2.2.1.3.cmml" xref="S4.E10.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.2.2.1.3.1.cmml" xref="S4.E10.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S4.E10.m1.2.2.2.2.1.3.2.cmml" xref="S4.E10.m1.2.2.2.2.1.3.2">bold-italic-ϕ</ci><ci id="S4.E10.m1.2.2.2.2.1.3.3.cmml" xref="S4.E10.m1.2.2.2.2.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E10.m1.8.8.1.1.2.cmml" xref="S4.E10.m1.8.8.1.1.2"><times id="S4.E10.m1.8.8.1.1.2.3.cmml" xref="S4.E10.m1.8.8.1.1.2.3"></times><ci id="S4.E10.m1.8.8.1.1.2.4.cmml" xref="S4.E10.m1.8.8.1.1.2.4">𝒩</ci><list id="S4.E10.m1.8.8.1.1.2.2.3.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2"><ci id="S4.E10.m1.7.7.cmml" xref="S4.E10.m1.7.7">bold-italic-ϕ</ci><apply id="S4.E10.m1.8.8.1.1.1.1.1.1.cmml" xref="S4.E10.m1.8.8.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E10.m1.8.8.1.1.1.1.1.1.1.cmml" xref="S4.E10.m1.8.8.1.1.1.1.1.1">superscript</csymbol><apply id="S4.E10.m1.8.8.1.1.1.1.1.1.2.cmml" xref="S4.E10.m1.8.8.1.1.1.1.1.1.2"><ci id="S4.E10.m1.8.8.1.1.1.1.1.1.2.1.cmml" xref="S4.E10.m1.8.8.1.1.1.1.1.1.2.1">¯</ci><ci id="S4.E10.m1.8.8.1.1.1.1.1.1.2.2.cmml" xref="S4.E10.m1.8.8.1.1.1.1.1.1.2.2">bold-italic-ϵ</ci></apply><apply id="S4.E10.m1.4.4.2.2.1.cmml" xref="S4.E10.m1.4.4.2.2"><ci id="S4.E10.m1.4.4.2.2.1.1.cmml" xref="S4.E10.m1.4.4.2.2.1.1">→</ci><apply id="S4.E10.m1.4.4.2.2.1.2.cmml" xref="S4.E10.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S4.E10.m1.4.4.2.2.1.2.1.cmml" xref="S4.E10.m1.4.4.2.2.1.2">superscript</csymbol><apply id="S4.E10.m1.4.4.2.2.1.2.2.cmml" xref="S4.E10.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S4.E10.m1.4.4.2.2.1.2.2.1.cmml" xref="S4.E10.m1.4.4.2.2.1.2">subscript</csymbol><ci id="S4.E10.m1.4.4.2.2.1.2.2.2.cmml" xref="S4.E10.m1.4.4.2.2.1.2.2.2">𝒁</ci><ci id="S4.E10.m1.4.4.2.2.1.2.2.3.cmml" xref="S4.E10.m1.4.4.2.2.1.2.2.3">𝑛</ci></apply><ci id="S4.E10.m1.3.3.1.1.1.1.cmml" xref="S4.E10.m1.3.3.1.1.1.1">𝑘</ci></apply><apply id="S4.E10.m1.4.4.2.2.1.3.cmml" xref="S4.E10.m1.4.4.2.2.1.3"><csymbol cd="ambiguous" id="S4.E10.m1.4.4.2.2.1.3.1.cmml" xref="S4.E10.m1.4.4.2.2.1.3">subscript</csymbol><ci id="S4.E10.m1.4.4.2.2.1.3.2.cmml" xref="S4.E10.m1.4.4.2.2.1.3.2">bold-italic-ϕ</ci><ci id="S4.E10.m1.4.4.2.2.1.3.3.cmml" xref="S4.E10.m1.4.4.2.2.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E10.m1.8.8.1.1.2.2.2.2.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S4.E10.m1.8.8.1.1.2.2.2.2.2.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2">superscript</csymbol><apply id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.1.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1">superscript</csymbol><apply id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2"><ci id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.1.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.1">¯</ci><apply id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2"><ci id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.1.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.1">¯</ci><ci id="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.2.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.1.1.1.2.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E10.m1.6.6.2.2.1.cmml" xref="S4.E10.m1.6.6.2.2"><ci id="S4.E10.m1.6.6.2.2.1.1.cmml" xref="S4.E10.m1.6.6.2.2.1.1">→</ci><apply id="S4.E10.m1.6.6.2.2.1.2.cmml" xref="S4.E10.m1.6.6.2.2.1.2"><csymbol cd="ambiguous" id="S4.E10.m1.6.6.2.2.1.2.1.cmml" xref="S4.E10.m1.6.6.2.2.1.2">superscript</csymbol><apply id="S4.E10.m1.6.6.2.2.1.2.2.cmml" xref="S4.E10.m1.6.6.2.2.1.2"><csymbol cd="ambiguous" id="S4.E10.m1.6.6.2.2.1.2.2.1.cmml" xref="S4.E10.m1.6.6.2.2.1.2">subscript</csymbol><ci id="S4.E10.m1.6.6.2.2.1.2.2.2.cmml" xref="S4.E10.m1.6.6.2.2.1.2.2.2">𝒁</ci><ci id="S4.E10.m1.6.6.2.2.1.2.2.3.cmml" xref="S4.E10.m1.6.6.2.2.1.2.2.3">𝑛</ci></apply><ci id="S4.E10.m1.5.5.1.1.1.1.cmml" xref="S4.E10.m1.5.5.1.1.1.1">𝑘</ci></apply><apply id="S4.E10.m1.6.6.2.2.1.3.cmml" xref="S4.E10.m1.6.6.2.2.1.3"><csymbol cd="ambiguous" id="S4.E10.m1.6.6.2.2.1.3.1.cmml" xref="S4.E10.m1.6.6.2.2.1.3">subscript</csymbol><ci id="S4.E10.m1.6.6.2.2.1.3.2.cmml" xref="S4.E10.m1.6.6.2.2.1.3.2">bold-italic-ϕ</ci><ci id="S4.E10.m1.6.6.2.2.1.3.3.cmml" xref="S4.E10.m1.6.6.2.2.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E10.m1.8.8.1.1.2.2.2.2.3.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.3"><minus id="S4.E10.m1.8.8.1.1.2.2.2.2.3.1.cmml" xref="S4.E10.m1.8.8.1.1.2.2.2.2.3"></minus><cn id="S4.E10.m1.8.8.1.1.2.2.2.2.3.2.cmml" type="integer" xref="S4.E10.m1.8.8.1.1.2.2.2.2.3.2">1</cn></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10.m1.8c">\epsilon^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}=\mathcal{N}\!\left(\bm{% \phi};\bar{\bm{\epsilon}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})},\left(% \bar{\bar{\bm{\epsilon}}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}\right)^% {-1}\right)\!,</annotation><annotation encoding="application/x-llamapun" id="S4.E10.m1.8d">italic_ϵ start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT = caligraphic_N ( bold_italic_ϕ ; over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT , ( over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 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">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.7">with mean <math alttext="\bar{\bm{\epsilon}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}" class="ltx_Math" display="inline" id="S4.p2.4.m1.2"><semantics id="S4.p2.4.m1.2a"><msup id="S4.p2.4.m1.2.3" xref="S4.p2.4.m1.2.3.cmml"><mover accent="true" id="S4.p2.4.m1.2.3.2" xref="S4.p2.4.m1.2.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.4.m1.2.3.2.2" mathvariant="bold-italic" xref="S4.p2.4.m1.2.3.2.2.cmml">ϵ</mi><mo id="S4.p2.4.m1.2.3.2.1" xref="S4.p2.4.m1.2.3.2.1.cmml">¯</mo></mover><mrow id="S4.p2.4.m1.2.2.2.2" xref="S4.p2.4.m1.2.2.2.2.1.cmml"><mo id="S4.p2.4.m1.2.2.2.2.2" stretchy="false" xref="S4.p2.4.m1.2.2.2.2.1.cmml">(</mo><mrow id="S4.p2.4.m1.2.2.2.2.1" xref="S4.p2.4.m1.2.2.2.2.1.cmml"><msubsup id="S4.p2.4.m1.2.2.2.2.1.2" xref="S4.p2.4.m1.2.2.2.2.1.2.cmml"><mi id="S4.p2.4.m1.2.2.2.2.1.2.2.2" xref="S4.p2.4.m1.2.2.2.2.1.2.2.2.cmml">𝒁</mi><mi id="S4.p2.4.m1.2.2.2.2.1.2.2.3" xref="S4.p2.4.m1.2.2.2.2.1.2.2.3.cmml">n</mi><mrow id="S4.p2.4.m1.1.1.1.1.1.3" xref="S4.p2.4.m1.2.2.2.2.1.2.cmml"><mo id="S4.p2.4.m1.1.1.1.1.1.3.1" stretchy="false" xref="S4.p2.4.m1.2.2.2.2.1.2.cmml">(</mo><mi id="S4.p2.4.m1.1.1.1.1.1.1" xref="S4.p2.4.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S4.p2.4.m1.1.1.1.1.1.3.2" stretchy="false" xref="S4.p2.4.m1.2.2.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="S4.p2.4.m1.2.2.2.2.1.1" stretchy="false" xref="S4.p2.4.m1.2.2.2.2.1.1.cmml">→</mo><msub id="S4.p2.4.m1.2.2.2.2.1.3" xref="S4.p2.4.m1.2.2.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.4.m1.2.2.2.2.1.3.2" mathvariant="bold-italic" xref="S4.p2.4.m1.2.2.2.2.1.3.2.cmml">ϕ</mi><mi id="S4.p2.4.m1.2.2.2.2.1.3.3" xref="S4.p2.4.m1.2.2.2.2.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.p2.4.m1.2.2.2.2.3" stretchy="false" xref="S4.p2.4.m1.2.2.2.2.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p2.4.m1.2b"><apply id="S4.p2.4.m1.2.3.cmml" xref="S4.p2.4.m1.2.3"><csymbol cd="ambiguous" id="S4.p2.4.m1.2.3.1.cmml" xref="S4.p2.4.m1.2.3">superscript</csymbol><apply id="S4.p2.4.m1.2.3.2.cmml" xref="S4.p2.4.m1.2.3.2"><ci id="S4.p2.4.m1.2.3.2.1.cmml" xref="S4.p2.4.m1.2.3.2.1">¯</ci><ci id="S4.p2.4.m1.2.3.2.2.cmml" xref="S4.p2.4.m1.2.3.2.2">bold-italic-ϵ</ci></apply><apply id="S4.p2.4.m1.2.2.2.2.1.cmml" xref="S4.p2.4.m1.2.2.2.2"><ci id="S4.p2.4.m1.2.2.2.2.1.1.cmml" xref="S4.p2.4.m1.2.2.2.2.1.1">→</ci><apply id="S4.p2.4.m1.2.2.2.2.1.2.cmml" xref="S4.p2.4.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.p2.4.m1.2.2.2.2.1.2.1.cmml" xref="S4.p2.4.m1.2.2.2.2.1.2">superscript</csymbol><apply id="S4.p2.4.m1.2.2.2.2.1.2.2.cmml" xref="S4.p2.4.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.p2.4.m1.2.2.2.2.1.2.2.1.cmml" xref="S4.p2.4.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S4.p2.4.m1.2.2.2.2.1.2.2.2.cmml" xref="S4.p2.4.m1.2.2.2.2.1.2.2.2">𝒁</ci><ci id="S4.p2.4.m1.2.2.2.2.1.2.2.3.cmml" xref="S4.p2.4.m1.2.2.2.2.1.2.2.3">𝑛</ci></apply><ci id="S4.p2.4.m1.1.1.1.1.1.1.cmml" xref="S4.p2.4.m1.1.1.1.1.1.1">𝑘</ci></apply><apply id="S4.p2.4.m1.2.2.2.2.1.3.cmml" xref="S4.p2.4.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.p2.4.m1.2.2.2.2.1.3.1.cmml" xref="S4.p2.4.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S4.p2.4.m1.2.2.2.2.1.3.2.cmml" xref="S4.p2.4.m1.2.2.2.2.1.3.2">bold-italic-ϕ</ci><ci id="S4.p2.4.m1.2.2.2.2.1.3.3.cmml" xref="S4.p2.4.m1.2.2.2.2.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m1.2c">\bar{\bm{\epsilon}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m1.2d">over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math> and covariance matrix <math alttext="\bar{\bar{\bm{\epsilon}}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}" class="ltx_Math" display="inline" id="S4.p2.5.m2.2"><semantics id="S4.p2.5.m2.2a"><msup id="S4.p2.5.m2.2.3" xref="S4.p2.5.m2.2.3.cmml"><mover accent="true" id="S4.p2.5.m2.2.3.2" xref="S4.p2.5.m2.2.3.2.cmml"><mover accent="true" id="S4.p2.5.m2.2.3.2.2" xref="S4.p2.5.m2.2.3.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.5.m2.2.3.2.2.2" mathvariant="bold-italic" xref="S4.p2.5.m2.2.3.2.2.2.cmml">ϵ</mi><mo id="S4.p2.5.m2.2.3.2.2.1" xref="S4.p2.5.m2.2.3.2.2.1.cmml">¯</mo></mover><mo id="S4.p2.5.m2.2.3.2.1" xref="S4.p2.5.m2.2.3.2.1.cmml">¯</mo></mover><mrow id="S4.p2.5.m2.2.2.2.2" xref="S4.p2.5.m2.2.2.2.2.1.cmml"><mo id="S4.p2.5.m2.2.2.2.2.2" stretchy="false" xref="S4.p2.5.m2.2.2.2.2.1.cmml">(</mo><mrow id="S4.p2.5.m2.2.2.2.2.1" xref="S4.p2.5.m2.2.2.2.2.1.cmml"><msubsup id="S4.p2.5.m2.2.2.2.2.1.2" xref="S4.p2.5.m2.2.2.2.2.1.2.cmml"><mi id="S4.p2.5.m2.2.2.2.2.1.2.2.2" xref="S4.p2.5.m2.2.2.2.2.1.2.2.2.cmml">𝒁</mi><mi id="S4.p2.5.m2.2.2.2.2.1.2.2.3" xref="S4.p2.5.m2.2.2.2.2.1.2.2.3.cmml">n</mi><mrow id="S4.p2.5.m2.1.1.1.1.1.3" xref="S4.p2.5.m2.2.2.2.2.1.2.cmml"><mo id="S4.p2.5.m2.1.1.1.1.1.3.1" stretchy="false" xref="S4.p2.5.m2.2.2.2.2.1.2.cmml">(</mo><mi id="S4.p2.5.m2.1.1.1.1.1.1" xref="S4.p2.5.m2.1.1.1.1.1.1.cmml">k</mi><mo id="S4.p2.5.m2.1.1.1.1.1.3.2" stretchy="false" xref="S4.p2.5.m2.2.2.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="S4.p2.5.m2.2.2.2.2.1.1" stretchy="false" xref="S4.p2.5.m2.2.2.2.2.1.1.cmml">→</mo><msub id="S4.p2.5.m2.2.2.2.2.1.3" xref="S4.p2.5.m2.2.2.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.5.m2.2.2.2.2.1.3.2" mathvariant="bold-italic" xref="S4.p2.5.m2.2.2.2.2.1.3.2.cmml">ϕ</mi><mi id="S4.p2.5.m2.2.2.2.2.1.3.3" xref="S4.p2.5.m2.2.2.2.2.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.p2.5.m2.2.2.2.2.3" stretchy="false" xref="S4.p2.5.m2.2.2.2.2.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p2.5.m2.2b"><apply id="S4.p2.5.m2.2.3.cmml" xref="S4.p2.5.m2.2.3"><csymbol cd="ambiguous" id="S4.p2.5.m2.2.3.1.cmml" xref="S4.p2.5.m2.2.3">superscript</csymbol><apply id="S4.p2.5.m2.2.3.2.cmml" xref="S4.p2.5.m2.2.3.2"><ci id="S4.p2.5.m2.2.3.2.1.cmml" xref="S4.p2.5.m2.2.3.2.1">¯</ci><apply id="S4.p2.5.m2.2.3.2.2.cmml" xref="S4.p2.5.m2.2.3.2.2"><ci id="S4.p2.5.m2.2.3.2.2.1.cmml" xref="S4.p2.5.m2.2.3.2.2.1">¯</ci><ci id="S4.p2.5.m2.2.3.2.2.2.cmml" xref="S4.p2.5.m2.2.3.2.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.p2.5.m2.2.2.2.2.1.cmml" xref="S4.p2.5.m2.2.2.2.2"><ci id="S4.p2.5.m2.2.2.2.2.1.1.cmml" xref="S4.p2.5.m2.2.2.2.2.1.1">→</ci><apply id="S4.p2.5.m2.2.2.2.2.1.2.cmml" xref="S4.p2.5.m2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.p2.5.m2.2.2.2.2.1.2.1.cmml" xref="S4.p2.5.m2.2.2.2.2.1.2">superscript</csymbol><apply id="S4.p2.5.m2.2.2.2.2.1.2.2.cmml" xref="S4.p2.5.m2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.p2.5.m2.2.2.2.2.1.2.2.1.cmml" xref="S4.p2.5.m2.2.2.2.2.1.2">subscript</csymbol><ci id="S4.p2.5.m2.2.2.2.2.1.2.2.2.cmml" xref="S4.p2.5.m2.2.2.2.2.1.2.2.2">𝒁</ci><ci id="S4.p2.5.m2.2.2.2.2.1.2.2.3.cmml" xref="S4.p2.5.m2.2.2.2.2.1.2.2.3">𝑛</ci></apply><ci id="S4.p2.5.m2.1.1.1.1.1.1.cmml" xref="S4.p2.5.m2.1.1.1.1.1.1">𝑘</ci></apply><apply id="S4.p2.5.m2.2.2.2.2.1.3.cmml" xref="S4.p2.5.m2.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.p2.5.m2.2.2.2.2.1.3.1.cmml" xref="S4.p2.5.m2.2.2.2.2.1.3">subscript</csymbol><ci id="S4.p2.5.m2.2.2.2.2.1.3.2.cmml" xref="S4.p2.5.m2.2.2.2.2.1.3.2">bold-italic-ϕ</ci><ci id="S4.p2.5.m2.2.2.2.2.1.3.3.cmml" xref="S4.p2.5.m2.2.2.2.2.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m2.2c">\bar{\bar{\bm{\epsilon}}}^{(\bm{Z}_{n}^{(k)}\rightarrow\bm{\phi}_{n})}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m2.2d">over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math>. The optimal approximation of the message is found by minimizing the KL divergence between <math alttext="p(\bm{Z}_{n}^{(k)}|\bm{\phi}_{n})" class="ltx_Math" display="inline" id="S4.p2.6.m3.2"><semantics id="S4.p2.6.m3.2a"><mrow id="S4.p2.6.m3.2.2" xref="S4.p2.6.m3.2.2.cmml"><mi id="S4.p2.6.m3.2.2.3" xref="S4.p2.6.m3.2.2.3.cmml">p</mi><mo id="S4.p2.6.m3.2.2.2" xref="S4.p2.6.m3.2.2.2.cmml">⁢</mo><mrow id="S4.p2.6.m3.2.2.1.1" xref="S4.p2.6.m3.2.2.1.1.1.cmml"><mo id="S4.p2.6.m3.2.2.1.1.2" stretchy="false" xref="S4.p2.6.m3.2.2.1.1.1.cmml">(</mo><mrow id="S4.p2.6.m3.2.2.1.1.1" xref="S4.p2.6.m3.2.2.1.1.1.cmml"><msubsup id="S4.p2.6.m3.2.2.1.1.1.2" xref="S4.p2.6.m3.2.2.1.1.1.2.cmml"><mi id="S4.p2.6.m3.2.2.1.1.1.2.2.2" xref="S4.p2.6.m3.2.2.1.1.1.2.2.2.cmml">𝒁</mi><mi id="S4.p2.6.m3.2.2.1.1.1.2.2.3" xref="S4.p2.6.m3.2.2.1.1.1.2.2.3.cmml">n</mi><mrow id="S4.p2.6.m3.1.1.1.3" xref="S4.p2.6.m3.2.2.1.1.1.2.cmml"><mo id="S4.p2.6.m3.1.1.1.3.1" stretchy="false" xref="S4.p2.6.m3.2.2.1.1.1.2.cmml">(</mo><mi id="S4.p2.6.m3.1.1.1.1" xref="S4.p2.6.m3.1.1.1.1.cmml">k</mi><mo id="S4.p2.6.m3.1.1.1.3.2" stretchy="false" xref="S4.p2.6.m3.2.2.1.1.1.2.cmml">)</mo></mrow></msubsup><mo fence="false" id="S4.p2.6.m3.2.2.1.1.1.1" xref="S4.p2.6.m3.2.2.1.1.1.1.cmml">|</mo><msub id="S4.p2.6.m3.2.2.1.1.1.3" xref="S4.p2.6.m3.2.2.1.1.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.6.m3.2.2.1.1.1.3.2" mathvariant="bold-italic" xref="S4.p2.6.m3.2.2.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.p2.6.m3.2.2.1.1.1.3.3" xref="S4.p2.6.m3.2.2.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.p2.6.m3.2.2.1.1.3" stretchy="false" xref="S4.p2.6.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m3.2b"><apply id="S4.p2.6.m3.2.2.cmml" xref="S4.p2.6.m3.2.2"><times id="S4.p2.6.m3.2.2.2.cmml" xref="S4.p2.6.m3.2.2.2"></times><ci id="S4.p2.6.m3.2.2.3.cmml" xref="S4.p2.6.m3.2.2.3">𝑝</ci><apply id="S4.p2.6.m3.2.2.1.1.1.cmml" xref="S4.p2.6.m3.2.2.1.1"><csymbol cd="latexml" id="S4.p2.6.m3.2.2.1.1.1.1.cmml" xref="S4.p2.6.m3.2.2.1.1.1.1">conditional</csymbol><apply id="S4.p2.6.m3.2.2.1.1.1.2.cmml" xref="S4.p2.6.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.p2.6.m3.2.2.1.1.1.2.1.cmml" xref="S4.p2.6.m3.2.2.1.1.1.2">superscript</csymbol><apply id="S4.p2.6.m3.2.2.1.1.1.2.2.cmml" xref="S4.p2.6.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.p2.6.m3.2.2.1.1.1.2.2.1.cmml" xref="S4.p2.6.m3.2.2.1.1.1.2">subscript</csymbol><ci id="S4.p2.6.m3.2.2.1.1.1.2.2.2.cmml" xref="S4.p2.6.m3.2.2.1.1.1.2.2.2">𝒁</ci><ci id="S4.p2.6.m3.2.2.1.1.1.2.2.3.cmml" xref="S4.p2.6.m3.2.2.1.1.1.2.2.3">𝑛</ci></apply><ci id="S4.p2.6.m3.1.1.1.1.cmml" xref="S4.p2.6.m3.1.1.1.1">𝑘</ci></apply><apply id="S4.p2.6.m3.2.2.1.1.1.3.cmml" xref="S4.p2.6.m3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.p2.6.m3.2.2.1.1.1.3.1.cmml" xref="S4.p2.6.m3.2.2.1.1.1.3">subscript</csymbol><ci id="S4.p2.6.m3.2.2.1.1.1.3.2.cmml" xref="S4.p2.6.m3.2.2.1.1.1.3.2">bold-italic-ϕ</ci><ci id="S4.p2.6.m3.2.2.1.1.1.3.3.cmml" xref="S4.p2.6.m3.2.2.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m3.2c">p(\bm{Z}_{n}^{(k)}|\bm{\phi}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m3.2d">italic_p ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> and a Gaussian in <math alttext="\bm{\phi}_{n}" class="ltx_Math" display="inline" id="S4.p2.7.m4.1"><semantics id="S4.p2.7.m4.1a"><msub id="S4.p2.7.m4.1.1" xref="S4.p2.7.m4.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.7.m4.1.1.2" mathvariant="bold-italic" xref="S4.p2.7.m4.1.1.2.cmml">ϕ</mi><mi id="S4.p2.7.m4.1.1.3" xref="S4.p2.7.m4.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.7.m4.1b"><apply id="S4.p2.7.m4.1.1.cmml" xref="S4.p2.7.m4.1.1"><csymbol cd="ambiguous" id="S4.p2.7.m4.1.1.1.cmml" xref="S4.p2.7.m4.1.1">subscript</csymbol><ci id="S4.p2.7.m4.1.1.2.cmml" xref="S4.p2.7.m4.1.1.2">bold-italic-ϕ</ci><ci id="S4.p2.7.m4.1.1.3.cmml" xref="S4.p2.7.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m4.1c">\bm{\phi}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m4.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx3"> <tbody id="S4.E11"><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\left\{\widehat{\overline{{\bm{\epsilon}}}}_{n}^{(k)},\widehat{% \overline{\overline{{\bm{\epsilon}}}}}_{n}^{(k)}\right\}=\underset{\overline{{% \bm{\epsilon}}},\overline{\overline{{\bm{\epsilon}}}}}{\arg\min}\ D_{KL}\!% \left(\mathcal{N}\!\left(\bm{\phi}_{n};\overline{\bm{\epsilon}},\overline{% \overline{\bm{\epsilon}}}^{-1}\right)\!\Big{\|}p\!\left(\!\bm{Z}_{n}^{(k)}|\bm% {\phi}_{n}\right)\!\right)\!," class="ltx_math_unparsed" display="inline" id="S4.E11.m1.6"><semantics id="S4.E11.m1.6a"><mrow id="S4.E11.m1.6b"><mrow id="S4.E11.m1.6.7"><mo id="S4.E11.m1.6.7.1">{</mo><msubsup id="S4.E11.m1.6.7.2"><mover accent="true" id="S4.E11.m1.6.7.2.2.2"><mover accent="true" id="S4.E11.m1.6.7.2.2.2.2"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.6.7.2.2.2.2.2" mathvariant="bold-italic">ϵ</mi><mo id="S4.E11.m1.6.7.2.2.2.2.1">¯</mo></mover><mo id="S4.E11.m1.6.7.2.2.2.1">^</mo></mover><mi id="S4.E11.m1.6.7.2.2.3">n</mi><mrow id="S4.E11.m1.1.1.1.3"><mo id="S4.E11.m1.1.1.1.3.1" stretchy="false">(</mo><mi id="S4.E11.m1.1.1.1.1">k</mi><mo id="S4.E11.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E11.m1.6.7.3">,</mo><msubsup id="S4.E11.m1.6.7.4"><mover accent="true" id="S4.E11.m1.6.7.4.2.2"><mover accent="true" id="S4.E11.m1.6.7.4.2.2.2"><mover accent="true" id="S4.E11.m1.6.7.4.2.2.2.2"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.6.7.4.2.2.2.2.2" mathvariant="bold-italic">ϵ</mi><mo id="S4.E11.m1.6.7.4.2.2.2.2.1">¯</mo></mover><mo id="S4.E11.m1.6.7.4.2.2.2.1">¯</mo></mover><mo id="S4.E11.m1.6.7.4.2.2.1">^</mo></mover><mi id="S4.E11.m1.6.7.4.2.3">n</mi><mrow id="S4.E11.m1.2.2.1.3"><mo id="S4.E11.m1.2.2.1.3.1" stretchy="false">(</mo><mi id="S4.E11.m1.2.2.1.1">k</mi><mo id="S4.E11.m1.2.2.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E11.m1.6.7.5">}</mo></mrow><mo id="S4.E11.m1.6.8">=</mo><munder accentunder="true" id="S4.E11.m1.4.4"><mrow id="S4.E11.m1.4.4.3"><mi id="S4.E11.m1.4.4.3.1">arg</mi><mo id="S4.E11.m1.4.4.3a" lspace="0.167em">⁡</mo><mi id="S4.E11.m1.4.4.3.2">min</mi></mrow><mrow id="S4.E11.m1.4.4.2.4"><mover accent="true" id="S4.E11.m1.3.3.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.3.3.1.1.2" mathvariant="bold-italic">ϵ</mi><mo id="S4.E11.m1.3.3.1.1.1">¯</mo></mover><mo id="S4.E11.m1.4.4.2.4.1">,</mo><mover accent="true" id="S4.E11.m1.4.4.2.2"><mover accent="true" id="S4.E11.m1.4.4.2.2.2"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.4.4.2.2.2.2" mathvariant="bold-italic">ϵ</mi><mo id="S4.E11.m1.4.4.2.2.2.1">¯</mo></mover><mo id="S4.E11.m1.4.4.2.2.1">¯</mo></mover></mrow></munder><msub id="S4.E11.m1.6.9"><mi id="S4.E11.m1.6.9.2">D</mi><mrow id="S4.E11.m1.6.9.3"><mi id="S4.E11.m1.6.9.3.2">K</mi><mo id="S4.E11.m1.6.9.3.1">⁢</mo><mi id="S4.E11.m1.6.9.3.3">L</mi></mrow></msub><mrow id="S4.E11.m1.6.10"><mo id="S4.E11.m1.6.10.1">(</mo><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E11.m1.6.10.2">𝒩</mi></mpadded><mrow id="S4.E11.m1.6.10.3"><mo id="S4.E11.m1.6.10.3.1">(</mo><msub id="S4.E11.m1.6.10.3.2"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.6.10.3.2.2" mathvariant="bold-italic">ϕ</mi><mi id="S4.E11.m1.6.10.3.2.3">n</mi></msub><mo id="S4.E11.m1.6.10.3.3">;</mo><mover accent="true" id="S4.E11.m1.6.6"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.6.6.2" mathvariant="bold-italic">ϵ</mi><mo id="S4.E11.m1.6.6.1">¯</mo></mover><mo id="S4.E11.m1.6.10.3.4">,</mo><msup id="S4.E11.m1.6.10.3.5"><mover accent="true" id="S4.E11.m1.6.10.3.5.2"><mover accent="true" id="S4.E11.m1.6.10.3.5.2.2"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.6.10.3.5.2.2.2" mathvariant="bold-italic">ϵ</mi><mo id="S4.E11.m1.6.10.3.5.2.2.1">¯</mo></mover><mo id="S4.E11.m1.6.10.3.5.2.1">¯</mo></mover><mrow id="S4.E11.m1.6.10.3.5.3"><mo id="S4.E11.m1.6.10.3.5.3a">−</mo><mn id="S4.E11.m1.6.10.3.5.3.2">1</mn></mrow></msup><mpadded width="0.288em"><mo id="S4.E11.m1.6.10.3.6">)</mo></mpadded></mrow><mo id="S4.E11.m1.6.10.4" mathsize="160%" rspace="0.167em">∥</mo><mpadded width="0.333em"><mi id="S4.E11.m1.6.10.5">p</mi></mpadded><mrow id="S4.E11.m1.6.10.6"><mpadded width="0.288em"><mo id="S4.E11.m1.6.10.6.1">(</mo></mpadded><msubsup id="S4.E11.m1.6.10.6.2"><mi id="S4.E11.m1.6.10.6.2.2.2">𝒁</mi><mi id="S4.E11.m1.6.10.6.2.2.3">n</mi><mrow id="S4.E11.m1.5.5.1.3"><mo id="S4.E11.m1.5.5.1.3.1" stretchy="false">(</mo><mi id="S4.E11.m1.5.5.1.1">k</mi><mo id="S4.E11.m1.5.5.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo fence="false" id="S4.E11.m1.6.10.6.3" rspace="0.167em" stretchy="false">|</mo><msub id="S4.E11.m1.6.10.6.4"><mi class="ltx_mathvariant_bold-italic" id="S4.E11.m1.6.10.6.4.2" mathvariant="bold-italic">ϕ</mi><mi id="S4.E11.m1.6.10.6.4.3">n</mi></msub><mpadded width="0.288em"><mo id="S4.E11.m1.6.10.6.5">)</mo></mpadded></mrow><mpadded width="0.288em"><mo id="S4.E11.m1.6.10.7">)</mo></mpadded></mrow><mo id="S4.E11.m1.6.11">,</mo></mrow><annotation encoding="application/x-tex" id="S4.E11.m1.6c">\displaystyle\left\{\widehat{\overline{{\bm{\epsilon}}}}_{n}^{(k)},\widehat{% \overline{\overline{{\bm{\epsilon}}}}}_{n}^{(k)}\right\}=\underset{\overline{{% \bm{\epsilon}}},\overline{\overline{{\bm{\epsilon}}}}}{\arg\min}\ D_{KL}\!% \left(\mathcal{N}\!\left(\bm{\phi}_{n};\overline{\bm{\epsilon}},\overline{% \overline{\bm{\epsilon}}}^{-1}\right)\!\Big{\|}p\!\left(\!\bm{Z}_{n}^{(k)}|\bm% {\phi}_{n}\right)\!\right)\!,</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m1.6d">{ over^ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT , over^ start_ARG over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT } = start_UNDERACCENT over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG end_UNDERACCENT start_ARG roman_arg roman_min end_ARG italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ( caligraphic_N ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ∥ italic_p ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n 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">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.15">where the superscript has been dropped for brevity. The KL divergence reads</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="D_{KL}=-\int_{\bm{\phi}_{n}}\mathcal{N}\!\left(\bm{\phi}_{n};\overline{\bm{% \epsilon}},\overline{\overline{\bm{\epsilon}}}^{-1}\right)\ln\!\left(p\!\left(% \bm{Z}_{n}^{(k)}|\bm{\phi}_{n}\right)\!\right)\text{d}\bm{\phi}_{n}\\ +\int_{\bm{\phi}_{n}}\mathcal{N}\!\left(\bm{\phi}_{n};\overline{\bm{\epsilon}}% ,\overline{\overline{\bm{\epsilon}}}^{-1}\right)\ln\!\left(\mathcal{N}\!\left(% \bm{\phi}_{n};\overline{\bm{\epsilon}},\overline{\overline{\bm{\epsilon}}}^{-1% }\right)\!\right)\text{d}\bm{\phi}_{n}" class="ltx_Math" display="block" id="S4.E12.m1.72"><semantics id="S4.E12.m1.72a"><mtable displaystyle="true" id="S4.E12.m1.72.72.12" rowspacing="0pt" xref="S4.E12.m1.66.66.6.cmml"><mtr id="S4.E12.m1.72.72.12a" xref="S4.E12.m1.66.66.6.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E12.m1.72.72.12b" xref="S4.E12.m1.66.66.6.cmml"><mrow id="S4.E12.m1.69.69.9.63.34.34" xref="S4.E12.m1.66.66.6.cmml"><msub id="S4.E12.m1.69.69.9.63.34.34.35" xref="S4.E12.m1.66.66.6.cmml"><mi id="S4.E12.m1.1.1.1.1.1.1" xref="S4.E12.m1.1.1.1.1.1.1.cmml">D</mi><mrow id="S4.E12.m1.2.2.2.2.2.2.1" xref="S4.E12.m1.2.2.2.2.2.2.1.cmml"><mi id="S4.E12.m1.2.2.2.2.2.2.1.2" xref="S4.E12.m1.2.2.2.2.2.2.1.2.cmml">K</mi><mo id="S4.E12.m1.2.2.2.2.2.2.1.1" xref="S4.E12.m1.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.E12.m1.2.2.2.2.2.2.1.3" xref="S4.E12.m1.2.2.2.2.2.2.1.3.cmml">L</mi></mrow></msub><mo id="S4.E12.m1.3.3.3.3.3.3" xref="S4.E12.m1.3.3.3.3.3.3.cmml">=</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.69.69.9.63.34.34.34a" xref="S4.E12.m1.66.66.6.cmml">−</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3" xref="S4.E12.m1.66.66.6.cmml"><msub id="S4.E12.m1.69.69.9.63.34.34.34.3.4" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.5.5.5.5.5.5" xref="S4.E12.m1.5.5.5.5.5.5.cmml">∫</mo><msub id="S4.E12.m1.6.6.6.6.6.6.1" xref="S4.E12.m1.6.6.6.6.6.6.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.6.6.6.6.6.6.1.2" mathvariant="bold-italic" xref="S4.E12.m1.6.6.6.6.6.6.1.2.cmml">ϕ</mi><mi id="S4.E12.m1.6.6.6.6.6.6.1.3" xref="S4.E12.m1.6.6.6.6.6.6.1.3.cmml">n</mi></msub></msub><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3.3" xref="S4.E12.m1.66.66.6.cmml"><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E12.m1.7.7.7.7.7.7" xref="S4.E12.m1.7.7.7.7.7.7.cmml">𝒩</mi></mpadded><mo id="S4.E12.m1.69.69.9.63.34.34.34.3.3.4" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mrow id="S4.E12.m1.68.68.8.62.33.33.33.2.2.2.2" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.8.8.8.8.8.8" xref="S4.E12.m1.66.66.6.cmml">(</mo><msub id="S4.E12.m1.67.67.7.61.32.32.32.1.1.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.9.9.9.9.9.9" mathvariant="bold-italic" xref="S4.E12.m1.9.9.9.9.9.9.cmml">ϕ</mi><mi id="S4.E12.m1.10.10.10.10.10.10.1" xref="S4.E12.m1.10.10.10.10.10.10.1.cmml">n</mi></msub><mo id="S4.E12.m1.11.11.11.11.11.11" xref="S4.E12.m1.66.66.6.cmml">;</mo><mover accent="true" id="S4.E12.m1.12.12.12.12.12.12" xref="S4.E12.m1.12.12.12.12.12.12.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.12.12.12.12.12.12.2" mathvariant="bold-italic" xref="S4.E12.m1.12.12.12.12.12.12.2.cmml">ϵ</mi><mo id="S4.E12.m1.12.12.12.12.12.12.1" xref="S4.E12.m1.12.12.12.12.12.12.1.cmml">¯</mo></mover><mo id="S4.E12.m1.13.13.13.13.13.13" xref="S4.E12.m1.66.66.6.cmml">,</mo><msup id="S4.E12.m1.68.68.8.62.33.33.33.2.2.2.2.2" xref="S4.E12.m1.66.66.6.cmml"><mover accent="true" id="S4.E12.m1.14.14.14.14.14.14" xref="S4.E12.m1.14.14.14.14.14.14.cmml"><mover accent="true" id="S4.E12.m1.14.14.14.14.14.14.2" xref="S4.E12.m1.14.14.14.14.14.14.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.14.14.14.14.14.14.2.2" mathvariant="bold-italic" xref="S4.E12.m1.14.14.14.14.14.14.2.2.cmml">ϵ</mi><mo id="S4.E12.m1.14.14.14.14.14.14.2.1" xref="S4.E12.m1.14.14.14.14.14.14.2.1.cmml">¯</mo></mover><mo id="S4.E12.m1.14.14.14.14.14.14.1" xref="S4.E12.m1.14.14.14.14.14.14.1.cmml">¯</mo></mover><mrow id="S4.E12.m1.15.15.15.15.15.15.1" xref="S4.E12.m1.15.15.15.15.15.15.1.cmml"><mo id="S4.E12.m1.15.15.15.15.15.15.1a" xref="S4.E12.m1.15.15.15.15.15.15.1.cmml">−</mo><mn id="S4.E12.m1.15.15.15.15.15.15.1.2" xref="S4.E12.m1.15.15.15.15.15.15.1.2.cmml">1</mn></mrow></msup><mo id="S4.E12.m1.16.16.16.16.16.16" xref="S4.E12.m1.66.66.6.cmml">)</mo></mrow><mo id="S4.E12.m1.69.69.9.63.34.34.34.3.3.4a" lspace="0.167em" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1" xref="S4.E12.m1.66.66.6.cmml"><mpadded width="0.748em"><mi id="S4.E12.m1.17.17.17.17.17.17" xref="S4.E12.m1.17.17.17.17.17.17.cmml">ln</mi></mpadded><mo id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1a" xref="S4.E12.m1.66.66.6.cmml">⁡</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.18.18.18.18.18.18" xref="S4.E12.m1.66.66.6.cmml">(</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mpadded width="0.333em"><mi id="S4.E12.m1.19.19.19.19.19.19" xref="S4.E12.m1.19.19.19.19.19.19.cmml">p</mi></mpadded><mo id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1.1.2" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.20.20.20.20.20.20" xref="S4.E12.m1.66.66.6.cmml">(</mo><mrow id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1.1.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><msubsup id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1.1.1.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mi id="S4.E12.m1.21.21.21.21.21.21" xref="S4.E12.m1.21.21.21.21.21.21.cmml">𝒁</mi><mi id="S4.E12.m1.22.22.22.22.22.22.1" xref="S4.E12.m1.22.22.22.22.22.22.1.cmml">n</mi><mrow id="S4.E12.m1.23.23.23.23.23.23.1.3" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.23.23.23.23.23.23.1.3.1" stretchy="false" xref="S4.E12.m1.66.66.6.cmml">(</mo><mi id="S4.E12.m1.23.23.23.23.23.23.1.1" xref="S4.E12.m1.23.23.23.23.23.23.1.1.cmml">k</mi><mo id="S4.E12.m1.23.23.23.23.23.23.1.3.2" stretchy="false" xref="S4.E12.m1.66.66.6.cmml">)</mo></mrow></msubsup><mo fence="false" id="S4.E12.m1.24.24.24.24.24.24" xref="S4.E12.m1.24.24.24.24.24.24.cmml">|</mo><msub id="S4.E12.m1.69.69.9.63.34.34.34.3.3.3.1.1.1.1.1.1.2" xref="S4.E12.m1.66.66.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.25.25.25.25.25.25" mathvariant="bold-italic" xref="S4.E12.m1.25.25.25.25.25.25.cmml">ϕ</mi><mi id="S4.E12.m1.26.26.26.26.26.26.1" xref="S4.E12.m1.26.26.26.26.26.26.1.cmml">n</mi></msub></mrow><mpadded width="0.288em"><mo id="S4.E12.m1.27.27.27.27.27.27" xref="S4.E12.m1.66.66.6.cmml">)</mo></mpadded></mrow></mrow><mo id="S4.E12.m1.28.28.28.28.28.28" xref="S4.E12.m1.66.66.6.cmml">)</mo></mrow></mrow><mo id="S4.E12.m1.69.69.9.63.34.34.34.3.3.4b" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mtext id="S4.E12.m1.29.29.29.29.29.29" xref="S4.E12.m1.29.29.29.29.29.29a.cmml">d</mtext><mo id="S4.E12.m1.69.69.9.63.34.34.34.3.3.4c" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><msub id="S4.E12.m1.69.69.9.63.34.34.34.3.3.5" xref="S4.E12.m1.66.66.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.30.30.30.30.30.30" mathvariant="bold-italic" xref="S4.E12.m1.30.30.30.30.30.30.cmml">ϕ</mi><mi id="S4.E12.m1.31.31.31.31.31.31.1" xref="S4.E12.m1.31.31.31.31.31.31.1.cmml">n</mi></msub></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E12.m1.72.72.12c" xref="S4.E12.m1.66.66.6.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E12.m1.72.72.12d" xref="S4.E12.m1.66.66.6.cmml"><mrow id="S4.E12.m1.72.72.12.66.32.32" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.72.72.12.66.32.32a" xref="S4.E12.m1.66.66.6.cmml">+</mo><mrow id="S4.E12.m1.72.72.12.66.32.32.32" xref="S4.E12.m1.66.66.6.cmml"><msub id="S4.E12.m1.72.72.12.66.32.32.32.4" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.33.33.33.2.2.2" xref="S4.E12.m1.33.33.33.2.2.2.cmml">∫</mo><msub id="S4.E12.m1.34.34.34.3.3.3.1" xref="S4.E12.m1.34.34.34.3.3.3.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.34.34.34.3.3.3.1.2" mathvariant="bold-italic" xref="S4.E12.m1.34.34.34.3.3.3.1.2.cmml">ϕ</mi><mi id="S4.E12.m1.34.34.34.3.3.3.1.3" xref="S4.E12.m1.34.34.34.3.3.3.1.3.cmml">n</mi></msub></msub><mrow id="S4.E12.m1.72.72.12.66.32.32.32.3" xref="S4.E12.m1.66.66.6.cmml"><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E12.m1.35.35.35.4.4.4" xref="S4.E12.m1.35.35.35.4.4.4.cmml">𝒩</mi></mpadded><mo id="S4.E12.m1.72.72.12.66.32.32.32.3.4" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mrow id="S4.E12.m1.71.71.11.65.31.31.31.2.2.2" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.36.36.36.5.5.5" xref="S4.E12.m1.66.66.6.cmml">(</mo><msub id="S4.E12.m1.70.70.10.64.30.30.30.1.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.37.37.37.6.6.6" mathvariant="bold-italic" xref="S4.E12.m1.37.37.37.6.6.6.cmml">ϕ</mi><mi id="S4.E12.m1.38.38.38.7.7.7.1" xref="S4.E12.m1.38.38.38.7.7.7.1.cmml">n</mi></msub><mo id="S4.E12.m1.39.39.39.8.8.8" xref="S4.E12.m1.66.66.6.cmml">;</mo><mover accent="true" id="S4.E12.m1.40.40.40.9.9.9" xref="S4.E12.m1.40.40.40.9.9.9.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.40.40.40.9.9.9.2" mathvariant="bold-italic" xref="S4.E12.m1.40.40.40.9.9.9.2.cmml">ϵ</mi><mo id="S4.E12.m1.40.40.40.9.9.9.1" xref="S4.E12.m1.40.40.40.9.9.9.1.cmml">¯</mo></mover><mo id="S4.E12.m1.41.41.41.10.10.10" xref="S4.E12.m1.66.66.6.cmml">,</mo><msup id="S4.E12.m1.71.71.11.65.31.31.31.2.2.2.2" xref="S4.E12.m1.66.66.6.cmml"><mover accent="true" id="S4.E12.m1.42.42.42.11.11.11" xref="S4.E12.m1.42.42.42.11.11.11.cmml"><mover accent="true" id="S4.E12.m1.42.42.42.11.11.11.2" xref="S4.E12.m1.42.42.42.11.11.11.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.42.42.42.11.11.11.2.2" mathvariant="bold-italic" xref="S4.E12.m1.42.42.42.11.11.11.2.2.cmml">ϵ</mi><mo id="S4.E12.m1.42.42.42.11.11.11.2.1" xref="S4.E12.m1.42.42.42.11.11.11.2.1.cmml">¯</mo></mover><mo id="S4.E12.m1.42.42.42.11.11.11.1" xref="S4.E12.m1.42.42.42.11.11.11.1.cmml">¯</mo></mover><mrow id="S4.E12.m1.43.43.43.12.12.12.1" xref="S4.E12.m1.43.43.43.12.12.12.1.cmml"><mo id="S4.E12.m1.43.43.43.12.12.12.1a" xref="S4.E12.m1.43.43.43.12.12.12.1.cmml">−</mo><mn id="S4.E12.m1.43.43.43.12.12.12.1.2" xref="S4.E12.m1.43.43.43.12.12.12.1.2.cmml">1</mn></mrow></msup><mo id="S4.E12.m1.44.44.44.13.13.13" xref="S4.E12.m1.66.66.6.cmml">)</mo></mrow><mo id="S4.E12.m1.72.72.12.66.32.32.32.3.4a" lspace="0.167em" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mrow id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1" xref="S4.E12.m1.66.66.6.cmml"><mpadded width="0.748em"><mi id="S4.E12.m1.45.45.45.14.14.14" xref="S4.E12.m1.45.45.45.14.14.14.cmml">ln</mi></mpadded><mo id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1a" xref="S4.E12.m1.66.66.6.cmml">⁡</mo><mrow id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1.1" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.46.46.46.15.15.15" xref="S4.E12.m1.66.66.6.cmml">(</mo><mrow id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E12.m1.47.47.47.16.16.16" xref="S4.E12.m1.47.47.47.16.16.16.cmml">𝒩</mi></mpadded><mo id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1.1.1.3" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mrow id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1.1.1.2.2" xref="S4.E12.m1.66.66.6.cmml"><mo id="S4.E12.m1.48.48.48.17.17.17" xref="S4.E12.m1.66.66.6.cmml">(</mo><msub id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1.1.1.1.1.1" xref="S4.E12.m1.66.66.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.49.49.49.18.18.18" mathvariant="bold-italic" xref="S4.E12.m1.49.49.49.18.18.18.cmml">ϕ</mi><mi id="S4.E12.m1.50.50.50.19.19.19.1" xref="S4.E12.m1.50.50.50.19.19.19.1.cmml">n</mi></msub><mo id="S4.E12.m1.51.51.51.20.20.20" xref="S4.E12.m1.66.66.6.cmml">;</mo><mover accent="true" id="S4.E12.m1.52.52.52.21.21.21" xref="S4.E12.m1.52.52.52.21.21.21.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.52.52.52.21.21.21.2" mathvariant="bold-italic" xref="S4.E12.m1.52.52.52.21.21.21.2.cmml">ϵ</mi><mo id="S4.E12.m1.52.52.52.21.21.21.1" xref="S4.E12.m1.52.52.52.21.21.21.1.cmml">¯</mo></mover><mo id="S4.E12.m1.53.53.53.22.22.22" xref="S4.E12.m1.66.66.6.cmml">,</mo><msup id="S4.E12.m1.72.72.12.66.32.32.32.3.3.1.1.1.2.2.2" xref="S4.E12.m1.66.66.6.cmml"><mover accent="true" id="S4.E12.m1.54.54.54.23.23.23" xref="S4.E12.m1.54.54.54.23.23.23.cmml"><mover accent="true" id="S4.E12.m1.54.54.54.23.23.23.2" xref="S4.E12.m1.54.54.54.23.23.23.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.54.54.54.23.23.23.2.2" mathvariant="bold-italic" xref="S4.E12.m1.54.54.54.23.23.23.2.2.cmml">ϵ</mi><mo id="S4.E12.m1.54.54.54.23.23.23.2.1" xref="S4.E12.m1.54.54.54.23.23.23.2.1.cmml">¯</mo></mover><mo id="S4.E12.m1.54.54.54.23.23.23.1" xref="S4.E12.m1.54.54.54.23.23.23.1.cmml">¯</mo></mover><mrow id="S4.E12.m1.55.55.55.24.24.24.1" xref="S4.E12.m1.55.55.55.24.24.24.1.cmml"><mo id="S4.E12.m1.55.55.55.24.24.24.1a" xref="S4.E12.m1.55.55.55.24.24.24.1.cmml">−</mo><mn id="S4.E12.m1.55.55.55.24.24.24.1.2" xref="S4.E12.m1.55.55.55.24.24.24.1.2.cmml">1</mn></mrow></msup><mpadded width="0.288em"><mo id="S4.E12.m1.56.56.56.25.25.25" xref="S4.E12.m1.66.66.6.cmml">)</mo></mpadded></mrow></mrow><mo id="S4.E12.m1.57.57.57.26.26.26" xref="S4.E12.m1.66.66.6.cmml">)</mo></mrow></mrow><mo id="S4.E12.m1.72.72.12.66.32.32.32.3.4b" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><mtext id="S4.E12.m1.58.58.58.27.27.27" xref="S4.E12.m1.58.58.58.27.27.27a.cmml">d</mtext><mo id="S4.E12.m1.72.72.12.66.32.32.32.3.4c" xref="S4.E12.m1.66.66.6.cmml">⁢</mo><msub id="S4.E12.m1.72.72.12.66.32.32.32.3.5" xref="S4.E12.m1.66.66.6.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E12.m1.59.59.59.28.28.28" mathvariant="bold-italic" xref="S4.E12.m1.59.59.59.28.28.28.cmml">ϕ</mi><mi id="S4.E12.m1.60.60.60.29.29.29.1" xref="S4.E12.m1.60.60.60.29.29.29.1.cmml">n</mi></msub></mrow></mrow></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E12.m1.72b"><apply id="S4.E12.m1.66.66.6.cmml" xref="S4.E12.m1.72.72.12"><eq id="S4.E12.m1.3.3.3.3.3.3.cmml" xref="S4.E12.m1.3.3.3.3.3.3"></eq><apply id="S4.E12.m1.66.66.6.8.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.66.66.6.8.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.1.1.1.1.1.1.cmml" xref="S4.E12.m1.1.1.1.1.1.1">𝐷</ci><apply id="S4.E12.m1.2.2.2.2.2.2.1.cmml" xref="S4.E12.m1.2.2.2.2.2.2.1"><times id="S4.E12.m1.2.2.2.2.2.2.1.1.cmml" xref="S4.E12.m1.2.2.2.2.2.2.1.1"></times><ci id="S4.E12.m1.2.2.2.2.2.2.1.2.cmml" xref="S4.E12.m1.2.2.2.2.2.2.1.2">𝐾</ci><ci id="S4.E12.m1.2.2.2.2.2.2.1.3.cmml" xref="S4.E12.m1.2.2.2.2.2.2.1.3">𝐿</ci></apply></apply><apply id="S4.E12.m1.66.66.6.6.cmml" xref="S4.E12.m1.72.72.12"><plus id="S4.E12.m1.32.32.32.1.1.1.cmml" xref="S4.E12.m1.72.72.12"></plus><apply id="S4.E12.m1.63.63.3.3.3.cmml" xref="S4.E12.m1.72.72.12"><minus id="S4.E12.m1.4.4.4.4.4.4.cmml" xref="S4.E12.m1.72.72.12"></minus><apply id="S4.E12.m1.63.63.3.3.3.3.cmml" xref="S4.E12.m1.72.72.12"><apply id="S4.E12.m1.63.63.3.3.3.3.4.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.63.63.3.3.3.3.4.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><int id="S4.E12.m1.5.5.5.5.5.5.cmml" xref="S4.E12.m1.5.5.5.5.5.5"></int><apply id="S4.E12.m1.6.6.6.6.6.6.1.cmml" xref="S4.E12.m1.6.6.6.6.6.6.1"><csymbol cd="ambiguous" id="S4.E12.m1.6.6.6.6.6.6.1.1.cmml" xref="S4.E12.m1.6.6.6.6.6.6.1">subscript</csymbol><ci id="S4.E12.m1.6.6.6.6.6.6.1.2.cmml" xref="S4.E12.m1.6.6.6.6.6.6.1.2">bold-italic-ϕ</ci><ci id="S4.E12.m1.6.6.6.6.6.6.1.3.cmml" xref="S4.E12.m1.6.6.6.6.6.6.1.3">𝑛</ci></apply></apply><apply id="S4.E12.m1.63.63.3.3.3.3.3.cmml" xref="S4.E12.m1.72.72.12"><times id="S4.E12.m1.63.63.3.3.3.3.3.4.cmml" xref="S4.E12.m1.72.72.12"></times><ci id="S4.E12.m1.7.7.7.7.7.7.cmml" xref="S4.E12.m1.7.7.7.7.7.7">𝒩</ci><list id="S4.E12.m1.62.62.2.2.2.2.2.2.3.cmml" xref="S4.E12.m1.72.72.12"><apply id="S4.E12.m1.61.61.1.1.1.1.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.61.61.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.9.9.9.9.9.9.cmml" xref="S4.E12.m1.9.9.9.9.9.9">bold-italic-ϕ</ci><ci id="S4.E12.m1.10.10.10.10.10.10.1.cmml" xref="S4.E12.m1.10.10.10.10.10.10.1">𝑛</ci></apply><apply id="S4.E12.m1.12.12.12.12.12.12.cmml" xref="S4.E12.m1.12.12.12.12.12.12"><ci id="S4.E12.m1.12.12.12.12.12.12.1.cmml" xref="S4.E12.m1.12.12.12.12.12.12.1">¯</ci><ci id="S4.E12.m1.12.12.12.12.12.12.2.cmml" xref="S4.E12.m1.12.12.12.12.12.12.2">bold-italic-ϵ</ci></apply><apply id="S4.E12.m1.62.62.2.2.2.2.2.2.2.2.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.62.62.2.2.2.2.2.2.2.2.1.cmml" xref="S4.E12.m1.72.72.12">superscript</csymbol><apply id="S4.E12.m1.14.14.14.14.14.14.cmml" xref="S4.E12.m1.14.14.14.14.14.14"><ci id="S4.E12.m1.14.14.14.14.14.14.1.cmml" xref="S4.E12.m1.14.14.14.14.14.14.1">¯</ci><apply id="S4.E12.m1.14.14.14.14.14.14.2.cmml" xref="S4.E12.m1.14.14.14.14.14.14.2"><ci id="S4.E12.m1.14.14.14.14.14.14.2.1.cmml" xref="S4.E12.m1.14.14.14.14.14.14.2.1">¯</ci><ci id="S4.E12.m1.14.14.14.14.14.14.2.2.cmml" xref="S4.E12.m1.14.14.14.14.14.14.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E12.m1.15.15.15.15.15.15.1.cmml" xref="S4.E12.m1.15.15.15.15.15.15.1"><minus id="S4.E12.m1.15.15.15.15.15.15.1.1.cmml" xref="S4.E12.m1.15.15.15.15.15.15.1"></minus><cn id="S4.E12.m1.15.15.15.15.15.15.1.2.cmml" type="integer" xref="S4.E12.m1.15.15.15.15.15.15.1.2">1</cn></apply></apply></list><apply id="S4.E12.m1.63.63.3.3.3.3.3.3.2.cmml" xref="S4.E12.m1.72.72.12"><ln id="S4.E12.m1.17.17.17.17.17.17.cmml" xref="S4.E12.m1.17.17.17.17.17.17"></ln><apply id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.cmml" xref="S4.E12.m1.72.72.12"><times id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.2.cmml" xref="S4.E12.m1.72.72.12"></times><ci id="S4.E12.m1.19.19.19.19.19.19.cmml" xref="S4.E12.m1.19.19.19.19.19.19">𝑝</ci><apply id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="latexml" id="S4.E12.m1.24.24.24.24.24.24.cmml" xref="S4.E12.m1.24.24.24.24.24.24">conditional</csymbol><apply id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.2.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.2.1.cmml" xref="S4.E12.m1.72.72.12">superscript</csymbol><apply id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.2.2.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.21.21.21.21.21.21.cmml" xref="S4.E12.m1.21.21.21.21.21.21">𝒁</ci><ci id="S4.E12.m1.22.22.22.22.22.22.1.cmml" xref="S4.E12.m1.22.22.22.22.22.22.1">𝑛</ci></apply><ci id="S4.E12.m1.23.23.23.23.23.23.1.1.cmml" xref="S4.E12.m1.23.23.23.23.23.23.1.1">𝑘</ci></apply><apply id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.3.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.63.63.3.3.3.3.3.3.1.1.1.1.1.1.3.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.25.25.25.25.25.25.cmml" xref="S4.E12.m1.25.25.25.25.25.25">bold-italic-ϕ</ci><ci id="S4.E12.m1.26.26.26.26.26.26.1.cmml" xref="S4.E12.m1.26.26.26.26.26.26.1">𝑛</ci></apply></apply></apply></apply><ci id="S4.E12.m1.29.29.29.29.29.29a.cmml" xref="S4.E12.m1.29.29.29.29.29.29"><mtext id="S4.E12.m1.29.29.29.29.29.29.cmml" xref="S4.E12.m1.29.29.29.29.29.29">d</mtext></ci><apply id="S4.E12.m1.63.63.3.3.3.3.3.7.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.63.63.3.3.3.3.3.7.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.30.30.30.30.30.30.cmml" xref="S4.E12.m1.30.30.30.30.30.30">bold-italic-ϕ</ci><ci id="S4.E12.m1.31.31.31.31.31.31.1.cmml" xref="S4.E12.m1.31.31.31.31.31.31.1">𝑛</ci></apply></apply></apply></apply><apply id="S4.E12.m1.66.66.6.6.6.cmml" xref="S4.E12.m1.72.72.12"><apply id="S4.E12.m1.66.66.6.6.6.4.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.66.66.6.6.6.4.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><int id="S4.E12.m1.33.33.33.2.2.2.cmml" xref="S4.E12.m1.33.33.33.2.2.2"></int><apply id="S4.E12.m1.34.34.34.3.3.3.1.cmml" xref="S4.E12.m1.34.34.34.3.3.3.1"><csymbol cd="ambiguous" id="S4.E12.m1.34.34.34.3.3.3.1.1.cmml" xref="S4.E12.m1.34.34.34.3.3.3.1">subscript</csymbol><ci id="S4.E12.m1.34.34.34.3.3.3.1.2.cmml" xref="S4.E12.m1.34.34.34.3.3.3.1.2">bold-italic-ϕ</ci><ci id="S4.E12.m1.34.34.34.3.3.3.1.3.cmml" xref="S4.E12.m1.34.34.34.3.3.3.1.3">𝑛</ci></apply></apply><apply id="S4.E12.m1.66.66.6.6.6.3.cmml" xref="S4.E12.m1.72.72.12"><times id="S4.E12.m1.66.66.6.6.6.3.4.cmml" xref="S4.E12.m1.72.72.12"></times><ci id="S4.E12.m1.35.35.35.4.4.4.cmml" xref="S4.E12.m1.35.35.35.4.4.4">𝒩</ci><list id="S4.E12.m1.65.65.5.5.5.2.2.3.cmml" xref="S4.E12.m1.72.72.12"><apply id="S4.E12.m1.64.64.4.4.4.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.64.64.4.4.4.1.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.37.37.37.6.6.6.cmml" xref="S4.E12.m1.37.37.37.6.6.6">bold-italic-ϕ</ci><ci id="S4.E12.m1.38.38.38.7.7.7.1.cmml" xref="S4.E12.m1.38.38.38.7.7.7.1">𝑛</ci></apply><apply id="S4.E12.m1.40.40.40.9.9.9.cmml" xref="S4.E12.m1.40.40.40.9.9.9"><ci id="S4.E12.m1.40.40.40.9.9.9.1.cmml" xref="S4.E12.m1.40.40.40.9.9.9.1">¯</ci><ci id="S4.E12.m1.40.40.40.9.9.9.2.cmml" xref="S4.E12.m1.40.40.40.9.9.9.2">bold-italic-ϵ</ci></apply><apply id="S4.E12.m1.65.65.5.5.5.2.2.2.2.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.65.65.5.5.5.2.2.2.2.1.cmml" xref="S4.E12.m1.72.72.12">superscript</csymbol><apply id="S4.E12.m1.42.42.42.11.11.11.cmml" xref="S4.E12.m1.42.42.42.11.11.11"><ci id="S4.E12.m1.42.42.42.11.11.11.1.cmml" xref="S4.E12.m1.42.42.42.11.11.11.1">¯</ci><apply id="S4.E12.m1.42.42.42.11.11.11.2.cmml" xref="S4.E12.m1.42.42.42.11.11.11.2"><ci id="S4.E12.m1.42.42.42.11.11.11.2.1.cmml" xref="S4.E12.m1.42.42.42.11.11.11.2.1">¯</ci><ci id="S4.E12.m1.42.42.42.11.11.11.2.2.cmml" xref="S4.E12.m1.42.42.42.11.11.11.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E12.m1.43.43.43.12.12.12.1.cmml" xref="S4.E12.m1.43.43.43.12.12.12.1"><minus id="S4.E12.m1.43.43.43.12.12.12.1.1.cmml" xref="S4.E12.m1.43.43.43.12.12.12.1"></minus><cn id="S4.E12.m1.43.43.43.12.12.12.1.2.cmml" type="integer" xref="S4.E12.m1.43.43.43.12.12.12.1.2">1</cn></apply></apply></list><apply id="S4.E12.m1.66.66.6.6.6.3.3.2.cmml" xref="S4.E12.m1.72.72.12"><ln id="S4.E12.m1.45.45.45.14.14.14.cmml" xref="S4.E12.m1.45.45.45.14.14.14"></ln><apply id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.cmml" xref="S4.E12.m1.72.72.12"><times id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.3.cmml" xref="S4.E12.m1.72.72.12"></times><ci id="S4.E12.m1.47.47.47.16.16.16.cmml" xref="S4.E12.m1.47.47.47.16.16.16">𝒩</ci><list id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.2.3.cmml" xref="S4.E12.m1.72.72.12"><apply id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.49.49.49.18.18.18.cmml" xref="S4.E12.m1.49.49.49.18.18.18">bold-italic-ϕ</ci><ci id="S4.E12.m1.50.50.50.19.19.19.1.cmml" xref="S4.E12.m1.50.50.50.19.19.19.1">𝑛</ci></apply><apply id="S4.E12.m1.52.52.52.21.21.21.cmml" xref="S4.E12.m1.52.52.52.21.21.21"><ci id="S4.E12.m1.52.52.52.21.21.21.1.cmml" xref="S4.E12.m1.52.52.52.21.21.21.1">¯</ci><ci id="S4.E12.m1.52.52.52.21.21.21.2.cmml" xref="S4.E12.m1.52.52.52.21.21.21.2">bold-italic-ϵ</ci></apply><apply id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.2.2.2.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.66.66.6.6.6.3.3.1.1.1.2.2.2.1.cmml" xref="S4.E12.m1.72.72.12">superscript</csymbol><apply id="S4.E12.m1.54.54.54.23.23.23.cmml" xref="S4.E12.m1.54.54.54.23.23.23"><ci id="S4.E12.m1.54.54.54.23.23.23.1.cmml" xref="S4.E12.m1.54.54.54.23.23.23.1">¯</ci><apply id="S4.E12.m1.54.54.54.23.23.23.2.cmml" xref="S4.E12.m1.54.54.54.23.23.23.2"><ci id="S4.E12.m1.54.54.54.23.23.23.2.1.cmml" xref="S4.E12.m1.54.54.54.23.23.23.2.1">¯</ci><ci id="S4.E12.m1.54.54.54.23.23.23.2.2.cmml" xref="S4.E12.m1.54.54.54.23.23.23.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E12.m1.55.55.55.24.24.24.1.cmml" xref="S4.E12.m1.55.55.55.24.24.24.1"><minus id="S4.E12.m1.55.55.55.24.24.24.1.1.cmml" xref="S4.E12.m1.55.55.55.24.24.24.1"></minus><cn id="S4.E12.m1.55.55.55.24.24.24.1.2.cmml" type="integer" xref="S4.E12.m1.55.55.55.24.24.24.1.2">1</cn></apply></apply></list></apply></apply><ci id="S4.E12.m1.58.58.58.27.27.27a.cmml" xref="S4.E12.m1.58.58.58.27.27.27"><mtext id="S4.E12.m1.58.58.58.27.27.27.cmml" xref="S4.E12.m1.58.58.58.27.27.27">d</mtext></ci><apply id="S4.E12.m1.66.66.6.6.6.3.7.cmml" xref="S4.E12.m1.72.72.12"><csymbol cd="ambiguous" id="S4.E12.m1.66.66.6.6.6.3.7.1.cmml" xref="S4.E12.m1.72.72.12">subscript</csymbol><ci id="S4.E12.m1.59.59.59.28.28.28.cmml" xref="S4.E12.m1.59.59.59.28.28.28">bold-italic-ϕ</ci><ci id="S4.E12.m1.60.60.60.29.29.29.1.cmml" xref="S4.E12.m1.60.60.60.29.29.29.1">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12.m1.72c">D_{KL}=-\int_{\bm{\phi}_{n}}\mathcal{N}\!\left(\bm{\phi}_{n};\overline{\bm{% \epsilon}},\overline{\overline{\bm{\epsilon}}}^{-1}\right)\ln\!\left(p\!\left(% \bm{Z}_{n}^{(k)}|\bm{\phi}_{n}\right)\!\right)\text{d}\bm{\phi}_{n}\\ +\int_{\bm{\phi}_{n}}\mathcal{N}\!\left(\bm{\phi}_{n};\overline{\bm{\epsilon}}% ,\overline{\overline{\bm{\epsilon}}}^{-1}\right)\ln\!\left(\mathcal{N}\!\left(% \bm{\phi}_{n};\overline{\bm{\epsilon}},\overline{\overline{\bm{\epsilon}}}^{-1% }\right)\!\right)\text{d}\bm{\phi}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.E12.m1.72d">start_ROW start_CELL italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT = - ∫ start_POSTSUBSCRIPT bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_N ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) roman_ln ( italic_p ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) d bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + ∫ start_POSTSUBSCRIPT bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_N ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) roman_ln ( caligraphic_N ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ) d bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT 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">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.9">where the second term is the entropy of a Gaussian distribution, <math alttext="\zeta(\bar{\bm{\epsilon}},\bar{\bar{\bm{\epsilon}}})" class="ltx_Math" display="inline" id="S4.p2.8.m1.2"><semantics id="S4.p2.8.m1.2a"><mrow id="S4.p2.8.m1.2.3" xref="S4.p2.8.m1.2.3.cmml"><mi id="S4.p2.8.m1.2.3.2" xref="S4.p2.8.m1.2.3.2.cmml">ζ</mi><mo id="S4.p2.8.m1.2.3.1" xref="S4.p2.8.m1.2.3.1.cmml">⁢</mo><mrow id="S4.p2.8.m1.2.3.3.2" xref="S4.p2.8.m1.2.3.3.1.cmml"><mo id="S4.p2.8.m1.2.3.3.2.1" stretchy="false" xref="S4.p2.8.m1.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.p2.8.m1.1.1" xref="S4.p2.8.m1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.8.m1.1.1.2" mathvariant="bold-italic" xref="S4.p2.8.m1.1.1.2.cmml">ϵ</mi><mo id="S4.p2.8.m1.1.1.1" xref="S4.p2.8.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.p2.8.m1.2.3.3.2.2" xref="S4.p2.8.m1.2.3.3.1.cmml">,</mo><mover accent="true" id="S4.p2.8.m1.2.2" xref="S4.p2.8.m1.2.2.cmml"><mover accent="true" id="S4.p2.8.m1.2.2.2" xref="S4.p2.8.m1.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.8.m1.2.2.2.2" mathvariant="bold-italic" xref="S4.p2.8.m1.2.2.2.2.cmml">ϵ</mi><mo id="S4.p2.8.m1.2.2.2.1" xref="S4.p2.8.m1.2.2.2.1.cmml">¯</mo></mover><mo id="S4.p2.8.m1.2.2.1" xref="S4.p2.8.m1.2.2.1.cmml">¯</mo></mover><mo id="S4.p2.8.m1.2.3.3.2.3" stretchy="false" xref="S4.p2.8.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.8.m1.2b"><apply id="S4.p2.8.m1.2.3.cmml" xref="S4.p2.8.m1.2.3"><times id="S4.p2.8.m1.2.3.1.cmml" xref="S4.p2.8.m1.2.3.1"></times><ci id="S4.p2.8.m1.2.3.2.cmml" xref="S4.p2.8.m1.2.3.2">𝜁</ci><interval closure="open" id="S4.p2.8.m1.2.3.3.1.cmml" xref="S4.p2.8.m1.2.3.3.2"><apply id="S4.p2.8.m1.1.1.cmml" xref="S4.p2.8.m1.1.1"><ci id="S4.p2.8.m1.1.1.1.cmml" xref="S4.p2.8.m1.1.1.1">¯</ci><ci id="S4.p2.8.m1.1.1.2.cmml" xref="S4.p2.8.m1.1.1.2">bold-italic-ϵ</ci></apply><apply id="S4.p2.8.m1.2.2.cmml" xref="S4.p2.8.m1.2.2"><ci id="S4.p2.8.m1.2.2.1.cmml" xref="S4.p2.8.m1.2.2.1">¯</ci><apply id="S4.p2.8.m1.2.2.2.cmml" xref="S4.p2.8.m1.2.2.2"><ci id="S4.p2.8.m1.2.2.2.1.cmml" xref="S4.p2.8.m1.2.2.2.1">¯</ci><ci id="S4.p2.8.m1.2.2.2.2.cmml" xref="S4.p2.8.m1.2.2.2.2">bold-italic-ϵ</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m1.2c">\zeta(\bar{\bm{\epsilon}},\bar{\bar{\bm{\epsilon}}})</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m1.2d">italic_ζ ( over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG )</annotation></semantics></math>. The first term is the expectation of <math alttext="\ln(p(\bm{Z}_{n}^{(k)}|\bm{\phi}_{n}))" class="ltx_Math" display="inline" id="S4.p2.9.m2.3"><semantics id="S4.p2.9.m2.3a"><mrow id="S4.p2.9.m2.3.3.1" xref="S4.p2.9.m2.3.3.2.cmml"><mi id="S4.p2.9.m2.2.2" xref="S4.p2.9.m2.2.2.cmml">ln</mi><mo id="S4.p2.9.m2.3.3.1a" xref="S4.p2.9.m2.3.3.2.cmml">⁡</mo><mrow id="S4.p2.9.m2.3.3.1.1" xref="S4.p2.9.m2.3.3.2.cmml"><mo id="S4.p2.9.m2.3.3.1.1.2" stretchy="false" xref="S4.p2.9.m2.3.3.2.cmml">(</mo><mrow id="S4.p2.9.m2.3.3.1.1.1" xref="S4.p2.9.m2.3.3.1.1.1.cmml"><mi id="S4.p2.9.m2.3.3.1.1.1.3" xref="S4.p2.9.m2.3.3.1.1.1.3.cmml">p</mi><mo id="S4.p2.9.m2.3.3.1.1.1.2" xref="S4.p2.9.m2.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.p2.9.m2.3.3.1.1.1.1.1" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.cmml"><mo id="S4.p2.9.m2.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.p2.9.m2.3.3.1.1.1.1.1.1" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.cmml"><msubsup id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.cmml"><mi id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.2" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.2.cmml">𝒁</mi><mi id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.3" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.3.cmml">n</mi><mrow id="S4.p2.9.m2.1.1.1.3" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.cmml"><mo id="S4.p2.9.m2.1.1.1.3.1" stretchy="false" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.cmml">(</mo><mi id="S4.p2.9.m2.1.1.1.1" xref="S4.p2.9.m2.1.1.1.1.cmml">k</mi><mo id="S4.p2.9.m2.1.1.1.3.2" stretchy="false" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.cmml">)</mo></mrow></msubsup><mo fence="false" id="S4.p2.9.m2.3.3.1.1.1.1.1.1.1" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.1.cmml">|</mo><msub id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.2" mathvariant="bold-italic" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.3" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.p2.9.m2.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.p2.9.m2.3.3.1.1.3" stretchy="false" xref="S4.p2.9.m2.3.3.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.9.m2.3b"><apply id="S4.p2.9.m2.3.3.2.cmml" xref="S4.p2.9.m2.3.3.1"><ln id="S4.p2.9.m2.2.2.cmml" xref="S4.p2.9.m2.2.2"></ln><apply id="S4.p2.9.m2.3.3.1.1.1.cmml" xref="S4.p2.9.m2.3.3.1.1.1"><times id="S4.p2.9.m2.3.3.1.1.1.2.cmml" xref="S4.p2.9.m2.3.3.1.1.1.2"></times><ci id="S4.p2.9.m2.3.3.1.1.1.3.cmml" xref="S4.p2.9.m2.3.3.1.1.1.3">𝑝</ci><apply id="S4.p2.9.m2.3.3.1.1.1.1.1.1.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S4.p2.9.m2.3.3.1.1.1.1.1.1.1.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.1">conditional</csymbol><apply id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.1.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2">superscript</csymbol><apply id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.1.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.2.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.2">𝒁</ci><ci id="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.3.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.2.2.3">𝑛</ci></apply><ci id="S4.p2.9.m2.1.1.1.1.cmml" xref="S4.p2.9.m2.1.1.1.1">𝑘</ci></apply><apply id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.1.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.2.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.2">bold-italic-ϕ</ci><ci id="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.3.cmml" xref="S4.p2.9.m2.3.3.1.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.9.m2.3c">\ln(p(\bm{Z}_{n}^{(k)}|\bm{\phi}_{n}))</annotation><annotation encoding="application/x-llamapun" id="S4.p2.9.m2.3d">roman_ln ( italic_p ( bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) )</annotation></semantics></math> w.r.t. a Gaussian and reads</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx4"> <tbody id="S4.E16"><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\begin{multlined}\mathbb{E}\!\left[-\!\left\langle\bm{S}_{n}^{(k)% }\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\bm{S}_{n}^{(% k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\right\rangle\!\right]\\ \propto-\!\left\langle\mathbb{E}\!\left[\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \right]-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\mathbb{E}\!\left[% \mathbf{S}_{n}^{(k)}\!(\bm{\phi}_{n})\right]-\bm{Z}_{n}^{(k)}\right\rangle\\ -\text{Tr}\!\left(\mathbb{E}\!\left[\!\big{|}\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \big{\rangle}\big{\langle}\bm{S}_{n}^{(k)}(\bm{\phi}_{n})\big{|}\right]\bm{% \Lambda}_{Z}\!\right)\!.\end{multlined}\mathbb{E}\!\left[-\!\left\langle\bm{S}% _{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\bm{% S}_{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\right\rangle\!\right]\\ \propto-\!\left\langle\mathbb{E}\!\left[\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \right]-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\mathbb{E}\!\left[% \mathbf{S}_{n}^{(k)}\!(\bm{\phi}_{n})\right]-\bm{Z}_{n}^{(k)}\right\rangle\\ -\text{Tr}\!\left(\mathbb{E}\!\left[\!\big{|}\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \big{\rangle}\big{\langle}\bm{S}_{n}^{(k)}(\bm{\phi}_{n})\big{|}\right]\bm{% \Lambda}_{Z}\!\right)\!." class="ltx_Math" display="inline" id="S4.E16.m1.1"><semantics id="S4.E16.m1.1a"><mtable id="S4.E16.m1.1.1.2" rowspacing="0pt"><mtr id="S4.E16.m1.1.1.2a"><mtd class="ltx_align_left" columnalign="left" id="S4.E16.m1.1.1.2b"><mrow id="S4.E13.33.33"><mpadded width="0.497em"><mi id="S4.E13.1.1.1" xref="S4.E13.1.1.1.cmml">𝔼</mi></mpadded><mo id="S4.E13.33.33.34" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E13.33.33.33.1"><mo id="S4.E13.2.2.2" xref="S4.E16.m1.1.1.1.1.1.cmml">[</mo><mrow id="S4.E13.33.33.33.1.1"><mo id="S4.E13.33.33.33.1.1a" xref="S4.E16.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E13.33.33.33.1.1.3.3"><mo id="S4.E13.4.4.4" xref="S4.E16.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E13.33.33.33.1.1.1.1.1"><mrow id="S4.E13.33.33.33.1.1.1.1.1.1"><msubsup id="S4.E13.33.33.33.1.1.1.1.1.1.3"><mi id="S4.E13.5.5.5" xref="S4.E13.5.5.5.cmml">𝑺</mi><mi id="S4.E13.6.6.6.1" xref="S4.E13.6.6.6.1.cmml">n</mi><mrow id="S4.E13.7.7.7.1.3"><mo id="S4.E13.7.7.7.1.3.1" stretchy="false">(</mo><mi id="S4.E13.7.7.7.1.1" xref="S4.E13.7.7.7.1.1.cmml">k</mi><mo id="S4.E13.7.7.7.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E13.33.33.33.1.1.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E13.33.33.33.1.1.1.1.1.1.1.1"><mo id="S4.E13.8.8.8" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E13.33.33.33.1.1.1.1.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E13.9.9.9" mathvariant="bold-italic" xref="S4.E13.9.9.9.cmml">ϕ</mi><mi id="S4.E13.10.10.10.1" xref="S4.E13.10.10.10.1.cmml">n</mi></msub><mo id="S4.E13.11.11.11" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E13.12.12.12" xref="S4.E13.12.12.12.cmml">−</mo><msubsup id="S4.E13.33.33.33.1.1.1.1.1.2"><mi id="S4.E13.13.13.13" xref="S4.E13.13.13.13.cmml">𝒁</mi><mi id="S4.E13.14.14.14.1" xref="S4.E13.14.14.14.1.cmml">n</mi><mrow id="S4.E13.15.15.15.1.3"><mo id="S4.E13.15.15.15.1.3.1" stretchy="false">(</mo><mi id="S4.E13.15.15.15.1.1" xref="S4.E13.15.15.15.1.1.cmml">k</mi><mo id="S4.E13.15.15.15.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E13.16.16.16" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E13.33.33.33.1.1.2.2.2"><mi id="S4.E13.17.17.17" xref="S4.E13.17.17.17.cmml">𝚲</mi><mi id="S4.E13.18.18.18.1" xref="S4.E13.18.18.18.1.cmml">Z</mi></msub><mo id="S4.E13.19.19.19" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E13.33.33.33.1.1.3.3.3"><mrow id="S4.E13.33.33.33.1.1.3.3.3.1"><msubsup id="S4.E13.33.33.33.1.1.3.3.3.1.3"><mi id="S4.E13.20.20.20" xref="S4.E13.20.20.20.cmml">𝑺</mi><mi id="S4.E13.21.21.21.1" xref="S4.E13.21.21.21.1.cmml">n</mi><mrow id="S4.E13.22.22.22.1.3"><mo id="S4.E13.22.22.22.1.3.1" stretchy="false">(</mo><mi id="S4.E13.22.22.22.1.1" xref="S4.E13.22.22.22.1.1.cmml">k</mi><mo id="S4.E13.22.22.22.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E13.33.33.33.1.1.3.3.3.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E13.33.33.33.1.1.3.3.3.1.1.1"><mo id="S4.E13.23.23.23" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E13.33.33.33.1.1.3.3.3.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E13.24.24.24" mathvariant="bold-italic" xref="S4.E13.24.24.24.cmml">ϕ</mi><mi id="S4.E13.25.25.25.1" xref="S4.E13.25.25.25.1.cmml">n</mi></msub><mo id="S4.E13.26.26.26" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E13.27.27.27" xref="S4.E13.27.27.27.cmml">−</mo><msubsup id="S4.E13.33.33.33.1.1.3.3.3.2"><mi id="S4.E13.28.28.28" xref="S4.E13.28.28.28.cmml">𝒁</mi><mi id="S4.E13.29.29.29.1" xref="S4.E13.29.29.29.1.cmml">n</mi><mrow id="S4.E13.30.30.30.1.3"><mo id="S4.E13.30.30.30.1.3.1" stretchy="false">(</mo><mi id="S4.E13.30.30.30.1.1" xref="S4.E13.30.30.30.1.1.cmml">k</mi><mo id="S4.E13.30.30.30.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mpadded width="0.219em"><mo id="S4.E13.31.31.31" xref="S4.E16.m1.1.1.1.1.1.cmml">⟩</mo></mpadded></mrow></mrow><mo id="S4.E13.32.32.32" xref="S4.E16.m1.1.1.1.1.1.cmml">]</mo></mrow></mrow></mtd></mtr><mtr id="S4.E16.m1.1.1.2c"><mtd id="S4.E16.m1.1.1.2d"><mrow id="S4.E14.39.39"><mi id="S4.E14.39.39.40" xref="S4.E16.m1.1.1.1.1.1.cmml"></mi><mo id="S4.E14.1.1.1" xref="S4.E14.1.1.1.cmml">∝</mo><mrow id="S4.E14.39.39.39"><mo id="S4.E14.39.39.39a" xref="S4.E16.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E14.39.39.39.3.3"><mo id="S4.E14.3.3.3" xref="S4.E16.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E14.37.37.37.1.1.1"><mrow id="S4.E14.37.37.37.1.1.1.1"><mpadded width="0.497em"><mi id="S4.E14.4.4.4" xref="S4.E14.4.4.4.cmml">𝔼</mi></mpadded><mo id="S4.E14.37.37.37.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E14.37.37.37.1.1.1.1.1.1"><mo id="S4.E14.5.5.5" xref="S4.E16.m1.1.1.1.1.1.cmml">[</mo><mrow id="S4.E14.37.37.37.1.1.1.1.1.1.1"><msubsup id="S4.E14.37.37.37.1.1.1.1.1.1.1.3"><mi id="S4.E14.6.6.6" xref="S4.E14.6.6.6.cmml">𝑺</mi><mi id="S4.E14.7.7.7.1" xref="S4.E14.7.7.7.1.cmml">n</mi><mrow id="S4.E14.8.8.8.1.3"><mo id="S4.E14.8.8.8.1.3.1" stretchy="false">(</mo><mi id="S4.E14.8.8.8.1.1" xref="S4.E14.8.8.8.1.1.cmml">k</mi><mo id="S4.E14.8.8.8.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E14.37.37.37.1.1.1.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E14.37.37.37.1.1.1.1.1.1.1.1.1"><mo id="S4.E14.9.9.9" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E14.37.37.37.1.1.1.1.1.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E14.10.10.10" mathvariant="bold-italic" xref="S4.E14.10.10.10.cmml">ϕ</mi><mi id="S4.E14.11.11.11.1" xref="S4.E14.11.11.11.1.cmml">n</mi></msub><mo id="S4.E14.12.12.12" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E14.13.13.13" xref="S4.E16.m1.1.1.1.1.1.cmml">]</mo></mrow></mrow><mo id="S4.E14.14.14.14" xref="S4.E14.14.14.14.cmml">−</mo><msubsup id="S4.E14.37.37.37.1.1.1.2"><mi id="S4.E14.15.15.15" xref="S4.E14.15.15.15.cmml">𝒁</mi><mi id="S4.E14.16.16.16.1" xref="S4.E14.16.16.16.1.cmml">n</mi><mrow id="S4.E14.17.17.17.1.3"><mo id="S4.E14.17.17.17.1.3.1" stretchy="false">(</mo><mi id="S4.E14.17.17.17.1.1" xref="S4.E14.17.17.17.1.1.cmml">k</mi><mo id="S4.E14.17.17.17.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E14.18.18.18" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E14.38.38.38.2.2.2"><mi id="S4.E14.19.19.19" xref="S4.E14.19.19.19.cmml">𝚲</mi><mi id="S4.E14.20.20.20.1" xref="S4.E14.20.20.20.1.cmml">Z</mi></msub><mo id="S4.E14.21.21.21" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E14.39.39.39.3.3.3"><mrow id="S4.E14.39.39.39.3.3.3.1"><mpadded width="0.497em"><mi id="S4.E14.22.22.22" xref="S4.E14.22.22.22.cmml">𝔼</mi></mpadded><mo id="S4.E14.39.39.39.3.3.3.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E14.39.39.39.3.3.3.1.1.1"><mo id="S4.E14.23.23.23" xref="S4.E16.m1.1.1.1.1.1.cmml">[</mo><mrow id="S4.E14.39.39.39.3.3.3.1.1.1.1"><msubsup id="S4.E14.39.39.39.3.3.3.1.1.1.1.3"><mi id="S4.E14.24.24.24" xref="S4.E14.24.24.24.cmml">𝐒</mi><mi id="S4.E14.25.25.25.1" xref="S4.E14.25.25.25.1.cmml">n</mi><mrow id="S4.E14.26.26.26.1.3"><mo id="S4.E14.26.26.26.1.3.1" stretchy="false">(</mo><mi id="S4.E14.26.26.26.1.1" xref="S4.E14.26.26.26.1.1.cmml">k</mi><mo id="S4.E14.26.26.26.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E14.39.39.39.3.3.3.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E14.39.39.39.3.3.3.1.1.1.1.1.1"><mo id="S4.E14.27.27.27" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E14.39.39.39.3.3.3.1.1.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E14.28.28.28" mathvariant="bold-italic" xref="S4.E14.28.28.28.cmml">ϕ</mi><mi id="S4.E14.29.29.29.1" xref="S4.E14.29.29.29.1.cmml">n</mi></msub><mo id="S4.E14.30.30.30" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E14.31.31.31" xref="S4.E16.m1.1.1.1.1.1.cmml">]</mo></mrow></mrow><mo id="S4.E14.32.32.32" xref="S4.E14.32.32.32.cmml">−</mo><msubsup id="S4.E14.39.39.39.3.3.3.2"><mi id="S4.E14.33.33.33" xref="S4.E14.33.33.33.cmml">𝒁</mi><mi id="S4.E14.34.34.34.1" xref="S4.E14.34.34.34.1.cmml">n</mi><mrow id="S4.E14.35.35.35.1.3"><mo id="S4.E14.35.35.35.1.3.1" stretchy="false">(</mo><mi id="S4.E14.35.35.35.1.1" xref="S4.E14.35.35.35.1.1.cmml">k</mi><mo id="S4.E14.35.35.35.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E14.36.36.36" xref="S4.E16.m1.1.1.1.1.1.cmml">⟩</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E16.m1.1.1.2e"><mtd class="ltx_align_right" columnalign="right" id="S4.E16.m1.1.1.2f"><mrow id="S4.E15.29.29.29"><mrow id="S4.E15.29.29.29.1"><mo id="S4.E15.29.29.29.1a" xref="S4.E16.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E15.29.29.29.1.1"><mpadded width="1.032em"><mtext id="S4.E15.2.2.2" xref="S4.E15.2.2.2a.cmml">Tr</mtext></mpadded><mo id="S4.E15.29.29.29.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E15.29.29.29.1.1.1.1"><mo id="S4.E15.3.3.3" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1"><mpadded width="0.497em"><mi id="S4.E15.4.4.4" xref="S4.E15.4.4.4.cmml">𝔼</mi></mpadded><mo id="S4.E15.29.29.29.1.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1"><mpadded width="0.247em"><mo id="S4.E15.5.5.5" xref="S4.E16.m1.1.1.1.1.1.cmml">[</mo></mpadded><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1"><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.1.1"><mo id="S4.E15.6.6.6" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.1.1.1"><msubsup id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.1.1.1.3"><mi id="S4.E15.7.7.7" xref="S4.E15.7.7.7.cmml">𝑺</mi><mi id="S4.E15.8.8.8.1" xref="S4.E15.8.8.8.1.cmml">n</mi><mrow id="S4.E15.9.9.9.1.3"><mo id="S4.E15.9.9.9.1.3.1" stretchy="false">(</mo><mi id="S4.E15.9.9.9.1.1" xref="S4.E15.9.9.9.1.1.cmml">k</mi><mo id="S4.E15.9.9.9.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.1.1.1.1.1"><mo id="S4.E15.10.10.10" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.1.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E15.11.11.11" mathvariant="bold-italic" xref="S4.E15.11.11.11.cmml">ϕ</mi><mi id="S4.E15.12.12.12.1" xref="S4.E15.12.12.12.1.cmml">n</mi></msub><mo id="S4.E15.13.13.13" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E15.14.14.14" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">⟩</mo></mrow><mo id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.3" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.2.1"><mo id="S4.E15.15.15.15" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.2.1.1"><msubsup id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.2.1.1.3"><mi id="S4.E15.16.16.16" xref="S4.E15.16.16.16.cmml">𝑺</mi><mi id="S4.E15.17.17.17.1" xref="S4.E15.17.17.17.1.cmml">n</mi><mrow id="S4.E15.18.18.18.1.3"><mo id="S4.E15.18.18.18.1.3.1" stretchy="false">(</mo><mi id="S4.E15.18.18.18.1.1" xref="S4.E15.18.18.18.1.1.cmml">k</mi><mo id="S4.E15.18.18.18.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.2.1.1.2" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.2.1.1.1.1"><mo id="S4.E15.19.19.19" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E15.29.29.29.1.1.1.1.1.1.1.1.2.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E15.20.20.20" mathvariant="bold-italic" xref="S4.E15.20.20.20.cmml">ϕ</mi><mi id="S4.E15.21.21.21.1" xref="S4.E15.21.21.21.1.cmml">n</mi></msub><mo id="S4.E15.22.22.22" stretchy="false" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E15.23.23.23" maxsize="120%" minsize="120%" xref="S4.E16.m1.1.1.1.1.1.cmml">|</mo></mrow></mrow><mo id="S4.E15.24.24.24" xref="S4.E16.m1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S4.E15.29.29.29.1.1.1.1.1.2a" xref="S4.E16.m1.1.1.1.1.1.cmml">⁢</mo><msub id="S4.E15.29.29.29.1.1.1.1.1.3"><mi id="S4.E15.25.25.25" xref="S4.E15.25.25.25.cmml">𝚲</mi><mi id="S4.E15.26.26.26.1" xref="S4.E15.26.26.26.1.cmml">Z</mi></msub></mrow><mpadded width="0.288em"><mo id="S4.E15.27.27.27" xref="S4.E16.m1.1.1.1.1.1.cmml">)</mo></mpadded></mrow></mrow></mrow><mo id="S4.E15.28.28.28" xref="S4.E16.m1.1.1.1.1.1.cmml">.</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E16.m1.1b"><apply id="S4.E16.m1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E14.1.1.1.cmml" xref="S4.E14.1.1.1">proportional-to</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><ci id="S4.E13.1.1.1.cmml" xref="S4.E13.1.1.1">𝔼</ci><apply id="S4.E16.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E13.33.33.34">delimited-[]</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><minus id="S4.E13.3.3.3.cmml" xref="S4.E13.33.33.34"></minus><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.4.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.4.1.cmml" xref="S4.E13.33.33.34">quantum-operator-product</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><minus id="S4.E13.12.12.12.cmml" xref="S4.E13.12.12.12"></minus><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.5.5.5.cmml" xref="S4.E13.5.5.5">𝑺</ci><ci id="S4.E13.6.6.6.1.cmml" xref="S4.E13.6.6.6.1">𝑛</ci></apply><ci id="S4.E13.7.7.7.1.1.cmml" xref="S4.E13.7.7.7.1.1">𝑘</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.9.9.9.cmml" xref="S4.E13.9.9.9">bold-italic-ϕ</ci><ci id="S4.E13.10.10.10.1.cmml" xref="S4.E13.10.10.10.1">𝑛</ci></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.13.13.13.cmml" xref="S4.E13.13.13.13">𝒁</ci><ci id="S4.E13.14.14.14.1.cmml" xref="S4.E13.14.14.14.1">𝑛</ci></apply><ci id="S4.E13.15.15.15.1.1.cmml" xref="S4.E13.15.15.15.1.1">𝑘</ci></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.17.17.17.cmml" xref="S4.E13.17.17.17">𝚲</ci><ci id="S4.E13.18.18.18.1.cmml" xref="S4.E13.18.18.18.1">𝑍</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="S4.E13.33.33.34"><minus id="S4.E13.27.27.27.cmml" xref="S4.E13.27.27.27"></minus><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.2.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.20.20.20.cmml" xref="S4.E13.20.20.20">𝑺</ci><ci id="S4.E13.21.21.21.1.cmml" xref="S4.E13.21.21.21.1">𝑛</ci></apply><ci id="S4.E13.22.22.22.1.1.cmml" xref="S4.E13.22.22.22.1.1">𝑘</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.1.1.1.1.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.24.24.24.cmml" xref="S4.E13.24.24.24">bold-italic-ϕ</ci><ci id="S4.E13.25.25.25.1.cmml" xref="S4.E13.25.25.25.1">𝑛</ci></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.1.1.1.1.3.3.3.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E13.28.28.28.cmml" xref="S4.E13.28.28.28">𝒁</ci><ci id="S4.E13.29.29.29.1.cmml" xref="S4.E13.29.29.29.1">𝑛</ci></apply><ci id="S4.E13.30.30.30.1.1.cmml" xref="S4.E13.30.30.30.1.1">𝑘</ci></apply></apply></apply></apply></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.5.cmml" xref="S4.E13.33.33.34"><minus id="S4.E15.1.1.1.cmml" xref="S4.E13.33.33.34"></minus><apply id="S4.E16.m1.1.1.1.1.1.4.3.cmml" xref="S4.E13.33.33.34"><minus id="S4.E14.2.2.2.cmml" xref="S4.E13.33.33.34"></minus><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.4.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.4.3.3.4.1.cmml" xref="S4.E13.33.33.34">quantum-operator-product</csymbol><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.cmml" xref="S4.E13.33.33.34"><minus id="S4.E14.14.14.14.cmml" xref="S4.E14.14.14.14"></minus><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><ci id="S4.E14.4.4.4.cmml" xref="S4.E14.4.4.4">𝔼</ci><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.2.1.cmml" xref="S4.E13.33.33.34">delimited-[]</csymbol><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.6.6.6.cmml" xref="S4.E14.6.6.6">𝑺</ci><ci id="S4.E14.7.7.7.1.cmml" xref="S4.E14.7.7.7.1">𝑛</ci></apply><ci id="S4.E14.8.8.8.1.1.cmml" xref="S4.E14.8.8.8.1.1">𝑘</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.10.10.10.cmml" xref="S4.E14.10.10.10">bold-italic-ϕ</ci><ci id="S4.E14.11.11.11.1.cmml" xref="S4.E14.11.11.11.1">𝑛</ci></apply></apply></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.2.1.1.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.15.15.15.cmml" xref="S4.E14.15.15.15">𝒁</ci><ci id="S4.E14.16.16.16.1.cmml" xref="S4.E14.16.16.16.1">𝑛</ci></apply><ci id="S4.E14.17.17.17.1.1.cmml" xref="S4.E14.17.17.17.1.1">𝑘</ci></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.3.2.2.2.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.3.2.2.2.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.19.19.19.cmml" xref="S4.E14.19.19.19">𝚲</ci><ci id="S4.E14.20.20.20.1.cmml" xref="S4.E14.20.20.20.1">𝑍</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.cmml" xref="S4.E13.33.33.34"><minus id="S4.E14.32.32.32.cmml" xref="S4.E14.32.32.32"></minus><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.2.cmml" xref="S4.E13.33.33.34"></times><ci id="S4.E14.22.22.22.cmml" xref="S4.E14.22.22.22">𝔼</ci><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.2.1.cmml" xref="S4.E13.33.33.34">delimited-[]</csymbol><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.24.24.24.cmml" xref="S4.E14.24.24.24">𝐒</ci><ci id="S4.E14.25.25.25.1.cmml" xref="S4.E14.25.25.25.1">𝑛</ci></apply><ci id="S4.E14.26.26.26.1.1.cmml" xref="S4.E14.26.26.26.1.1">𝑘</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.28.28.28.cmml" xref="S4.E14.28.28.28">bold-italic-ϕ</ci><ci id="S4.E14.29.29.29.1.cmml" xref="S4.E14.29.29.29.1">𝑛</ci></apply></apply></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.4.3.3.3.3.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E14.33.33.33.cmml" xref="S4.E14.33.33.33">𝒁</ci><ci id="S4.E14.34.34.34.1.cmml" xref="S4.E14.34.34.34.1">𝑛</ci></apply><ci id="S4.E14.35.35.35.1.1.cmml" xref="S4.E14.35.35.35.1.1">𝑘</ci></apply></apply></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.5.4.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.5.4.2.cmml" xref="S4.E13.33.33.34"></times><ci id="S4.E15.2.2.2a.cmml" xref="S4.E15.2.2.2"><mtext id="S4.E15.2.2.2.cmml" xref="S4.E15.2.2.2">Tr</mtext></ci><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><ci id="S4.E15.4.4.4.cmml" xref="S4.E15.4.4.4">𝔼</ci><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.2.1.cmml" xref="S4.E13.33.33.34">delimited-[]</csymbol><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E13.33.33.34">ket</csymbol><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E15.7.7.7.cmml" xref="S4.E15.7.7.7">𝑺</ci><ci id="S4.E15.8.8.8.1.cmml" xref="S4.E15.8.8.8.1">𝑛</ci></apply><ci id="S4.E15.9.9.9.1.1.cmml" xref="S4.E15.9.9.9.1.1">𝑘</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E15.11.11.11.cmml" xref="S4.E15.11.11.11">bold-italic-ϕ</ci><ci id="S4.E15.12.12.12.1.cmml" xref="S4.E15.12.12.12.1">𝑛</ci></apply></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="latexml" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E13.33.33.34">bra</csymbol><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.cmml" xref="S4.E13.33.33.34"><times id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.2.cmml" xref="S4.E13.33.33.34"></times><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.3.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.3.1.cmml" xref="S4.E13.33.33.34">superscript</csymbol><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.3.2.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.3.2.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E15.16.16.16.cmml" xref="S4.E15.16.16.16">𝑺</ci><ci id="S4.E15.17.17.17.1.cmml" xref="S4.E15.17.17.17.1">𝑛</ci></apply><ci id="S4.E15.18.18.18.1.1.cmml" xref="S4.E15.18.18.18.1.1">𝑘</ci></apply><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.1.1.1.2.1.1.1.1.1.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E15.20.20.20.cmml" xref="S4.E15.20.20.20">bold-italic-ϕ</ci><ci id="S4.E15.21.21.21.1.cmml" xref="S4.E15.21.21.21.1">𝑛</ci></apply></apply></apply></apply></apply><apply id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.4.cmml" xref="S4.E13.33.33.34"><csymbol cd="ambiguous" id="S4.E16.m1.1.1.1.1.1.5.4.1.1.1.4.1.cmml" xref="S4.E13.33.33.34">subscript</csymbol><ci id="S4.E15.25.25.25.cmml" xref="S4.E15.25.25.25">𝚲</ci><ci id="S4.E15.26.26.26.1.cmml" xref="S4.E15.26.26.26.1">𝑍</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m1.1c">\displaystyle\begin{multlined}\mathbb{E}\!\left[-\!\left\langle\bm{S}_{n}^{(k)% }\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\bm{S}_{n}^{(% k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\right\rangle\!\right]\\ \propto-\!\left\langle\mathbb{E}\!\left[\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \right]-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\mathbb{E}\!\left[% \mathbf{S}_{n}^{(k)}\!(\bm{\phi}_{n})\right]-\bm{Z}_{n}^{(k)}\right\rangle\\ -\text{Tr}\!\left(\mathbb{E}\!\left[\!\big{|}\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \big{\rangle}\big{\langle}\bm{S}_{n}^{(k)}(\bm{\phi}_{n})\big{|}\right]\bm{% \Lambda}_{Z}\!\right)\!.\end{multlined}\mathbb{E}\!\left[-\!\left\langle\bm{S}% _{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\bm{% S}_{n}^{(k)}\!(\bm{\phi}_{n})-\bm{Z}_{n}^{(k)}\right\rangle\!\right]\\ \propto-\!\left\langle\mathbb{E}\!\left[\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \right]-\bm{Z}_{n}^{(k)}\big{|}\bm{\Lambda}_{Z}\big{|}\mathbb{E}\!\left[% \mathbf{S}_{n}^{(k)}\!(\bm{\phi}_{n})\right]-\bm{Z}_{n}^{(k)}\right\rangle\\ -\text{Tr}\!\left(\mathbb{E}\!\left[\!\big{|}\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})% \big{\rangle}\big{\langle}\bm{S}_{n}^{(k)}(\bm{\phi}_{n})\big{|}\right]\bm{% \Lambda}_{Z}\!\right)\!.</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.1d">start_ROW start_CELL blackboard_E [ - ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ⟩ ] end_CELL end_ROW start_ROW start_CELL ∝ - ⟨ blackboard_E [ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] - bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | blackboard_E [ bold_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] - bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ⟩ end_CELL end_ROW start_ROW start_CELL - Tr ( blackboard_E [ | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ⟩ ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | ] bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ) . 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">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.16">These expectations are intractable, but can be approximated by the Delta method as</p> <table class="ltx_equation ltx_eqn_table" id="S4.E17"> <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="\mathbb{E}\!\left[\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})\right]\approx\bm{S}_{n}^{(% k)}\!(\bar{\bm{\epsilon}})," class="ltx_Math" display="block" id="S4.E17.m1.4"><semantics id="S4.E17.m1.4a"><mrow id="S4.E17.m1.4.4.1" xref="S4.E17.m1.4.4.1.1.cmml"><mrow id="S4.E17.m1.4.4.1.1" xref="S4.E17.m1.4.4.1.1.cmml"><mrow id="S4.E17.m1.4.4.1.1.1" xref="S4.E17.m1.4.4.1.1.1.cmml"><mpadded width="0.497em"><mi id="S4.E17.m1.4.4.1.1.1.3" xref="S4.E17.m1.4.4.1.1.1.3.cmml">𝔼</mi></mpadded><mo id="S4.E17.m1.4.4.1.1.1.2" xref="S4.E17.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.E17.m1.4.4.1.1.1.1.1" xref="S4.E17.m1.4.4.1.1.1.1.2.cmml"><mo id="S4.E17.m1.4.4.1.1.1.1.1.2" xref="S4.E17.m1.4.4.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.E17.m1.4.4.1.1.1.1.1.1" xref="S4.E17.m1.4.4.1.1.1.1.1.1.cmml"><msubsup id="S4.E17.m1.4.4.1.1.1.1.1.1.3" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.cmml"><mi id="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.2" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.2.cmml">𝑺</mi><mi id="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.3" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.3.cmml">n</mi><mrow id="S4.E17.m1.1.1.1.3" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.cmml"><mo id="S4.E17.m1.1.1.1.3.1" stretchy="false" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.cmml">(</mo><mi id="S4.E17.m1.1.1.1.1" xref="S4.E17.m1.1.1.1.1.cmml">k</mi><mo id="S4.E17.m1.1.1.1.3.2" stretchy="false" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.cmml">)</mo></mrow></msubsup><mo id="S4.E17.m1.4.4.1.1.1.1.1.1.2" xref="S4.E17.m1.4.4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.3" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E17.m1.4.4.1.1.1.1.1.3" xref="S4.E17.m1.4.4.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S4.E17.m1.4.4.1.1.2" xref="S4.E17.m1.4.4.1.1.2.cmml">≈</mo><mrow id="S4.E17.m1.4.4.1.1.3" xref="S4.E17.m1.4.4.1.1.3.cmml"><msubsup id="S4.E17.m1.4.4.1.1.3.2" xref="S4.E17.m1.4.4.1.1.3.2.cmml"><mi id="S4.E17.m1.4.4.1.1.3.2.2.2" xref="S4.E17.m1.4.4.1.1.3.2.2.2.cmml">𝑺</mi><mi id="S4.E17.m1.4.4.1.1.3.2.2.3" xref="S4.E17.m1.4.4.1.1.3.2.2.3.cmml">n</mi><mrow id="S4.E17.m1.2.2.1.3" xref="S4.E17.m1.4.4.1.1.3.2.cmml"><mo id="S4.E17.m1.2.2.1.3.1" stretchy="false" xref="S4.E17.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S4.E17.m1.2.2.1.1" xref="S4.E17.m1.2.2.1.1.cmml">k</mi><mo id="S4.E17.m1.2.2.1.3.2" stretchy="false" xref="S4.E17.m1.4.4.1.1.3.2.cmml">)</mo></mrow></msubsup><mo id="S4.E17.m1.4.4.1.1.3.1" xref="S4.E17.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S4.E17.m1.4.4.1.1.3.3.2" xref="S4.E17.m1.3.3.cmml"><mo id="S4.E17.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S4.E17.m1.3.3.cmml">(</mo><mover accent="true" id="S4.E17.m1.3.3" xref="S4.E17.m1.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E17.m1.3.3.2" mathvariant="bold-italic" xref="S4.E17.m1.3.3.2.cmml">ϵ</mi><mo id="S4.E17.m1.3.3.1" xref="S4.E17.m1.3.3.1.cmml">¯</mo></mover><mo id="S4.E17.m1.4.4.1.1.3.3.2.2" stretchy="false" xref="S4.E17.m1.3.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E17.m1.4.4.1.2" xref="S4.E17.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E17.m1.4b"><apply id="S4.E17.m1.4.4.1.1.cmml" xref="S4.E17.m1.4.4.1"><approx id="S4.E17.m1.4.4.1.1.2.cmml" xref="S4.E17.m1.4.4.1.1.2"></approx><apply id="S4.E17.m1.4.4.1.1.1.cmml" xref="S4.E17.m1.4.4.1.1.1"><times id="S4.E17.m1.4.4.1.1.1.2.cmml" xref="S4.E17.m1.4.4.1.1.1.2"></times><ci id="S4.E17.m1.4.4.1.1.1.3.cmml" xref="S4.E17.m1.4.4.1.1.1.3">𝔼</ci><apply id="S4.E17.m1.4.4.1.1.1.1.2.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1"><csymbol cd="latexml" id="S4.E17.m1.4.4.1.1.1.1.2.1.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.E17.m1.4.4.1.1.1.1.1.1.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1"><times id="S4.E17.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.2"></times><apply id="S4.E17.m1.4.4.1.1.1.1.1.1.3.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E17.m1.4.4.1.1.1.1.1.1.3.1.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3">superscript</csymbol><apply id="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.2.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.2">𝑺</ci><ci id="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.3.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.3.2.3">𝑛</ci></apply><ci id="S4.E17.m1.1.1.1.1.cmml" xref="S4.E17.m1.1.1.1.1">𝑘</ci></apply><apply id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E17.m1.4.4.1.1.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply><apply id="S4.E17.m1.4.4.1.1.3.cmml" xref="S4.E17.m1.4.4.1.1.3"><times id="S4.E17.m1.4.4.1.1.3.1.cmml" xref="S4.E17.m1.4.4.1.1.3.1"></times><apply id="S4.E17.m1.4.4.1.1.3.2.cmml" xref="S4.E17.m1.4.4.1.1.3.2"><csymbol cd="ambiguous" id="S4.E17.m1.4.4.1.1.3.2.1.cmml" xref="S4.E17.m1.4.4.1.1.3.2">superscript</csymbol><apply id="S4.E17.m1.4.4.1.1.3.2.2.cmml" xref="S4.E17.m1.4.4.1.1.3.2"><csymbol cd="ambiguous" id="S4.E17.m1.4.4.1.1.3.2.2.1.cmml" xref="S4.E17.m1.4.4.1.1.3.2">subscript</csymbol><ci id="S4.E17.m1.4.4.1.1.3.2.2.2.cmml" xref="S4.E17.m1.4.4.1.1.3.2.2.2">𝑺</ci><ci id="S4.E17.m1.4.4.1.1.3.2.2.3.cmml" xref="S4.E17.m1.4.4.1.1.3.2.2.3">𝑛</ci></apply><ci id="S4.E17.m1.2.2.1.1.cmml" xref="S4.E17.m1.2.2.1.1">𝑘</ci></apply><apply id="S4.E17.m1.3.3.cmml" xref="S4.E17.m1.4.4.1.1.3.3.2"><ci id="S4.E17.m1.3.3.1.cmml" xref="S4.E17.m1.3.3.1">¯</ci><ci id="S4.E17.m1.3.3.2.cmml" xref="S4.E17.m1.3.3.2">bold-italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E17.m1.4c">\mathbb{E}\!\left[\bm{S}_{n}^{(k)}\!(\bm{\phi}_{n})\right]\approx\bm{S}_{n}^{(% k)}\!(\bar{\bm{\epsilon}}),</annotation><annotation encoding="application/x-llamapun" id="S4.E17.m1.4d">blackboard_E [ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] ≈ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ 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> </table> <table class="ltx_equation ltx_eqn_table" id="S4.E18"> <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="\mathbb{E}\!\left[\!\big{|}\bm{S}_{n}^{(k)}\!(\bm{\phi})\big{\rangle}\big{% \langle}\bm{S}_{n}^{(k)}\!(\bm{\phi})\big{|}\right]\approx\big{|}\bm{S}_{n}^{(% k)}\!(\bar{\bm{\epsilon}})\big{\rangle}\big{\langle}\bm{S}_{n}^{(k)}\!(\bar{% \bm{\epsilon}})\big{|}\\ +\big{|}\nabla_{\phi}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}% \bar{\bar{\bm{\epsilon}}}\big{\langle}\nabla_{\phi}\bm{S}_{n}^{(k)}\!(\bar{\bm% {\epsilon}}_{n})\big{|}," class="ltx_Math" display="block" id="S4.E18.m1.66"><semantics id="S4.E18.m1.66a"><mtable displaystyle="true" id="S4.E18.m1.66.66.5" rowspacing="0pt"><mtr id="S4.E18.m1.66.66.5a"><mtd class="ltx_align_left" columnalign="left" id="S4.E18.m1.66.66.5b"><mrow id="S4.E18.m1.65.65.4.64.39.39"><mrow id="S4.E18.m1.63.63.2.62.37.37.37"><mpadded width="0.497em"><mi id="S4.E18.m1.1.1.1.1.1.1" xref="S4.E18.m1.1.1.1.1.1.1.cmml">𝔼</mi></mpadded><mo id="S4.E18.m1.63.63.2.62.37.37.37.2" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1"><mpadded width="0.247em"><mo id="S4.E18.m1.2.2.2.2.2.2" xref="S4.E18.m1.62.62.1.1.1.cmml">[</mo></mpadded><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1"><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.1.1"><mo id="S4.E18.m1.3.3.3.3.3.3" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">|</mo><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.1.1.1"><msubsup id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.1.1.1.2"><mi id="S4.E18.m1.4.4.4.4.4.4" xref="S4.E18.m1.4.4.4.4.4.4.cmml">𝑺</mi><mi id="S4.E18.m1.5.5.5.5.5.5.1" xref="S4.E18.m1.5.5.5.5.5.5.1.cmml">n</mi><mrow id="S4.E18.m1.6.6.6.6.6.6.1.3"><mo id="S4.E18.m1.6.6.6.6.6.6.1.3.1" stretchy="false">(</mo><mi id="S4.E18.m1.6.6.6.6.6.6.1.1" xref="S4.E18.m1.6.6.6.6.6.6.1.1.cmml">k</mi><mo id="S4.E18.m1.6.6.6.6.6.6.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.1.1.1.1" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.1.1.1.3"><mo id="S4.E18.m1.7.7.7.7.7.7" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">(</mo><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.8.8.8.8.8.8" mathvariant="bold-italic" xref="S4.E18.m1.8.8.8.8.8.8.cmml">ϕ</mi><mo id="S4.E18.m1.9.9.9.9.9.9" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E18.m1.10.10.10.10.10.10" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">⟩</mo></mrow><mo id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.3" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.2.1"><mo id="S4.E18.m1.11.11.11.11.11.11" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">⟨</mo><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.2.1.1"><msubsup id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.2.1.1.2"><mi id="S4.E18.m1.12.12.12.12.12.12" xref="S4.E18.m1.12.12.12.12.12.12.cmml">𝑺</mi><mi id="S4.E18.m1.13.13.13.13.13.13.1" xref="S4.E18.m1.13.13.13.13.13.13.1.cmml">n</mi><mrow id="S4.E18.m1.14.14.14.14.14.14.1.3"><mo id="S4.E18.m1.14.14.14.14.14.14.1.3.1" stretchy="false">(</mo><mi id="S4.E18.m1.14.14.14.14.14.14.1.1" xref="S4.E18.m1.14.14.14.14.14.14.1.1.cmml">k</mi><mo id="S4.E18.m1.14.14.14.14.14.14.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.2.1.1.1" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.63.63.2.62.37.37.37.1.1.1.2.1.1.3"><mo id="S4.E18.m1.15.15.15.15.15.15" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">(</mo><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.16.16.16.16.16.16" mathvariant="bold-italic" xref="S4.E18.m1.16.16.16.16.16.16.cmml">ϕ</mi><mo id="S4.E18.m1.17.17.17.17.17.17" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E18.m1.18.18.18.18.18.18" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">|</mo></mrow></mrow><mo id="S4.E18.m1.19.19.19.19.19.19" xref="S4.E18.m1.62.62.1.1.1.cmml">]</mo></mrow></mrow><mo id="S4.E18.m1.20.20.20.20.20.20" xref="S4.E18.m1.20.20.20.20.20.20.cmml">≈</mo><mrow id="S4.E18.m1.65.65.4.64.39.39.39"><mrow id="S4.E18.m1.64.64.3.63.38.38.38.1.1"><mo id="S4.E18.m1.21.21.21.21.21.21" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">|</mo><mrow id="S4.E18.m1.64.64.3.63.38.38.38.1.1.1"><msubsup id="S4.E18.m1.64.64.3.63.38.38.38.1.1.1.2"><mi id="S4.E18.m1.22.22.22.22.22.22" xref="S4.E18.m1.22.22.22.22.22.22.cmml">𝑺</mi><mi id="S4.E18.m1.23.23.23.23.23.23.1" xref="S4.E18.m1.23.23.23.23.23.23.1.cmml">n</mi><mrow id="S4.E18.m1.24.24.24.24.24.24.1.3"><mo id="S4.E18.m1.24.24.24.24.24.24.1.3.1" stretchy="false">(</mo><mi id="S4.E18.m1.24.24.24.24.24.24.1.1" xref="S4.E18.m1.24.24.24.24.24.24.1.1.cmml">k</mi><mo id="S4.E18.m1.24.24.24.24.24.24.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E18.m1.64.64.3.63.38.38.38.1.1.1.1" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.64.64.3.63.38.38.38.1.1.1.3"><mo id="S4.E18.m1.25.25.25.25.25.25" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">(</mo><mover accent="true" id="S4.E18.m1.26.26.26.26.26.26" xref="S4.E18.m1.26.26.26.26.26.26.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.26.26.26.26.26.26.2" mathvariant="bold-italic" xref="S4.E18.m1.26.26.26.26.26.26.2.cmml">ϵ</mi><mo id="S4.E18.m1.26.26.26.26.26.26.1" xref="S4.E18.m1.26.26.26.26.26.26.1.cmml">¯</mo></mover><mo id="S4.E18.m1.27.27.27.27.27.27" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E18.m1.28.28.28.28.28.28" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">⟩</mo></mrow><mo id="S4.E18.m1.65.65.4.64.39.39.39.3" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.65.65.4.64.39.39.39.2.1"><mo id="S4.E18.m1.29.29.29.29.29.29" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">⟨</mo><mrow id="S4.E18.m1.65.65.4.64.39.39.39.2.1.1"><msubsup id="S4.E18.m1.65.65.4.64.39.39.39.2.1.1.2"><mi id="S4.E18.m1.30.30.30.30.30.30" xref="S4.E18.m1.30.30.30.30.30.30.cmml">𝑺</mi><mi id="S4.E18.m1.31.31.31.31.31.31.1" xref="S4.E18.m1.31.31.31.31.31.31.1.cmml">n</mi><mrow id="S4.E18.m1.32.32.32.32.32.32.1.3"><mo id="S4.E18.m1.32.32.32.32.32.32.1.3.1" stretchy="false">(</mo><mi id="S4.E18.m1.32.32.32.32.32.32.1.1" xref="S4.E18.m1.32.32.32.32.32.32.1.1.cmml">k</mi><mo id="S4.E18.m1.32.32.32.32.32.32.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E18.m1.65.65.4.64.39.39.39.2.1.1.1" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.65.65.4.64.39.39.39.2.1.1.3"><mo id="S4.E18.m1.33.33.33.33.33.33" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">(</mo><mover accent="true" id="S4.E18.m1.34.34.34.34.34.34" xref="S4.E18.m1.34.34.34.34.34.34.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.34.34.34.34.34.34.2" mathvariant="bold-italic" xref="S4.E18.m1.34.34.34.34.34.34.2.cmml">ϵ</mi><mo id="S4.E18.m1.34.34.34.34.34.34.1" xref="S4.E18.m1.34.34.34.34.34.34.1.cmml">¯</mo></mover><mo id="S4.E18.m1.35.35.35.35.35.35" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E18.m1.36.36.36.36.36.36" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">|</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E18.m1.66.66.5c"><mtd class="ltx_align_right" columnalign="right" id="S4.E18.m1.66.66.5d"><mrow id="S4.E18.m1.66.66.5.65.26.26.26"><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1"><mo id="S4.E18.m1.66.66.5.65.26.26.26.1a" xref="S4.E18.m1.62.62.1.1.1.cmml">+</mo><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.2"><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1"><mo id="S4.E18.m1.38.38.38.2.2.2" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">|</mo><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1"><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1.3"><msub id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1.3.1"><mo id="S4.E18.m1.39.39.39.3.3.3" rspace="0.167em" xref="S4.E18.m1.39.39.39.3.3.3.cmml">∇</mo><mi id="S4.E18.m1.40.40.40.4.4.4.1" xref="S4.E18.m1.40.40.40.4.4.4.1.cmml">ϕ</mi></msub><msubsup id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1.3.2"><mi id="S4.E18.m1.41.41.41.5.5.5" xref="S4.E18.m1.41.41.41.5.5.5.cmml">𝑺</mi><mi id="S4.E18.m1.42.42.42.6.6.6.1" xref="S4.E18.m1.42.42.42.6.6.6.1.cmml">n</mi><mrow id="S4.E18.m1.43.43.43.7.7.7.1.3"><mo id="S4.E18.m1.43.43.43.7.7.7.1.3.1" stretchy="false">(</mo><mi id="S4.E18.m1.43.43.43.7.7.7.1.1" xref="S4.E18.m1.43.43.43.7.7.7.1.1.cmml">k</mi><mo id="S4.E18.m1.43.43.43.7.7.7.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1.2" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1.1.1"><mo id="S4.E18.m1.44.44.44.8.8.8" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">(</mo><msub id="S4.E18.m1.66.66.5.65.26.26.26.1.1.1.1.1.1.1.1"><mover accent="true" id="S4.E18.m1.45.45.45.9.9.9" xref="S4.E18.m1.45.45.45.9.9.9.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.45.45.45.9.9.9.2" mathvariant="bold-italic" xref="S4.E18.m1.45.45.45.9.9.9.2.cmml">ϵ</mi><mo id="S4.E18.m1.45.45.45.9.9.9.1" xref="S4.E18.m1.45.45.45.9.9.9.1.cmml">¯</mo></mover><mi id="S4.E18.m1.46.46.46.10.10.10.1" xref="S4.E18.m1.46.46.46.10.10.10.1.cmml">n</mi></msub><mo id="S4.E18.m1.47.47.47.11.11.11" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E18.m1.48.48.48.12.12.12" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">⟩</mo></mrow><mo id="S4.E18.m1.66.66.5.65.26.26.26.1.2.3" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mover accent="true" id="S4.E18.m1.49.49.49.13.13.13" xref="S4.E18.m1.49.49.49.13.13.13.cmml"><mover accent="true" id="S4.E18.m1.49.49.49.13.13.13.2" xref="S4.E18.m1.49.49.49.13.13.13.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.49.49.49.13.13.13.2.2" mathvariant="bold-italic" xref="S4.E18.m1.49.49.49.13.13.13.2.2.cmml">ϵ</mi><mo id="S4.E18.m1.49.49.49.13.13.13.2.1" xref="S4.E18.m1.49.49.49.13.13.13.2.1.cmml">¯</mo></mover><mo id="S4.E18.m1.49.49.49.13.13.13.1" xref="S4.E18.m1.49.49.49.13.13.13.1.cmml">¯</mo></mover><mo id="S4.E18.m1.66.66.5.65.26.26.26.1.2.3a" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1"><mo id="S4.E18.m1.50.50.50.14.14.14" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">⟨</mo><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1"><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1.3"><msub id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1.3.1"><mo id="S4.E18.m1.51.51.51.15.15.15" rspace="0.167em" xref="S4.E18.m1.51.51.51.15.15.15.cmml">∇</mo><mi id="S4.E18.m1.52.52.52.16.16.16.1" xref="S4.E18.m1.52.52.52.16.16.16.1.cmml">ϕ</mi></msub><msubsup id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1.3.2"><mi id="S4.E18.m1.53.53.53.17.17.17" xref="S4.E18.m1.53.53.53.17.17.17.cmml">𝑺</mi><mi id="S4.E18.m1.54.54.54.18.18.18.1" xref="S4.E18.m1.54.54.54.18.18.18.1.cmml">n</mi><mrow id="S4.E18.m1.55.55.55.19.19.19.1.3"><mo id="S4.E18.m1.55.55.55.19.19.19.1.3.1" stretchy="false">(</mo><mi id="S4.E18.m1.55.55.55.19.19.19.1.1" xref="S4.E18.m1.55.55.55.19.19.19.1.1.cmml">k</mi><mo id="S4.E18.m1.55.55.55.19.19.19.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1.2" xref="S4.E18.m1.62.62.1.1.1.cmml">⁢</mo><mrow id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1.1.1"><mo id="S4.E18.m1.56.56.56.20.20.20" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">(</mo><msub id="S4.E18.m1.66.66.5.65.26.26.26.1.2.2.1.1.1.1.1"><mover accent="true" id="S4.E18.m1.57.57.57.21.21.21" xref="S4.E18.m1.57.57.57.21.21.21.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E18.m1.57.57.57.21.21.21.2" mathvariant="bold-italic" xref="S4.E18.m1.57.57.57.21.21.21.2.cmml">ϵ</mi><mo id="S4.E18.m1.57.57.57.21.21.21.1" xref="S4.E18.m1.57.57.57.21.21.21.1.cmml">¯</mo></mover><mi id="S4.E18.m1.58.58.58.22.22.22.1" xref="S4.E18.m1.58.58.58.22.22.22.1.cmml">n</mi></msub><mo id="S4.E18.m1.59.59.59.23.23.23" stretchy="false" xref="S4.E18.m1.62.62.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E18.m1.60.60.60.24.24.24" maxsize="120%" minsize="120%" xref="S4.E18.m1.62.62.1.1.1.cmml">|</mo></mrow></mrow></mrow><mo id="S4.E18.m1.61.61.61.25.25.25" xref="S4.E18.m1.62.62.1.1.1.cmml">,</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E18.m1.66b"><apply id="S4.E18.m1.62.62.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><approx id="S4.E18.m1.20.20.20.20.20.20.cmml" xref="S4.E18.m1.20.20.20.20.20.20"></approx><apply id="S4.E18.m1.62.62.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><ci id="S4.E18.m1.1.1.1.1.1.1.cmml" xref="S4.E18.m1.1.1.1.1.1.1">𝔼</ci><apply id="S4.E18.m1.62.62.1.1.1.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">delimited-[]</csymbol><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.1.1.1.1.3.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">ket</csymbol><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">superscript</csymbol><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.4.4.4.4.4.4.cmml" xref="S4.E18.m1.4.4.4.4.4.4">𝑺</ci><ci id="S4.E18.m1.5.5.5.5.5.5.1.cmml" xref="S4.E18.m1.5.5.5.5.5.5.1">𝑛</ci></apply><ci id="S4.E18.m1.6.6.6.6.6.6.1.1.cmml" xref="S4.E18.m1.6.6.6.6.6.6.1.1">𝑘</ci></apply><ci id="S4.E18.m1.8.8.8.8.8.8.cmml" xref="S4.E18.m1.8.8.8.8.8.8">bold-italic-ϕ</ci></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">bra</csymbol><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">superscript</csymbol><apply id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.1.1.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.1.1.1.1.2.1.1.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.12.12.12.12.12.12.cmml" xref="S4.E18.m1.12.12.12.12.12.12">𝑺</ci><ci id="S4.E18.m1.13.13.13.13.13.13.1.cmml" xref="S4.E18.m1.13.13.13.13.13.13.1">𝑛</ci></apply><ci id="S4.E18.m1.14.14.14.14.14.14.1.1.cmml" xref="S4.E18.m1.14.14.14.14.14.14.1.1">𝑘</ci></apply><ci id="S4.E18.m1.16.16.16.16.16.16.cmml" xref="S4.E18.m1.16.16.16.16.16.16">bold-italic-ϕ</ci></apply></apply></apply></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.5.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><plus id="S4.E18.m1.37.37.37.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></plus><apply id="S4.E18.m1.62.62.1.1.1.3.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.3.2.3.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.2.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.2.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">ket</csymbol><apply id="S4.E18.m1.62.62.1.1.1.2.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.2.1.1.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">superscript</csymbol><apply id="S4.E18.m1.62.62.1.1.1.2.1.1.1.1.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.2.1.1.1.1.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.22.22.22.22.22.22.cmml" xref="S4.E18.m1.22.22.22.22.22.22">𝑺</ci><ci id="S4.E18.m1.23.23.23.23.23.23.1.cmml" xref="S4.E18.m1.23.23.23.23.23.23.1">𝑛</ci></apply><ci id="S4.E18.m1.24.24.24.24.24.24.1.1.cmml" xref="S4.E18.m1.24.24.24.24.24.24.1.1">𝑘</ci></apply><apply id="S4.E18.m1.26.26.26.26.26.26.cmml" xref="S4.E18.m1.26.26.26.26.26.26"><ci id="S4.E18.m1.26.26.26.26.26.26.1.cmml" xref="S4.E18.m1.26.26.26.26.26.26.1">¯</ci><ci id="S4.E18.m1.26.26.26.26.26.26.2.cmml" xref="S4.E18.m1.26.26.26.26.26.26.2">bold-italic-ϵ</ci></apply></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.3.2.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.3.2.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">bra</csymbol><apply id="S4.E18.m1.62.62.1.1.1.3.2.2.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.3.2.2.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.3.2.2.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.3.2.2.1.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">superscript</csymbol><apply id="S4.E18.m1.62.62.1.1.1.3.2.2.1.1.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.3.2.2.1.1.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.30.30.30.30.30.30.cmml" xref="S4.E18.m1.30.30.30.30.30.30">𝑺</ci><ci id="S4.E18.m1.31.31.31.31.31.31.1.cmml" xref="S4.E18.m1.31.31.31.31.31.31.1">𝑛</ci></apply><ci id="S4.E18.m1.32.32.32.32.32.32.1.1.cmml" xref="S4.E18.m1.32.32.32.32.32.32.1.1">𝑘</ci></apply><apply id="S4.E18.m1.34.34.34.34.34.34.cmml" xref="S4.E18.m1.34.34.34.34.34.34"><ci id="S4.E18.m1.34.34.34.34.34.34.1.cmml" xref="S4.E18.m1.34.34.34.34.34.34.1">¯</ci><ci id="S4.E18.m1.34.34.34.34.34.34.2.cmml" xref="S4.E18.m1.34.34.34.34.34.34.2">bold-italic-ϵ</ci></apply></apply></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.5.4.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.5.4.3.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.4.3.1.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">ket</csymbol><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.39.39.39.3.3.3.cmml" xref="S4.E18.m1.39.39.39.3.3.3">∇</ci><ci id="S4.E18.m1.40.40.40.4.4.4.1.cmml" xref="S4.E18.m1.40.40.40.4.4.4.1">italic-ϕ</ci></apply><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">superscript</csymbol><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.3.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.41.41.41.5.5.5.cmml" xref="S4.E18.m1.41.41.41.5.5.5">𝑺</ci><ci id="S4.E18.m1.42.42.42.6.6.6.1.cmml" xref="S4.E18.m1.42.42.42.6.6.6.1">𝑛</ci></apply><ci id="S4.E18.m1.43.43.43.7.7.7.1.1.cmml" xref="S4.E18.m1.43.43.43.7.7.7.1.1">𝑘</ci></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.4.3.1.1.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><apply id="S4.E18.m1.45.45.45.9.9.9.cmml" xref="S4.E18.m1.45.45.45.9.9.9"><ci id="S4.E18.m1.45.45.45.9.9.9.1.cmml" xref="S4.E18.m1.45.45.45.9.9.9.1">¯</ci><ci id="S4.E18.m1.45.45.45.9.9.9.2.cmml" xref="S4.E18.m1.45.45.45.9.9.9.2">bold-italic-ϵ</ci></apply><ci id="S4.E18.m1.46.46.46.10.10.10.1.cmml" xref="S4.E18.m1.46.46.46.10.10.10.1">𝑛</ci></apply></apply></apply><apply id="S4.E18.m1.49.49.49.13.13.13.cmml" xref="S4.E18.m1.49.49.49.13.13.13"><ci id="S4.E18.m1.49.49.49.13.13.13.1.cmml" xref="S4.E18.m1.49.49.49.13.13.13.1">¯</ci><apply id="S4.E18.m1.49.49.49.13.13.13.2.cmml" xref="S4.E18.m1.49.49.49.13.13.13.2"><ci id="S4.E18.m1.49.49.49.13.13.13.2.1.cmml" xref="S4.E18.m1.49.49.49.13.13.13.2.1">¯</ci><ci id="S4.E18.m1.49.49.49.13.13.13.2.2.cmml" xref="S4.E18.m1.49.49.49.13.13.13.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="latexml" id="S4.E18.m1.62.62.1.1.1.5.4.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">bra</csymbol><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><times id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"></times><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.51.51.51.15.15.15.cmml" xref="S4.E18.m1.51.51.51.15.15.15">∇</ci><ci id="S4.E18.m1.52.52.52.16.16.16.1.cmml" xref="S4.E18.m1.52.52.52.16.16.16.1">italic-ϕ</ci></apply><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">superscript</csymbol><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.2.2.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.3.2.2.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><ci id="S4.E18.m1.53.53.53.17.17.17.cmml" xref="S4.E18.m1.53.53.53.17.17.17">𝑺</ci><ci id="S4.E18.m1.54.54.54.18.18.18.1.cmml" xref="S4.E18.m1.54.54.54.18.18.18.1">𝑛</ci></apply><ci id="S4.E18.m1.55.55.55.19.19.19.1.1.cmml" xref="S4.E18.m1.55.55.55.19.19.19.1.1">𝑘</ci></apply></apply><apply id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2"><csymbol cd="ambiguous" id="S4.E18.m1.62.62.1.1.1.5.4.2.1.1.1.1.1.1.cmml" xref="S4.E18.m1.63.63.2.62.37.37.37.2">subscript</csymbol><apply id="S4.E18.m1.57.57.57.21.21.21.cmml" xref="S4.E18.m1.57.57.57.21.21.21"><ci id="S4.E18.m1.57.57.57.21.21.21.1.cmml" xref="S4.E18.m1.57.57.57.21.21.21.1">¯</ci><ci id="S4.E18.m1.57.57.57.21.21.21.2.cmml" xref="S4.E18.m1.57.57.57.21.21.21.2">bold-italic-ϵ</ci></apply><ci id="S4.E18.m1.58.58.58.22.22.22.1.cmml" xref="S4.E18.m1.58.58.58.22.22.22.1">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E18.m1.66c">\mathbb{E}\!\left[\!\big{|}\bm{S}_{n}^{(k)}\!(\bm{\phi})\big{\rangle}\big{% \langle}\bm{S}_{n}^{(k)}\!(\bm{\phi})\big{|}\right]\approx\big{|}\bm{S}_{n}^{(% k)}\!(\bar{\bm{\epsilon}})\big{\rangle}\big{\langle}\bm{S}_{n}^{(k)}\!(\bar{% \bm{\epsilon}})\big{|}\\ +\big{|}\nabla_{\phi}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}% \bar{\bar{\bm{\epsilon}}}\big{\langle}\nabla_{\phi}\bm{S}_{n}^{(k)}\!(\bar{\bm% {\epsilon}}_{n})\big{|},</annotation><annotation encoding="application/x-llamapun" id="S4.E18.m1.66d">start_ROW start_CELL blackboard_E [ | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ ) ⟩ ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( bold_italic_ϕ ) | ] ≈ | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG ) ⟩ ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG ) | end_CELL end_ROW start_ROW start_CELL + | ∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ⟩ over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG ⟨ ∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | , 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">(18)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.12">where <math alttext="\nabla_{\phi}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}})" class="ltx_Math" display="inline" id="S4.p2.10.m1.2"><semantics id="S4.p2.10.m1.2a"><mrow id="S4.p2.10.m1.2.3" xref="S4.p2.10.m1.2.3.cmml"><mrow id="S4.p2.10.m1.2.3.2" xref="S4.p2.10.m1.2.3.2.cmml"><msub id="S4.p2.10.m1.2.3.2.1" xref="S4.p2.10.m1.2.3.2.1.cmml"><mo id="S4.p2.10.m1.2.3.2.1.2" xref="S4.p2.10.m1.2.3.2.1.2.cmml">∇</mo><mi id="S4.p2.10.m1.2.3.2.1.3" xref="S4.p2.10.m1.2.3.2.1.3.cmml">ϕ</mi></msub><msubsup id="S4.p2.10.m1.2.3.2.2" xref="S4.p2.10.m1.2.3.2.2.cmml"><mi id="S4.p2.10.m1.2.3.2.2.2.2" xref="S4.p2.10.m1.2.3.2.2.2.2.cmml">𝑺</mi><mi id="S4.p2.10.m1.2.3.2.2.2.3" xref="S4.p2.10.m1.2.3.2.2.2.3.cmml">n</mi><mrow id="S4.p2.10.m1.1.1.1.3" xref="S4.p2.10.m1.2.3.2.2.cmml"><mo id="S4.p2.10.m1.1.1.1.3.1" stretchy="false" xref="S4.p2.10.m1.2.3.2.2.cmml">(</mo><mi id="S4.p2.10.m1.1.1.1.1" xref="S4.p2.10.m1.1.1.1.1.cmml">k</mi><mo id="S4.p2.10.m1.1.1.1.3.2" stretchy="false" xref="S4.p2.10.m1.2.3.2.2.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.p2.10.m1.2.3.1" xref="S4.p2.10.m1.2.3.1.cmml">⁢</mo><mrow id="S4.p2.10.m1.2.3.3.2" xref="S4.p2.10.m1.2.2.cmml"><mo id="S4.p2.10.m1.2.3.3.2.1" stretchy="false" xref="S4.p2.10.m1.2.2.cmml">(</mo><mover accent="true" id="S4.p2.10.m1.2.2" xref="S4.p2.10.m1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.10.m1.2.2.2" mathvariant="bold-italic" xref="S4.p2.10.m1.2.2.2.cmml">ϵ</mi><mo id="S4.p2.10.m1.2.2.1" xref="S4.p2.10.m1.2.2.1.cmml">¯</mo></mover><mo id="S4.p2.10.m1.2.3.3.2.2" stretchy="false" xref="S4.p2.10.m1.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.10.m1.2b"><apply id="S4.p2.10.m1.2.3.cmml" xref="S4.p2.10.m1.2.3"><times id="S4.p2.10.m1.2.3.1.cmml" xref="S4.p2.10.m1.2.3.1"></times><apply id="S4.p2.10.m1.2.3.2.cmml" xref="S4.p2.10.m1.2.3.2"><apply id="S4.p2.10.m1.2.3.2.1.cmml" xref="S4.p2.10.m1.2.3.2.1"><csymbol cd="ambiguous" id="S4.p2.10.m1.2.3.2.1.1.cmml" xref="S4.p2.10.m1.2.3.2.1">subscript</csymbol><ci id="S4.p2.10.m1.2.3.2.1.2.cmml" xref="S4.p2.10.m1.2.3.2.1.2">∇</ci><ci id="S4.p2.10.m1.2.3.2.1.3.cmml" xref="S4.p2.10.m1.2.3.2.1.3">italic-ϕ</ci></apply><apply id="S4.p2.10.m1.2.3.2.2.cmml" xref="S4.p2.10.m1.2.3.2.2"><csymbol cd="ambiguous" id="S4.p2.10.m1.2.3.2.2.1.cmml" xref="S4.p2.10.m1.2.3.2.2">superscript</csymbol><apply id="S4.p2.10.m1.2.3.2.2.2.cmml" xref="S4.p2.10.m1.2.3.2.2"><csymbol cd="ambiguous" id="S4.p2.10.m1.2.3.2.2.2.1.cmml" xref="S4.p2.10.m1.2.3.2.2">subscript</csymbol><ci id="S4.p2.10.m1.2.3.2.2.2.2.cmml" xref="S4.p2.10.m1.2.3.2.2.2.2">𝑺</ci><ci id="S4.p2.10.m1.2.3.2.2.2.3.cmml" xref="S4.p2.10.m1.2.3.2.2.2.3">𝑛</ci></apply><ci id="S4.p2.10.m1.1.1.1.1.cmml" xref="S4.p2.10.m1.1.1.1.1">𝑘</ci></apply></apply><apply id="S4.p2.10.m1.2.2.cmml" xref="S4.p2.10.m1.2.3.3.2"><ci id="S4.p2.10.m1.2.2.1.cmml" xref="S4.p2.10.m1.2.2.1">¯</ci><ci id="S4.p2.10.m1.2.2.2.cmml" xref="S4.p2.10.m1.2.2.2">bold-italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.10.m1.2c">\nabla_{\phi}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}})</annotation><annotation encoding="application/x-llamapun" id="S4.p2.10.m1.2d">∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG )</annotation></semantics></math> is the Jacobian of <math alttext="\mathbf{S}_{n}^{(k)}" class="ltx_Math" display="inline" id="S4.p2.11.m2.1"><semantics id="S4.p2.11.m2.1a"><msubsup id="S4.p2.11.m2.1.2" xref="S4.p2.11.m2.1.2.cmml"><mi id="S4.p2.11.m2.1.2.2.2" xref="S4.p2.11.m2.1.2.2.2.cmml">𝐒</mi><mi id="S4.p2.11.m2.1.2.2.3" xref="S4.p2.11.m2.1.2.2.3.cmml">n</mi><mrow id="S4.p2.11.m2.1.1.1.3" xref="S4.p2.11.m2.1.2.cmml"><mo id="S4.p2.11.m2.1.1.1.3.1" stretchy="false" xref="S4.p2.11.m2.1.2.cmml">(</mo><mi id="S4.p2.11.m2.1.1.1.1" xref="S4.p2.11.m2.1.1.1.1.cmml">k</mi><mo id="S4.p2.11.m2.1.1.1.3.2" stretchy="false" xref="S4.p2.11.m2.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.p2.11.m2.1b"><apply id="S4.p2.11.m2.1.2.cmml" xref="S4.p2.11.m2.1.2"><csymbol cd="ambiguous" id="S4.p2.11.m2.1.2.1.cmml" xref="S4.p2.11.m2.1.2">superscript</csymbol><apply id="S4.p2.11.m2.1.2.2.cmml" xref="S4.p2.11.m2.1.2"><csymbol cd="ambiguous" id="S4.p2.11.m2.1.2.2.1.cmml" xref="S4.p2.11.m2.1.2">subscript</csymbol><ci id="S4.p2.11.m2.1.2.2.2.cmml" xref="S4.p2.11.m2.1.2.2.2">𝐒</ci><ci id="S4.p2.11.m2.1.2.2.3.cmml" xref="S4.p2.11.m2.1.2.2.3">𝑛</ci></apply><ci id="S4.p2.11.m2.1.1.1.1.cmml" xref="S4.p2.11.m2.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.11.m2.1c">\mathbf{S}_{n}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.11.m2.1d">bold_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> evaluated in <math alttext="\bar{\bm{\epsilon}}" class="ltx_Math" display="inline" id="S4.p2.12.m3.1"><semantics id="S4.p2.12.m3.1a"><mover accent="true" id="S4.p2.12.m3.1.1" xref="S4.p2.12.m3.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.12.m3.1.1.2" mathvariant="bold-italic" xref="S4.p2.12.m3.1.1.2.cmml">ϵ</mi><mo id="S4.p2.12.m3.1.1.1" xref="S4.p2.12.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.p2.12.m3.1b"><apply id="S4.p2.12.m3.1.1.cmml" xref="S4.p2.12.m3.1.1"><ci id="S4.p2.12.m3.1.1.1.cmml" xref="S4.p2.12.m3.1.1.1">¯</ci><ci id="S4.p2.12.m3.1.1.2.cmml" xref="S4.p2.12.m3.1.1.2">bold-italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.12.m3.1c">\bar{\bm{\epsilon}}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.12.m3.1d">over¯ start_ARG bold_italic_ϵ end_ARG</annotation></semantics></math>. Inserting this in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E12" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">12</span></a>) gives,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx5"> <tbody id="S4.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\begin{multlined}D_{KL}\propto-2\text{Re}\left\{\!\big{\langle}% \bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big{|}\bm{Z% }_{n}^{(k)}\big{\rangle}\!\right\}\\ +\big{\langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{% Z}\big{|}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\\ +\text{Tr}\left(\bar{\bar{\bm{\epsilon}}}_{n}\big{\langle}\nabla_{\phi}\bm{S}_% {n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi% }\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\right)-\zeta(\bar{% \bar{\bm{\epsilon}}}).\end{multlined}D_{KL}\propto-2\text{Re}\left\{\!\big{% \langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big% {|}\bm{Z}_{n}^{(k)}\big{\rangle}\!\right\}\\ +\big{\langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{% Z}\big{|}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\\ +\text{Tr}\left(\bar{\bar{\bm{\epsilon}}}_{n}\big{\langle}\nabla_{\phi}\bm{S}_% {n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi% }\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\right)-\zeta(\bar{% \bar{\bm{\epsilon}}})." class="ltx_Math" display="inline" id="S4.E22.m1.1"><semantics id="S4.E22.m1.1a"><mtable id="S4.E22.m1.1.1.2" rowspacing="0pt"><mtr id="S4.E22.m1.1.1.2a"><mtd class="ltx_align_left" columnalign="left" id="S4.E22.m1.1.1.2b"><mrow id="S4.E19.25.25"><msub id="S4.E19.25.25.26"><mi id="S4.E19.1.1.1" xref="S4.E19.1.1.1.cmml">D</mi><mrow id="S4.E19.2.2.2.1" xref="S4.E19.2.2.2.1.cmml"><mi id="S4.E19.2.2.2.1.2" xref="S4.E19.2.2.2.1.2.cmml">K</mi><mo id="S4.E19.2.2.2.1.1" xref="S4.E19.2.2.2.1.1.cmml">⁢</mo><mi id="S4.E19.2.2.2.1.3" xref="S4.E19.2.2.2.1.3.cmml">L</mi></mrow></msub><mo id="S4.E19.3.3.3" xref="S4.E19.3.3.3.cmml">∝</mo><mrow id="S4.E19.25.25.25"><mo id="S4.E19.25.25.25a" xref="S4.E22.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E19.25.25.25.1"><mn id="S4.E19.5.5.5" xref="S4.E19.5.5.5.cmml">2</mn><mo id="S4.E19.25.25.25.1.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mtext id="S4.E19.6.6.6" xref="S4.E19.6.6.6a.cmml">Re</mtext><mo id="S4.E19.25.25.25.1.2a" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E19.25.25.25.1.1.1"><mpadded width="0.330em"><mo id="S4.E19.7.7.7" xref="S4.E22.m1.1.1.1.1.1.cmml">{</mo></mpadded><mrow id="S4.E19.25.25.25.1.1.1.1"><mo id="S4.E19.8.8.8" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E19.25.25.25.1.1.1.1.1.1"><msubsup id="S4.E19.25.25.25.1.1.1.1.1.1.3"><mi id="S4.E19.9.9.9" xref="S4.E19.9.9.9.cmml">𝑺</mi><mi id="S4.E19.10.10.10.1" xref="S4.E19.10.10.10.1.cmml">n</mi><mrow id="S4.E19.11.11.11.1.3"><mo id="S4.E19.11.11.11.1.3.1" stretchy="false">(</mo><mi id="S4.E19.11.11.11.1.1" xref="S4.E19.11.11.11.1.1.cmml">k</mi><mo id="S4.E19.11.11.11.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E19.25.25.25.1.1.1.1.1.1.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E19.25.25.25.1.1.1.1.1.1.1.1"><mo id="S4.E19.12.12.12" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E19.25.25.25.1.1.1.1.1.1.1.1.1"><mover accent="true" id="S4.E19.13.13.13" xref="S4.E19.13.13.13.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E19.13.13.13.2" mathvariant="bold-italic" xref="S4.E19.13.13.13.2.cmml">ϵ</mi><mo id="S4.E19.13.13.13.1" xref="S4.E19.13.13.13.1.cmml">¯</mo></mover><mi id="S4.E19.14.14.14.1" xref="S4.E19.14.14.14.1.cmml">n</mi></msub><mo id="S4.E19.15.15.15" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E19.16.16.16" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E19.25.25.25.1.1.1.1.2.2"><mi id="S4.E19.17.17.17" xref="S4.E19.17.17.17.cmml">𝚲</mi><mi id="S4.E19.18.18.18.1" xref="S4.E19.18.18.18.1.cmml">Z</mi></msub><mo id="S4.E19.19.19.19" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">|</mo><msubsup id="S4.E19.25.25.25.1.1.1.1.3.3"><mi id="S4.E19.20.20.20" xref="S4.E19.20.20.20.cmml">𝒁</mi><mi id="S4.E19.21.21.21.1" xref="S4.E19.21.21.21.1.cmml">n</mi><mrow id="S4.E19.22.22.22.1.3"><mo id="S4.E19.22.22.22.1.3.1" stretchy="false">(</mo><mi id="S4.E19.22.22.22.1.1" xref="S4.E19.22.22.22.1.1.cmml">k</mi><mo id="S4.E19.22.22.22.1.3.2" stretchy="false">)</mo></mrow></msubsup><mpadded width="0.219em"><mo id="S4.E19.23.23.23" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">⟩</mo></mpadded></mrow><mo id="S4.E19.24.24.24" xref="S4.E22.m1.1.1.1.1.1.cmml">}</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E22.m1.1.1.2c"><mtd id="S4.E22.m1.1.1.2d"><mrow id="S4.E20.24.24"><mo id="S4.E20.24.24a" xref="S4.E22.m1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E20.24.24.24.3"><mo id="S4.E20.2.2.2" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E20.22.22.22.1.1"><msubsup id="S4.E20.22.22.22.1.1.3"><mi id="S4.E20.3.3.3" xref="S4.E20.3.3.3.cmml">𝑺</mi><mi id="S4.E20.4.4.4.1" xref="S4.E20.4.4.4.1.cmml">n</mi><mrow id="S4.E20.5.5.5.1.3"><mo id="S4.E20.5.5.5.1.3.1" stretchy="false">(</mo><mi id="S4.E20.5.5.5.1.1" xref="S4.E20.5.5.5.1.1.cmml">k</mi><mo id="S4.E20.5.5.5.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E20.22.22.22.1.1.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E20.22.22.22.1.1.1.1"><mo id="S4.E20.6.6.6" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E20.22.22.22.1.1.1.1.1"><mover accent="true" id="S4.E20.7.7.7" xref="S4.E20.7.7.7.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E20.7.7.7.2" mathvariant="bold-italic" xref="S4.E20.7.7.7.2.cmml">ϵ</mi><mo id="S4.E20.7.7.7.1" xref="S4.E20.7.7.7.1.cmml">¯</mo></mover><mi id="S4.E20.8.8.8.1" xref="S4.E20.8.8.8.1.cmml">n</mi></msub><mo id="S4.E20.9.9.9" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E20.10.10.10" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E20.23.23.23.2.2"><mi id="S4.E20.11.11.11" xref="S4.E20.11.11.11.cmml">𝚲</mi><mi id="S4.E20.12.12.12.1" xref="S4.E20.12.12.12.1.cmml">Z</mi></msub><mo id="S4.E20.13.13.13" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E20.24.24.24.3.3"><msubsup id="S4.E20.24.24.24.3.3.3"><mi id="S4.E20.14.14.14" xref="S4.E20.14.14.14.cmml">𝑺</mi><mi id="S4.E20.15.15.15.1" xref="S4.E20.15.15.15.1.cmml">n</mi><mrow id="S4.E20.16.16.16.1.3"><mo id="S4.E20.16.16.16.1.3.1" stretchy="false">(</mo><mi id="S4.E20.16.16.16.1.1" xref="S4.E20.16.16.16.1.1.cmml">k</mi><mo id="S4.E20.16.16.16.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E20.24.24.24.3.3.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E20.24.24.24.3.3.1.1"><mo id="S4.E20.17.17.17" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E20.24.24.24.3.3.1.1.1"><mover accent="true" id="S4.E20.18.18.18" xref="S4.E20.18.18.18.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E20.18.18.18.2" mathvariant="bold-italic" xref="S4.E20.18.18.18.2.cmml">ϵ</mi><mo id="S4.E20.18.18.18.1" xref="S4.E20.18.18.18.1.cmml">¯</mo></mover><mi id="S4.E20.19.19.19.1" xref="S4.E20.19.19.19.1.cmml">n</mi></msub><mo id="S4.E20.20.20.20" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E20.21.21.21" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">⟩</mo></mrow></mrow></mtd></mtr><mtr id="S4.E22.m1.1.1.2e"><mtd class="ltx_align_right" columnalign="right" id="S4.E22.m1.1.1.2f"><mrow id="S4.E21.37.37.37"><mrow id="S4.E21.37.37.37.1"><mrow id="S4.E21.37.37.37.1.1"><mo id="S4.E21.37.37.37.1.1a" xref="S4.E22.m1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E21.37.37.37.1.1.1"><mtext id="S4.E21.2.2.2" xref="S4.E21.2.2.2a.cmml">Tr</mtext><mo id="S4.E21.37.37.37.1.1.1.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1"><mo id="S4.E21.3.3.3" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1.1"><msub id="S4.E21.37.37.37.1.1.1.1.1.1.5"><mover accent="true" id="S4.E21.4.4.4" xref="S4.E21.4.4.4.cmml"><mover accent="true" id="S4.E21.4.4.4.2" xref="S4.E21.4.4.4.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E21.4.4.4.2.2" mathvariant="bold-italic" xref="S4.E21.4.4.4.2.2.cmml">ϵ</mi><mo id="S4.E21.4.4.4.2.1" xref="S4.E21.4.4.4.2.1.cmml">¯</mo></mover><mo id="S4.E21.4.4.4.1" xref="S4.E21.4.4.4.1.cmml">¯</mo></mover><mi id="S4.E21.5.5.5.1" xref="S4.E21.5.5.5.1.cmml">n</mi></msub><mo id="S4.E21.37.37.37.1.1.1.1.1.1.4" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.3.3"><mo id="S4.E21.6.6.6" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1"><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1.3"><msub id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1.3.1"><mo id="S4.E21.7.7.7" rspace="0.167em" xref="S4.E21.7.7.7.cmml">∇</mo><mi id="S4.E21.8.8.8.1" xref="S4.E21.8.8.8.1.cmml">ϕ</mi></msub><msubsup id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1.3.2"><mi id="S4.E21.9.9.9" xref="S4.E21.9.9.9.cmml">𝑺</mi><mi id="S4.E21.10.10.10.1" xref="S4.E21.10.10.10.1.cmml">n</mi><mrow id="S4.E21.11.11.11.1.3"><mo id="S4.E21.11.11.11.1.3.1" stretchy="false">(</mo><mi id="S4.E21.11.11.11.1.1" xref="S4.E21.11.11.11.1.1.cmml">k</mi><mo id="S4.E21.11.11.11.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1.1.1"><mo id="S4.E21.12.12.12" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E21.37.37.37.1.1.1.1.1.1.1.1.1.1.1.1"><mover accent="true" id="S4.E21.13.13.13" xref="S4.E21.13.13.13.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E21.13.13.13.2" mathvariant="bold-italic" xref="S4.E21.13.13.13.2.cmml">ϵ</mi><mo id="S4.E21.13.13.13.1" xref="S4.E21.13.13.13.1.cmml">¯</mo></mover><mi id="S4.E21.14.14.14.1" xref="S4.E21.14.14.14.1.cmml">n</mi></msub><mo id="S4.E21.15.15.15" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E21.16.16.16" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E21.37.37.37.1.1.1.1.1.1.2.2.2"><mi id="S4.E21.17.17.17" xref="S4.E21.17.17.17.cmml">𝚲</mi><mi id="S4.E21.18.18.18.1" xref="S4.E21.18.18.18.1.cmml">Z</mi></msub><mo id="S4.E21.19.19.19" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3"><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3.3"><msub id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3.3.1"><mo id="S4.E21.20.20.20" rspace="0.167em" xref="S4.E21.20.20.20.cmml">∇</mo><mi id="S4.E21.21.21.21.1" xref="S4.E21.21.21.21.1.cmml">ϕ</mi></msub><msubsup id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3.3.2"><mi id="S4.E21.22.22.22" xref="S4.E21.22.22.22.cmml">𝑺</mi><mi id="S4.E21.23.23.23.1" xref="S4.E21.23.23.23.1.cmml">n</mi><mrow id="S4.E21.24.24.24.1.3"><mo id="S4.E21.24.24.24.1.3.1" stretchy="false">(</mo><mi id="S4.E21.24.24.24.1.1" xref="S4.E21.24.24.24.1.1.cmml">k</mi><mo id="S4.E21.24.24.24.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3.2" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3.1.1"><mo id="S4.E21.25.25.25" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E21.37.37.37.1.1.1.1.1.1.3.3.3.1.1.1"><mover accent="true" id="S4.E21.26.26.26" xref="S4.E21.26.26.26.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E21.26.26.26.2" mathvariant="bold-italic" xref="S4.E21.26.26.26.2.cmml">ϵ</mi><mo id="S4.E21.26.26.26.1" xref="S4.E21.26.26.26.1.cmml">¯</mo></mover><mi id="S4.E21.27.27.27.1" xref="S4.E21.27.27.27.1.cmml">n</mi></msub><mo id="S4.E21.28.28.28" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E21.29.29.29" maxsize="120%" minsize="120%" xref="S4.E22.m1.1.1.1.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.E21.30.30.30" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E21.31.31.31" xref="S4.E21.31.31.31.cmml">−</mo><mrow id="S4.E21.37.37.37.1.2"><mi id="S4.E21.32.32.32" xref="S4.E21.32.32.32.cmml">ζ</mi><mo id="S4.E21.37.37.37.1.2.1" xref="S4.E22.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E21.37.37.37.1.2.2"><mo id="S4.E21.33.33.33" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">(</mo><mover accent="true" id="S4.E21.34.34.34" xref="S4.E21.34.34.34.cmml"><mover accent="true" id="S4.E21.34.34.34.2" xref="S4.E21.34.34.34.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E21.34.34.34.2.2" mathvariant="bold-italic" xref="S4.E21.34.34.34.2.2.cmml">ϵ</mi><mo id="S4.E21.34.34.34.2.1" xref="S4.E21.34.34.34.2.1.cmml">¯</mo></mover><mo id="S4.E21.34.34.34.1" xref="S4.E21.34.34.34.1.cmml">¯</mo></mover><mo id="S4.E21.35.35.35" stretchy="false" xref="S4.E22.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E21.36.36.36" lspace="0em" xref="S4.E22.m1.1.1.1.1.1.cmml">.</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E22.m1.1b"><apply id="S4.E22.m1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="latexml" id="S4.E19.3.3.3.cmml" xref="S4.E19.3.3.3">proportional-to</csymbol><apply id="S4.E22.m1.1.1.1.1.1.7.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.7.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E19.1.1.1.cmml" xref="S4.E19.1.1.1">𝐷</ci><apply id="S4.E19.2.2.2.1.cmml" xref="S4.E19.2.2.2.1"><times id="S4.E19.2.2.2.1.1.cmml" xref="S4.E19.2.2.2.1.1"></times><ci id="S4.E19.2.2.2.1.2.cmml" xref="S4.E19.2.2.2.1.2">𝐾</ci><ci id="S4.E19.2.2.2.1.3.cmml" xref="S4.E19.2.2.2.1.3">𝐿</ci></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.5.cmml" xref="S4.E19.25.25.25a"><minus id="S4.E21.31.31.31.cmml" xref="S4.E21.31.31.31"></minus><apply id="S4.E22.m1.1.1.1.1.1.5.5.cmml" xref="S4.E19.25.25.25a"><plus id="S4.E20.1.1.1.cmml" xref="S4.E19.25.25.25a"></plus><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><minus id="S4.E19.4.4.4.cmml" xref="S4.E19.25.25.25a"></minus><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E19.25.25.25a"></times><cn id="S4.E19.5.5.5.cmml" type="integer" xref="S4.E19.5.5.5">2</cn><ci id="S4.E19.6.6.6a.cmml" xref="S4.E19.6.6.6"><mtext id="S4.E19.6.6.6.cmml" xref="S4.E19.6.6.6">Re</mtext></ci><set id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E19.25.25.25a"><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E19.25.25.25a"><csymbol cd="latexml" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.E19.25.25.25a">quantum-operator-product</csymbol><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E19.25.25.25a"></times><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E19.25.25.25a">superscript</csymbol><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E19.9.9.9.cmml" xref="S4.E19.9.9.9">𝑺</ci><ci id="S4.E19.10.10.10.1.cmml" xref="S4.E19.10.10.10.1">𝑛</ci></apply><ci id="S4.E19.11.11.11.1.1.cmml" xref="S4.E19.11.11.11.1.1">𝑘</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><apply id="S4.E19.13.13.13.cmml" xref="S4.E19.13.13.13"><ci id="S4.E19.13.13.13.1.cmml" xref="S4.E19.13.13.13.1">¯</ci><ci id="S4.E19.13.13.13.2.cmml" xref="S4.E19.13.13.13.2">bold-italic-ϵ</ci></apply><ci id="S4.E19.14.14.14.1.cmml" xref="S4.E19.14.14.14.1">𝑛</ci></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E19.17.17.17.cmml" xref="S4.E19.17.17.17">𝚲</ci><ci id="S4.E19.18.18.18.1.cmml" xref="S4.E19.18.18.18.1">𝑍</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="S4.E19.25.25.25a">superscript</csymbol><apply id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E19.20.20.20.cmml" xref="S4.E19.20.20.20">𝒁</ci><ci id="S4.E19.21.21.21.1.cmml" xref="S4.E19.21.21.21.1">𝑛</ci></apply><ci id="S4.E19.22.22.22.1.1.cmml" xref="S4.E19.22.22.22.1.1">𝑘</ci></apply></apply></set></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.4.4.4.4.cmml" xref="S4.E19.25.25.25a"><csymbol cd="latexml" id="S4.E22.m1.1.1.1.1.1.4.4.4.4.1.cmml" xref="S4.E19.25.25.25a">quantum-operator-product</csymbol><apply id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.2.cmml" xref="S4.E19.25.25.25a"></times><apply id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.3.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.3.1.cmml" xref="S4.E19.25.25.25a">superscript</csymbol><apply id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.3.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.3.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E20.3.3.3.cmml" xref="S4.E20.3.3.3">𝑺</ci><ci id="S4.E20.4.4.4.1.cmml" xref="S4.E20.4.4.4.1">𝑛</ci></apply><ci id="S4.E20.5.5.5.1.1.cmml" xref="S4.E20.5.5.5.1.1">𝑘</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><apply id="S4.E20.7.7.7.cmml" xref="S4.E20.7.7.7"><ci id="S4.E20.7.7.7.1.cmml" xref="S4.E20.7.7.7.1">¯</ci><ci id="S4.E20.7.7.7.2.cmml" xref="S4.E20.7.7.7.2">bold-italic-ϵ</ci></apply><ci id="S4.E20.8.8.8.1.cmml" xref="S4.E20.8.8.8.1">𝑛</ci></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.3.3.3.2.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.3.3.3.2.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E20.11.11.11.cmml" xref="S4.E20.11.11.11">𝚲</ci><ci id="S4.E20.12.12.12.1.cmml" xref="S4.E20.12.12.12.1">𝑍</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.2.cmml" xref="S4.E19.25.25.25a"></times><apply id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.3.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.3.1.cmml" xref="S4.E19.25.25.25a">superscript</csymbol><apply id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.3.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.3.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E20.14.14.14.cmml" xref="S4.E20.14.14.14">𝑺</ci><ci id="S4.E20.15.15.15.1.cmml" xref="S4.E20.15.15.15.1">𝑛</ci></apply><ci id="S4.E20.16.16.16.1.1.cmml" xref="S4.E20.16.16.16.1.1">𝑘</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.1.1.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.4.4.4.3.3.1.1.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><apply id="S4.E20.18.18.18.cmml" xref="S4.E20.18.18.18"><ci id="S4.E20.18.18.18.1.cmml" xref="S4.E20.18.18.18.1">¯</ci><ci id="S4.E20.18.18.18.2.cmml" xref="S4.E20.18.18.18.2">bold-italic-ϵ</ci></apply><ci id="S4.E20.19.19.19.1.cmml" xref="S4.E20.19.19.19.1">𝑛</ci></apply></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.5.5.5.2.cmml" xref="S4.E19.25.25.25a"></times><ci id="S4.E21.2.2.2a.cmml" xref="S4.E21.2.2.2"><mtext id="S4.E21.2.2.2.cmml" xref="S4.E21.2.2.2">Tr</mtext></ci><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.4.cmml" xref="S4.E19.25.25.25a"></times><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.5.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.5.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><apply id="S4.E21.4.4.4.cmml" xref="S4.E21.4.4.4"><ci id="S4.E21.4.4.4.1.cmml" xref="S4.E21.4.4.4.1">¯</ci><apply id="S4.E21.4.4.4.2.cmml" xref="S4.E21.4.4.4.2"><ci id="S4.E21.4.4.4.2.1.cmml" xref="S4.E21.4.4.4.2.1">¯</ci><ci id="S4.E21.4.4.4.2.2.cmml" xref="S4.E21.4.4.4.2.2">bold-italic-ϵ</ci></apply></apply><ci id="S4.E21.5.5.5.1.cmml" xref="S4.E21.5.5.5.1">𝑛</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.4.cmml" xref="S4.E19.25.25.25a"><csymbol cd="latexml" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.4.1.cmml" xref="S4.E19.25.25.25a">quantum-operator-product</csymbol><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.2.cmml" xref="S4.E19.25.25.25a"></times><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.cmml" xref="S4.E19.25.25.25a"><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E21.7.7.7.cmml" xref="S4.E21.7.7.7">∇</ci><ci id="S4.E21.8.8.8.1.cmml" xref="S4.E21.8.8.8.1">italic-ϕ</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E19.25.25.25a">superscript</csymbol><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.2.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.3.2.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E21.9.9.9.cmml" xref="S4.E21.9.9.9">𝑺</ci><ci id="S4.E21.10.10.10.1.cmml" xref="S4.E21.10.10.10.1">𝑛</ci></apply><ci id="S4.E21.11.11.11.1.1.cmml" xref="S4.E21.11.11.11.1.1">𝑘</ci></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><apply id="S4.E21.13.13.13.cmml" xref="S4.E21.13.13.13"><ci id="S4.E21.13.13.13.1.cmml" xref="S4.E21.13.13.13.1">¯</ci><ci id="S4.E21.13.13.13.2.cmml" xref="S4.E21.13.13.13.2">bold-italic-ϵ</ci></apply><ci id="S4.E21.14.14.14.1.cmml" xref="S4.E21.14.14.14.1">𝑛</ci></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.2.2.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.2.2.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E21.17.17.17.cmml" xref="S4.E21.17.17.17">𝚲</ci><ci id="S4.E21.18.18.18.1.cmml" xref="S4.E21.18.18.18.1">𝑍</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.2.cmml" xref="S4.E19.25.25.25a"></times><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.cmml" xref="S4.E19.25.25.25a"><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E21.20.20.20.cmml" xref="S4.E21.20.20.20">∇</ci><ci id="S4.E21.21.21.21.1.cmml" xref="S4.E21.21.21.21.1">italic-ϕ</ci></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.2.1.cmml" xref="S4.E19.25.25.25a">superscript</csymbol><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.2.2.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.3.2.2.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><ci id="S4.E21.22.22.22.cmml" xref="S4.E21.22.22.22">𝑺</ci><ci id="S4.E21.23.23.23.1.cmml" xref="S4.E21.23.23.23.1">𝑛</ci></apply><ci id="S4.E21.24.24.24.1.1.cmml" xref="S4.E21.24.24.24.1.1">𝑘</ci></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.1.1.1.cmml" xref="S4.E19.25.25.25a"><csymbol cd="ambiguous" id="S4.E22.m1.1.1.1.1.1.5.5.5.1.1.1.3.3.3.1.1.1.1.cmml" xref="S4.E19.25.25.25a">subscript</csymbol><apply id="S4.E21.26.26.26.cmml" xref="S4.E21.26.26.26"><ci id="S4.E21.26.26.26.1.cmml" xref="S4.E21.26.26.26.1">¯</ci><ci id="S4.E21.26.26.26.2.cmml" xref="S4.E21.26.26.26.2">bold-italic-ϵ</ci></apply><ci id="S4.E21.27.27.27.1.cmml" xref="S4.E21.27.27.27.1">𝑛</ci></apply></apply></apply></apply></apply></apply><apply id="S4.E22.m1.1.1.1.1.1.5.7.cmml" xref="S4.E19.25.25.25a"><times id="S4.E22.m1.1.1.1.1.1.5.7.1.cmml" xref="S4.E19.25.25.25a"></times><ci id="S4.E21.32.32.32.cmml" xref="S4.E21.32.32.32">𝜁</ci><apply id="S4.E21.34.34.34.cmml" xref="S4.E21.34.34.34"><ci id="S4.E21.34.34.34.1.cmml" xref="S4.E21.34.34.34.1">¯</ci><apply id="S4.E21.34.34.34.2.cmml" xref="S4.E21.34.34.34.2"><ci id="S4.E21.34.34.34.2.1.cmml" xref="S4.E21.34.34.34.2.1">¯</ci><ci id="S4.E21.34.34.34.2.2.cmml" xref="S4.E21.34.34.34.2.2">bold-italic-ϵ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E22.m1.1c">\displaystyle\begin{multlined}D_{KL}\propto-2\text{Re}\left\{\!\big{\langle}% \bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big{|}\bm{Z% }_{n}^{(k)}\big{\rangle}\!\right\}\\ +\big{\langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{% Z}\big{|}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\\ +\text{Tr}\left(\bar{\bar{\bm{\epsilon}}}_{n}\big{\langle}\nabla_{\phi}\bm{S}_% {n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi% }\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\right)-\zeta(\bar{% \bar{\bm{\epsilon}}}).\end{multlined}D_{KL}\propto-2\text{Re}\left\{\!\big{% \langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big% {|}\bm{Z}_{n}^{(k)}\big{\rangle}\!\right\}\\ +\big{\langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{% Z}\big{|}\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\\ +\text{Tr}\left(\bar{\bar{\bm{\epsilon}}}_{n}\big{\langle}\nabla_{\phi}\bm{S}_% {n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi% }\bm{S}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{\rangle}\right)-\zeta(\bar{% \bar{\bm{\epsilon}}}).</annotation><annotation encoding="application/x-llamapun" id="S4.E22.m1.1d">start_ROW start_CELL italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ∝ - 2 Re { ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ⟩ } end_CELL end_ROW start_ROW start_CELL + ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ⟩ end_CELL end_ROW start_ROW start_CELL + Tr ( over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟨ ∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | ∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ⟩ ) - italic_ζ ( over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG ) . 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">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.14">Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E22" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">22</span></a>) can then be minimized in <math alttext="\bar{\bm{\epsilon}}" class="ltx_Math" display="inline" id="S4.p2.13.m1.1"><semantics id="S4.p2.13.m1.1a"><mover accent="true" id="S4.p2.13.m1.1.1" xref="S4.p2.13.m1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.13.m1.1.1.2" mathvariant="bold-italic" xref="S4.p2.13.m1.1.1.2.cmml">ϵ</mi><mo id="S4.p2.13.m1.1.1.1" xref="S4.p2.13.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.p2.13.m1.1b"><apply id="S4.p2.13.m1.1.1.cmml" xref="S4.p2.13.m1.1.1"><ci id="S4.p2.13.m1.1.1.1.cmml" xref="S4.p2.13.m1.1.1.1">¯</ci><ci id="S4.p2.13.m1.1.1.2.cmml" xref="S4.p2.13.m1.1.1.2">bold-italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.13.m1.1c">\bar{\bm{\epsilon}}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.13.m1.1d">over¯ start_ARG bold_italic_ϵ end_ARG</annotation></semantics></math> and <math alttext="\bar{\bar{\bm{\epsilon}}}" class="ltx_Math" display="inline" id="S4.p2.14.m2.1"><semantics id="S4.p2.14.m2.1a"><mover accent="true" id="S4.p2.14.m2.1.1" xref="S4.p2.14.m2.1.1.cmml"><mover accent="true" id="S4.p2.14.m2.1.1.2" xref="S4.p2.14.m2.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p2.14.m2.1.1.2.2" mathvariant="bold-italic" xref="S4.p2.14.m2.1.1.2.2.cmml">ϵ</mi><mo id="S4.p2.14.m2.1.1.2.1" xref="S4.p2.14.m2.1.1.2.1.cmml">¯</mo></mover><mo id="S4.p2.14.m2.1.1.1" xref="S4.p2.14.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.p2.14.m2.1b"><apply id="S4.p2.14.m2.1.1.cmml" xref="S4.p2.14.m2.1.1"><ci id="S4.p2.14.m2.1.1.1.cmml" xref="S4.p2.14.m2.1.1.1">¯</ci><apply id="S4.p2.14.m2.1.1.2.cmml" xref="S4.p2.14.m2.1.1.2"><ci id="S4.p2.14.m2.1.1.2.1.cmml" xref="S4.p2.14.m2.1.1.2.1">¯</ci><ci id="S4.p2.14.m2.1.1.2.2.cmml" xref="S4.p2.14.m2.1.1.2.2">bold-italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.14.m2.1c">\bar{\bar{\bm{\epsilon}}}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.14.m2.1d">over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.2">A problem arises with the path loss <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mi id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><ci id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_α</annotation></semantics></math> as it contains information about the reflectivity of the target. Hence, <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mi id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><ci id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">italic_α</annotation></semantics></math> will be estimated using a maximum likelihood estimate using the previous mean of the kinematics,</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="\hat{\alpha}^{(k)}=\frac{\big{\langle}\bm{S}_{n}^{(k)}\!\big{(}\bar{\bm{\phi}}% _{n-1}\big{)}\big{|}\bm{\Lambda}_{Z}\big{|}\bm{Z}_{n}^{(k)}\big{\rangle}}{\big% {\langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\phi}}_{n-1})\big{|}\bm{\Lambda}_{Z}\big{% |}\bm{S}_{n}^{(k)}\big{(}\bar{\bm{\phi}}_{n-1}\big{)}\big{\rangle}}." class="ltx_Math" display="block" id="S4.E23.m1.12"><semantics id="S4.E23.m1.12a"><mrow id="S4.E23.m1.12.12.1" xref="S4.E23.m1.12.12.1.1.cmml"><mrow id="S4.E23.m1.12.12.1.1" xref="S4.E23.m1.12.12.1.1.cmml"><msup id="S4.E23.m1.12.12.1.1.2" xref="S4.E23.m1.12.12.1.1.2.cmml"><mover accent="true" id="S4.E23.m1.12.12.1.1.2.2" xref="S4.E23.m1.12.12.1.1.2.2.cmml"><mi id="S4.E23.m1.12.12.1.1.2.2.2" xref="S4.E23.m1.12.12.1.1.2.2.2.cmml">α</mi><mo id="S4.E23.m1.12.12.1.1.2.2.1" xref="S4.E23.m1.12.12.1.1.2.2.1.cmml">^</mo></mover><mrow id="S4.E23.m1.1.1.1.3" xref="S4.E23.m1.12.12.1.1.2.cmml"><mo id="S4.E23.m1.1.1.1.3.1" stretchy="false" xref="S4.E23.m1.12.12.1.1.2.cmml">(</mo><mi id="S4.E23.m1.1.1.1.1" xref="S4.E23.m1.1.1.1.1.cmml">k</mi><mo id="S4.E23.m1.1.1.1.3.2" stretchy="false" xref="S4.E23.m1.12.12.1.1.2.cmml">)</mo></mrow></msup><mo id="S4.E23.m1.12.12.1.1.1" xref="S4.E23.m1.12.12.1.1.1.cmml">=</mo><mfrac id="S4.E23.m1.11.11" xref="S4.E23.m1.11.11.cmml"><mrow id="S4.E23.m1.6.6.5.5" xref="S4.E23.m1.6.6.5.6.cmml"><mo id="S4.E23.m1.6.6.5.5.4" maxsize="120%" minsize="120%" xref="S4.E23.m1.6.6.5.6.1.cmml">⟨</mo><mrow id="S4.E23.m1.4.4.3.3.1" xref="S4.E23.m1.4.4.3.3.1.cmml"><msubsup id="S4.E23.m1.4.4.3.3.1.3" xref="S4.E23.m1.4.4.3.3.1.3.cmml"><mi id="S4.E23.m1.4.4.3.3.1.3.2.2" xref="S4.E23.m1.4.4.3.3.1.3.2.2.cmml">𝑺</mi><mi id="S4.E23.m1.4.4.3.3.1.3.2.3" xref="S4.E23.m1.4.4.3.3.1.3.2.3.cmml">n</mi><mrow id="S4.E23.m1.2.2.1.1.1.3" xref="S4.E23.m1.4.4.3.3.1.3.cmml"><mo id="S4.E23.m1.2.2.1.1.1.3.1" stretchy="false" xref="S4.E23.m1.4.4.3.3.1.3.cmml">(</mo><mi id="S4.E23.m1.2.2.1.1.1.1" xref="S4.E23.m1.2.2.1.1.1.1.cmml">k</mi><mo id="S4.E23.m1.2.2.1.1.1.3.2" stretchy="false" xref="S4.E23.m1.4.4.3.3.1.3.cmml">)</mo></mrow></msubsup><mo id="S4.E23.m1.4.4.3.3.1.2" xref="S4.E23.m1.4.4.3.3.1.2.cmml">⁢</mo><mrow id="S4.E23.m1.4.4.3.3.1.1.1" xref="S4.E23.m1.4.4.3.3.1.1.1.1.cmml"><mo id="S4.E23.m1.4.4.3.3.1.1.1.2" maxsize="120%" minsize="120%" xref="S4.E23.m1.4.4.3.3.1.1.1.1.cmml">(</mo><msub id="S4.E23.m1.4.4.3.3.1.1.1.1" xref="S4.E23.m1.4.4.3.3.1.1.1.1.cmml"><mover accent="true" id="S4.E23.m1.4.4.3.3.1.1.1.1.2" xref="S4.E23.m1.4.4.3.3.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E23.m1.4.4.3.3.1.1.1.1.2.2" mathvariant="bold-italic" xref="S4.E23.m1.4.4.3.3.1.1.1.1.2.2.cmml">ϕ</mi><mo id="S4.E23.m1.4.4.3.3.1.1.1.1.2.1" xref="S4.E23.m1.4.4.3.3.1.1.1.1.2.1.cmml">¯</mo></mover><mrow id="S4.E23.m1.4.4.3.3.1.1.1.1.3" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.cmml"><mi id="S4.E23.m1.4.4.3.3.1.1.1.1.3.2" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.2.cmml">n</mi><mo id="S4.E23.m1.4.4.3.3.1.1.1.1.3.1" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.1.cmml">−</mo><mn id="S4.E23.m1.4.4.3.3.1.1.1.1.3.3" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.E23.m1.4.4.3.3.1.1.1.3" maxsize="120%" minsize="120%" xref="S4.E23.m1.4.4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E23.m1.6.6.5.5.5" maxsize="120%" minsize="120%" xref="S4.E23.m1.6.6.5.6.1.cmml">|</mo><msub id="S4.E23.m1.5.5.4.4.2" xref="S4.E23.m1.5.5.4.4.2.cmml"><mi id="S4.E23.m1.5.5.4.4.2.2" xref="S4.E23.m1.5.5.4.4.2.2.cmml">𝚲</mi><mi id="S4.E23.m1.5.5.4.4.2.3" xref="S4.E23.m1.5.5.4.4.2.3.cmml">Z</mi></msub><mo id="S4.E23.m1.6.6.5.5.6" maxsize="120%" minsize="120%" xref="S4.E23.m1.6.6.5.6.1.cmml">|</mo><msubsup id="S4.E23.m1.6.6.5.5.3" xref="S4.E23.m1.6.6.5.5.3.cmml"><mi id="S4.E23.m1.6.6.5.5.3.2.2" xref="S4.E23.m1.6.6.5.5.3.2.2.cmml">𝒁</mi><mi id="S4.E23.m1.6.6.5.5.3.2.3" xref="S4.E23.m1.6.6.5.5.3.2.3.cmml">n</mi><mrow id="S4.E23.m1.3.3.2.2.1.3" xref="S4.E23.m1.6.6.5.5.3.cmml"><mo id="S4.E23.m1.3.3.2.2.1.3.1" stretchy="false" xref="S4.E23.m1.6.6.5.5.3.cmml">(</mo><mi id="S4.E23.m1.3.3.2.2.1.1" xref="S4.E23.m1.3.3.2.2.1.1.cmml">k</mi><mo id="S4.E23.m1.3.3.2.2.1.3.2" stretchy="false" xref="S4.E23.m1.6.6.5.5.3.cmml">)</mo></mrow></msubsup><mo id="S4.E23.m1.6.6.5.5.7" maxsize="120%" minsize="120%" xref="S4.E23.m1.6.6.5.6.1.cmml">⟩</mo></mrow><mrow id="S4.E23.m1.11.11.10.5" xref="S4.E23.m1.11.11.10.6.cmml"><mo id="S4.E23.m1.11.11.10.5.4" maxsize="120%" minsize="120%" xref="S4.E23.m1.11.11.10.6.1.cmml">⟨</mo><mrow id="S4.E23.m1.9.9.8.3.1" xref="S4.E23.m1.9.9.8.3.1.cmml"><msubsup id="S4.E23.m1.9.9.8.3.1.3" xref="S4.E23.m1.9.9.8.3.1.3.cmml"><mi id="S4.E23.m1.9.9.8.3.1.3.2.2" xref="S4.E23.m1.9.9.8.3.1.3.2.2.cmml">𝑺</mi><mi id="S4.E23.m1.9.9.8.3.1.3.2.3" xref="S4.E23.m1.9.9.8.3.1.3.2.3.cmml">n</mi><mrow id="S4.E23.m1.7.7.6.1.1.3" xref="S4.E23.m1.9.9.8.3.1.3.cmml"><mo id="S4.E23.m1.7.7.6.1.1.3.1" stretchy="false" xref="S4.E23.m1.9.9.8.3.1.3.cmml">(</mo><mi id="S4.E23.m1.7.7.6.1.1.1" xref="S4.E23.m1.7.7.6.1.1.1.cmml">k</mi><mo id="S4.E23.m1.7.7.6.1.1.3.2" stretchy="false" xref="S4.E23.m1.9.9.8.3.1.3.cmml">)</mo></mrow></msubsup><mo id="S4.E23.m1.9.9.8.3.1.2" xref="S4.E23.m1.9.9.8.3.1.2.cmml">⁢</mo><mrow id="S4.E23.m1.9.9.8.3.1.1.1" xref="S4.E23.m1.9.9.8.3.1.1.1.1.cmml"><mo id="S4.E23.m1.9.9.8.3.1.1.1.2" stretchy="false" xref="S4.E23.m1.9.9.8.3.1.1.1.1.cmml">(</mo><msub id="S4.E23.m1.9.9.8.3.1.1.1.1" xref="S4.E23.m1.9.9.8.3.1.1.1.1.cmml"><mover accent="true" id="S4.E23.m1.9.9.8.3.1.1.1.1.2" xref="S4.E23.m1.9.9.8.3.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E23.m1.9.9.8.3.1.1.1.1.2.2" mathvariant="bold-italic" xref="S4.E23.m1.9.9.8.3.1.1.1.1.2.2.cmml">ϕ</mi><mo id="S4.E23.m1.9.9.8.3.1.1.1.1.2.1" xref="S4.E23.m1.9.9.8.3.1.1.1.1.2.1.cmml">¯</mo></mover><mrow id="S4.E23.m1.9.9.8.3.1.1.1.1.3" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.cmml"><mi id="S4.E23.m1.9.9.8.3.1.1.1.1.3.2" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.2.cmml">n</mi><mo id="S4.E23.m1.9.9.8.3.1.1.1.1.3.1" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.1.cmml">−</mo><mn id="S4.E23.m1.9.9.8.3.1.1.1.1.3.3" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.E23.m1.9.9.8.3.1.1.1.3" stretchy="false" xref="S4.E23.m1.9.9.8.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E23.m1.11.11.10.5.5" maxsize="120%" minsize="120%" xref="S4.E23.m1.11.11.10.6.1.cmml">|</mo><msub id="S4.E23.m1.10.10.9.4.2" xref="S4.E23.m1.10.10.9.4.2.cmml"><mi id="S4.E23.m1.10.10.9.4.2.2" xref="S4.E23.m1.10.10.9.4.2.2.cmml">𝚲</mi><mi id="S4.E23.m1.10.10.9.4.2.3" xref="S4.E23.m1.10.10.9.4.2.3.cmml">Z</mi></msub><mo id="S4.E23.m1.11.11.10.5.6" maxsize="120%" minsize="120%" xref="S4.E23.m1.11.11.10.6.1.cmml">|</mo><mrow id="S4.E23.m1.11.11.10.5.3" xref="S4.E23.m1.11.11.10.5.3.cmml"><msubsup id="S4.E23.m1.11.11.10.5.3.3" xref="S4.E23.m1.11.11.10.5.3.3.cmml"><mi id="S4.E23.m1.11.11.10.5.3.3.2.2" xref="S4.E23.m1.11.11.10.5.3.3.2.2.cmml">𝑺</mi><mi id="S4.E23.m1.11.11.10.5.3.3.2.3" xref="S4.E23.m1.11.11.10.5.3.3.2.3.cmml">n</mi><mrow id="S4.E23.m1.8.8.7.2.1.3" xref="S4.E23.m1.11.11.10.5.3.3.cmml"><mo id="S4.E23.m1.8.8.7.2.1.3.1" stretchy="false" xref="S4.E23.m1.11.11.10.5.3.3.cmml">(</mo><mi id="S4.E23.m1.8.8.7.2.1.1" xref="S4.E23.m1.8.8.7.2.1.1.cmml">k</mi><mo id="S4.E23.m1.8.8.7.2.1.3.2" stretchy="false" xref="S4.E23.m1.11.11.10.5.3.3.cmml">)</mo></mrow></msubsup><mo id="S4.E23.m1.11.11.10.5.3.2" xref="S4.E23.m1.11.11.10.5.3.2.cmml">⁢</mo><mrow id="S4.E23.m1.11.11.10.5.3.1.1" xref="S4.E23.m1.11.11.10.5.3.1.1.1.cmml"><mo id="S4.E23.m1.11.11.10.5.3.1.1.2" maxsize="120%" minsize="120%" xref="S4.E23.m1.11.11.10.5.3.1.1.1.cmml">(</mo><msub id="S4.E23.m1.11.11.10.5.3.1.1.1" xref="S4.E23.m1.11.11.10.5.3.1.1.1.cmml"><mover accent="true" id="S4.E23.m1.11.11.10.5.3.1.1.1.2" xref="S4.E23.m1.11.11.10.5.3.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E23.m1.11.11.10.5.3.1.1.1.2.2" mathvariant="bold-italic" xref="S4.E23.m1.11.11.10.5.3.1.1.1.2.2.cmml">ϕ</mi><mo id="S4.E23.m1.11.11.10.5.3.1.1.1.2.1" xref="S4.E23.m1.11.11.10.5.3.1.1.1.2.1.cmml">¯</mo></mover><mrow id="S4.E23.m1.11.11.10.5.3.1.1.1.3" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.cmml"><mi id="S4.E23.m1.11.11.10.5.3.1.1.1.3.2" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.2.cmml">n</mi><mo id="S4.E23.m1.11.11.10.5.3.1.1.1.3.1" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.1.cmml">−</mo><mn id="S4.E23.m1.11.11.10.5.3.1.1.1.3.3" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.E23.m1.11.11.10.5.3.1.1.3" maxsize="120%" minsize="120%" xref="S4.E23.m1.11.11.10.5.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E23.m1.11.11.10.5.7" maxsize="120%" minsize="120%" xref="S4.E23.m1.11.11.10.6.1.cmml">⟩</mo></mrow></mfrac></mrow><mo id="S4.E23.m1.12.12.1.2" lspace="0em" xref="S4.E23.m1.12.12.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E23.m1.12b"><apply id="S4.E23.m1.12.12.1.1.cmml" xref="S4.E23.m1.12.12.1"><eq id="S4.E23.m1.12.12.1.1.1.cmml" xref="S4.E23.m1.12.12.1.1.1"></eq><apply id="S4.E23.m1.12.12.1.1.2.cmml" xref="S4.E23.m1.12.12.1.1.2"><csymbol cd="ambiguous" id="S4.E23.m1.12.12.1.1.2.1.cmml" xref="S4.E23.m1.12.12.1.1.2">superscript</csymbol><apply id="S4.E23.m1.12.12.1.1.2.2.cmml" xref="S4.E23.m1.12.12.1.1.2.2"><ci id="S4.E23.m1.12.12.1.1.2.2.1.cmml" xref="S4.E23.m1.12.12.1.1.2.2.1">^</ci><ci id="S4.E23.m1.12.12.1.1.2.2.2.cmml" xref="S4.E23.m1.12.12.1.1.2.2.2">𝛼</ci></apply><ci id="S4.E23.m1.1.1.1.1.cmml" xref="S4.E23.m1.1.1.1.1">𝑘</ci></apply><apply id="S4.E23.m1.11.11.cmml" xref="S4.E23.m1.11.11"><divide id="S4.E23.m1.11.11.11.cmml" xref="S4.E23.m1.11.11"></divide><apply id="S4.E23.m1.6.6.5.6.cmml" xref="S4.E23.m1.6.6.5.5"><csymbol cd="latexml" id="S4.E23.m1.6.6.5.6.1.cmml" xref="S4.E23.m1.6.6.5.5.4">quantum-operator-product</csymbol><apply id="S4.E23.m1.4.4.3.3.1.cmml" xref="S4.E23.m1.4.4.3.3.1"><times id="S4.E23.m1.4.4.3.3.1.2.cmml" xref="S4.E23.m1.4.4.3.3.1.2"></times><apply id="S4.E23.m1.4.4.3.3.1.3.cmml" xref="S4.E23.m1.4.4.3.3.1.3"><csymbol cd="ambiguous" id="S4.E23.m1.4.4.3.3.1.3.1.cmml" xref="S4.E23.m1.4.4.3.3.1.3">superscript</csymbol><apply id="S4.E23.m1.4.4.3.3.1.3.2.cmml" xref="S4.E23.m1.4.4.3.3.1.3"><csymbol cd="ambiguous" id="S4.E23.m1.4.4.3.3.1.3.2.1.cmml" xref="S4.E23.m1.4.4.3.3.1.3">subscript</csymbol><ci id="S4.E23.m1.4.4.3.3.1.3.2.2.cmml" xref="S4.E23.m1.4.4.3.3.1.3.2.2">𝑺</ci><ci id="S4.E23.m1.4.4.3.3.1.3.2.3.cmml" xref="S4.E23.m1.4.4.3.3.1.3.2.3">𝑛</ci></apply><ci id="S4.E23.m1.2.2.1.1.1.1.cmml" xref="S4.E23.m1.2.2.1.1.1.1">𝑘</ci></apply><apply id="S4.E23.m1.4.4.3.3.1.1.1.1.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.E23.m1.4.4.3.3.1.1.1.1.1.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1">subscript</csymbol><apply id="S4.E23.m1.4.4.3.3.1.1.1.1.2.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1.1.2"><ci id="S4.E23.m1.4.4.3.3.1.1.1.1.2.1.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1.1.2.1">¯</ci><ci id="S4.E23.m1.4.4.3.3.1.1.1.1.2.2.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1.1.2.2">bold-italic-ϕ</ci></apply><apply id="S4.E23.m1.4.4.3.3.1.1.1.1.3.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3"><minus id="S4.E23.m1.4.4.3.3.1.1.1.1.3.1.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.1"></minus><ci id="S4.E23.m1.4.4.3.3.1.1.1.1.3.2.cmml" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.2">𝑛</ci><cn id="S4.E23.m1.4.4.3.3.1.1.1.1.3.3.cmml" type="integer" xref="S4.E23.m1.4.4.3.3.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="S4.E23.m1.5.5.4.4.2.cmml" xref="S4.E23.m1.5.5.4.4.2"><csymbol cd="ambiguous" id="S4.E23.m1.5.5.4.4.2.1.cmml" xref="S4.E23.m1.5.5.4.4.2">subscript</csymbol><ci id="S4.E23.m1.5.5.4.4.2.2.cmml" xref="S4.E23.m1.5.5.4.4.2.2">𝚲</ci><ci id="S4.E23.m1.5.5.4.4.2.3.cmml" xref="S4.E23.m1.5.5.4.4.2.3">𝑍</ci></apply><apply id="S4.E23.m1.6.6.5.5.3.cmml" xref="S4.E23.m1.6.6.5.5.3"><csymbol cd="ambiguous" id="S4.E23.m1.6.6.5.5.3.1.cmml" xref="S4.E23.m1.6.6.5.5.3">superscript</csymbol><apply id="S4.E23.m1.6.6.5.5.3.2.cmml" xref="S4.E23.m1.6.6.5.5.3"><csymbol cd="ambiguous" id="S4.E23.m1.6.6.5.5.3.2.1.cmml" xref="S4.E23.m1.6.6.5.5.3">subscript</csymbol><ci id="S4.E23.m1.6.6.5.5.3.2.2.cmml" xref="S4.E23.m1.6.6.5.5.3.2.2">𝒁</ci><ci id="S4.E23.m1.6.6.5.5.3.2.3.cmml" xref="S4.E23.m1.6.6.5.5.3.2.3">𝑛</ci></apply><ci id="S4.E23.m1.3.3.2.2.1.1.cmml" xref="S4.E23.m1.3.3.2.2.1.1">𝑘</ci></apply></apply><apply id="S4.E23.m1.11.11.10.6.cmml" xref="S4.E23.m1.11.11.10.5"><csymbol cd="latexml" id="S4.E23.m1.11.11.10.6.1.cmml" xref="S4.E23.m1.11.11.10.5.4">quantum-operator-product</csymbol><apply id="S4.E23.m1.9.9.8.3.1.cmml" xref="S4.E23.m1.9.9.8.3.1"><times id="S4.E23.m1.9.9.8.3.1.2.cmml" xref="S4.E23.m1.9.9.8.3.1.2"></times><apply id="S4.E23.m1.9.9.8.3.1.3.cmml" xref="S4.E23.m1.9.9.8.3.1.3"><csymbol cd="ambiguous" id="S4.E23.m1.9.9.8.3.1.3.1.cmml" xref="S4.E23.m1.9.9.8.3.1.3">superscript</csymbol><apply id="S4.E23.m1.9.9.8.3.1.3.2.cmml" xref="S4.E23.m1.9.9.8.3.1.3"><csymbol cd="ambiguous" id="S4.E23.m1.9.9.8.3.1.3.2.1.cmml" xref="S4.E23.m1.9.9.8.3.1.3">subscript</csymbol><ci id="S4.E23.m1.9.9.8.3.1.3.2.2.cmml" xref="S4.E23.m1.9.9.8.3.1.3.2.2">𝑺</ci><ci id="S4.E23.m1.9.9.8.3.1.3.2.3.cmml" xref="S4.E23.m1.9.9.8.3.1.3.2.3">𝑛</ci></apply><ci id="S4.E23.m1.7.7.6.1.1.1.cmml" xref="S4.E23.m1.7.7.6.1.1.1">𝑘</ci></apply><apply id="S4.E23.m1.9.9.8.3.1.1.1.1.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1"><csymbol cd="ambiguous" id="S4.E23.m1.9.9.8.3.1.1.1.1.1.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1">subscript</csymbol><apply id="S4.E23.m1.9.9.8.3.1.1.1.1.2.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1.1.2"><ci id="S4.E23.m1.9.9.8.3.1.1.1.1.2.1.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1.1.2.1">¯</ci><ci id="S4.E23.m1.9.9.8.3.1.1.1.1.2.2.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1.1.2.2">bold-italic-ϕ</ci></apply><apply id="S4.E23.m1.9.9.8.3.1.1.1.1.3.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3"><minus id="S4.E23.m1.9.9.8.3.1.1.1.1.3.1.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.1"></minus><ci id="S4.E23.m1.9.9.8.3.1.1.1.1.3.2.cmml" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.2">𝑛</ci><cn id="S4.E23.m1.9.9.8.3.1.1.1.1.3.3.cmml" type="integer" xref="S4.E23.m1.9.9.8.3.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="S4.E23.m1.10.10.9.4.2.cmml" xref="S4.E23.m1.10.10.9.4.2"><csymbol cd="ambiguous" id="S4.E23.m1.10.10.9.4.2.1.cmml" xref="S4.E23.m1.10.10.9.4.2">subscript</csymbol><ci id="S4.E23.m1.10.10.9.4.2.2.cmml" xref="S4.E23.m1.10.10.9.4.2.2">𝚲</ci><ci id="S4.E23.m1.10.10.9.4.2.3.cmml" xref="S4.E23.m1.10.10.9.4.2.3">𝑍</ci></apply><apply id="S4.E23.m1.11.11.10.5.3.cmml" xref="S4.E23.m1.11.11.10.5.3"><times id="S4.E23.m1.11.11.10.5.3.2.cmml" xref="S4.E23.m1.11.11.10.5.3.2"></times><apply id="S4.E23.m1.11.11.10.5.3.3.cmml" xref="S4.E23.m1.11.11.10.5.3.3"><csymbol cd="ambiguous" id="S4.E23.m1.11.11.10.5.3.3.1.cmml" xref="S4.E23.m1.11.11.10.5.3.3">superscript</csymbol><apply id="S4.E23.m1.11.11.10.5.3.3.2.cmml" xref="S4.E23.m1.11.11.10.5.3.3"><csymbol cd="ambiguous" id="S4.E23.m1.11.11.10.5.3.3.2.1.cmml" xref="S4.E23.m1.11.11.10.5.3.3">subscript</csymbol><ci id="S4.E23.m1.11.11.10.5.3.3.2.2.cmml" xref="S4.E23.m1.11.11.10.5.3.3.2.2">𝑺</ci><ci id="S4.E23.m1.11.11.10.5.3.3.2.3.cmml" xref="S4.E23.m1.11.11.10.5.3.3.2.3">𝑛</ci></apply><ci id="S4.E23.m1.8.8.7.2.1.1.cmml" xref="S4.E23.m1.8.8.7.2.1.1">𝑘</ci></apply><apply id="S4.E23.m1.11.11.10.5.3.1.1.1.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1"><csymbol cd="ambiguous" id="S4.E23.m1.11.11.10.5.3.1.1.1.1.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1">subscript</csymbol><apply id="S4.E23.m1.11.11.10.5.3.1.1.1.2.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1.1.2"><ci id="S4.E23.m1.11.11.10.5.3.1.1.1.2.1.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1.1.2.1">¯</ci><ci id="S4.E23.m1.11.11.10.5.3.1.1.1.2.2.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1.1.2.2">bold-italic-ϕ</ci></apply><apply id="S4.E23.m1.11.11.10.5.3.1.1.1.3.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3"><minus id="S4.E23.m1.11.11.10.5.3.1.1.1.3.1.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.1"></minus><ci id="S4.E23.m1.11.11.10.5.3.1.1.1.3.2.cmml" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.2">𝑛</ci><cn id="S4.E23.m1.11.11.10.5.3.1.1.1.3.3.cmml" type="integer" xref="S4.E23.m1.11.11.10.5.3.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E23.m1.12c">\hat{\alpha}^{(k)}=\frac{\big{\langle}\bm{S}_{n}^{(k)}\!\big{(}\bar{\bm{\phi}}% _{n-1}\big{)}\big{|}\bm{\Lambda}_{Z}\big{|}\bm{Z}_{n}^{(k)}\big{\rangle}}{\big% {\langle}\bm{S}_{n}^{(k)}\!(\bar{\bm{\phi}}_{n-1})\big{|}\bm{\Lambda}_{Z}\big{% |}\bm{S}_{n}^{(k)}\big{(}\bar{\bm{\phi}}_{n-1}\big{)}\big{\rangle}}.</annotation><annotation encoding="application/x-llamapun" id="S4.E23.m1.12d">over^ start_ARG italic_α end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = divide start_ARG ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ⟩ end_ARG start_ARG ⟨ bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) ⟩ 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">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p3.4">Lastly the real operator in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E22" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">22</span></a>) causes instability when the optimization is performed numerically, due to the phase of <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.p3.3.m1.1"><semantics id="S4.p3.3.m1.1a"><mi id="S4.p3.3.m1.1.1" xref="S4.p3.3.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.p3.3.m1.1b"><ci id="S4.p3.3.m1.1.1.cmml" xref="S4.p3.3.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m1.1d">italic_α</annotation></semantics></math> being proportional to <math alttext="\omega_{c}\sqrt{x^{2}+y^{2}}" class="ltx_Math" display="inline" id="S4.p3.4.m2.1"><semantics id="S4.p3.4.m2.1a"><mrow id="S4.p3.4.m2.1.1" xref="S4.p3.4.m2.1.1.cmml"><msub id="S4.p3.4.m2.1.1.2" xref="S4.p3.4.m2.1.1.2.cmml"><mi id="S4.p3.4.m2.1.1.2.2" xref="S4.p3.4.m2.1.1.2.2.cmml">ω</mi><mi id="S4.p3.4.m2.1.1.2.3" xref="S4.p3.4.m2.1.1.2.3.cmml">c</mi></msub><mo id="S4.p3.4.m2.1.1.1" xref="S4.p3.4.m2.1.1.1.cmml">⁢</mo><msqrt id="S4.p3.4.m2.1.1.3" xref="S4.p3.4.m2.1.1.3.cmml"><mrow id="S4.p3.4.m2.1.1.3.2" xref="S4.p3.4.m2.1.1.3.2.cmml"><msup id="S4.p3.4.m2.1.1.3.2.2" xref="S4.p3.4.m2.1.1.3.2.2.cmml"><mi id="S4.p3.4.m2.1.1.3.2.2.2" xref="S4.p3.4.m2.1.1.3.2.2.2.cmml">x</mi><mn id="S4.p3.4.m2.1.1.3.2.2.3" xref="S4.p3.4.m2.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S4.p3.4.m2.1.1.3.2.1" xref="S4.p3.4.m2.1.1.3.2.1.cmml">+</mo><msup id="S4.p3.4.m2.1.1.3.2.3" xref="S4.p3.4.m2.1.1.3.2.3.cmml"><mi id="S4.p3.4.m2.1.1.3.2.3.2" xref="S4.p3.4.m2.1.1.3.2.3.2.cmml">y</mi><mn id="S4.p3.4.m2.1.1.3.2.3.3" xref="S4.p3.4.m2.1.1.3.2.3.3.cmml">2</mn></msup></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.4.m2.1b"><apply id="S4.p3.4.m2.1.1.cmml" xref="S4.p3.4.m2.1.1"><times id="S4.p3.4.m2.1.1.1.cmml" xref="S4.p3.4.m2.1.1.1"></times><apply id="S4.p3.4.m2.1.1.2.cmml" xref="S4.p3.4.m2.1.1.2"><csymbol cd="ambiguous" id="S4.p3.4.m2.1.1.2.1.cmml" xref="S4.p3.4.m2.1.1.2">subscript</csymbol><ci id="S4.p3.4.m2.1.1.2.2.cmml" xref="S4.p3.4.m2.1.1.2.2">𝜔</ci><ci id="S4.p3.4.m2.1.1.2.3.cmml" xref="S4.p3.4.m2.1.1.2.3">𝑐</ci></apply><apply id="S4.p3.4.m2.1.1.3.cmml" xref="S4.p3.4.m2.1.1.3"><root id="S4.p3.4.m2.1.1.3a.cmml" xref="S4.p3.4.m2.1.1.3"></root><apply id="S4.p3.4.m2.1.1.3.2.cmml" xref="S4.p3.4.m2.1.1.3.2"><plus id="S4.p3.4.m2.1.1.3.2.1.cmml" xref="S4.p3.4.m2.1.1.3.2.1"></plus><apply id="S4.p3.4.m2.1.1.3.2.2.cmml" xref="S4.p3.4.m2.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.p3.4.m2.1.1.3.2.2.1.cmml" xref="S4.p3.4.m2.1.1.3.2.2">superscript</csymbol><ci id="S4.p3.4.m2.1.1.3.2.2.2.cmml" xref="S4.p3.4.m2.1.1.3.2.2.2">𝑥</ci><cn id="S4.p3.4.m2.1.1.3.2.2.3.cmml" type="integer" xref="S4.p3.4.m2.1.1.3.2.2.3">2</cn></apply><apply id="S4.p3.4.m2.1.1.3.2.3.cmml" xref="S4.p3.4.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.p3.4.m2.1.1.3.2.3.1.cmml" xref="S4.p3.4.m2.1.1.3.2.3">superscript</csymbol><ci id="S4.p3.4.m2.1.1.3.2.3.2.cmml" xref="S4.p3.4.m2.1.1.3.2.3.2">𝑦</ci><cn id="S4.p3.4.m2.1.1.3.2.3.3.cmml" type="integer" xref="S4.p3.4.m2.1.1.3.2.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m2.1c">\omega_{c}\sqrt{x^{2}+y^{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m2.1d">italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT square-root start_ARG italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>. Hence, any estimation error arising from (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E23" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">23</span></a>) will be overlaid as a fast oscillating cosine. It was found that discarding this phase instability by replacing the operator by the absolute value greatly improved estimation accuracy, hence the objective function used in the algorithm is,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx6"> <tbody id="S4.E27"><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\begin{multlined}D_{KL}\propto-\left|\hat{\alpha}^{(k)}\big{% \langle}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}% _{Z}\big{|}\bm{Z}_{n}^{(k)}\big{\rangle}\right|\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\big{\langle}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{|}\bm{\Lambda}_{Z}\big{|}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{\rangle}\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\text{Tr}\!\left(\bar{\bar{\bm{\epsilon}}% }\big{\langle}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}})\big% {|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{% \epsilon}})\big{\rangle}\!\right)-\zeta(\bar{\bm{\epsilon}},\bar{\bar{\bm{% \epsilon}}}).\end{multlined}D_{KL}\propto-\left|\hat{\alpha}^{(k)}\big{\langle% }\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}% \big{|}\bm{Z}_{n}^{(k)}\big{\rangle}\right|\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\big{\langle}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{|}\bm{\Lambda}_{Z}\big{|}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{\rangle}\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\text{Tr}\!\left(\bar{\bar{\bm{\epsilon}}% }\big{\langle}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}})\big% {|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{% \epsilon}})\big{\rangle}\!\right)-\zeta(\bar{\bm{\epsilon}},\bar{\bar{\bm{% \epsilon}}})." class="ltx_Math" display="inline" id="S4.E27.m1.1"><semantics id="S4.E27.m1.1a"><mtable id="S4.E27.m1.1.1.2" rowspacing="0pt"><mtr id="S4.E27.m1.1.1.2a"><mtd class="ltx_align_left" columnalign="left" id="S4.E27.m1.1.1.2b"><mrow id="S4.E24.25.25"><msub id="S4.E24.25.25.26"><mi id="S4.E24.1.1.1" xref="S4.E24.1.1.1.cmml">D</mi><mrow id="S4.E24.2.2.2.1" xref="S4.E24.2.2.2.1.cmml"><mi id="S4.E24.2.2.2.1.2" xref="S4.E24.2.2.2.1.2.cmml">K</mi><mo id="S4.E24.2.2.2.1.1" xref="S4.E24.2.2.2.1.1.cmml">⁢</mo><mi id="S4.E24.2.2.2.1.3" xref="S4.E24.2.2.2.1.3.cmml">L</mi></mrow></msub><mo id="S4.E24.3.3.3" xref="S4.E24.3.3.3.cmml">∝</mo><mrow id="S4.E24.25.25.25"><mo id="S4.E24.25.25.25a" xref="S4.E27.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E24.25.25.25.1.1"><mo id="S4.E24.5.5.5" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E24.25.25.25.1.1.1"><msup id="S4.E24.25.25.25.1.1.1.5"><mover accent="true" id="S4.E24.6.6.6" xref="S4.E24.6.6.6.cmml"><mi id="S4.E24.6.6.6.2" xref="S4.E24.6.6.6.2.cmml">α</mi><mo id="S4.E24.6.6.6.1" xref="S4.E24.6.6.6.1.cmml">^</mo></mover><mrow id="S4.E24.7.7.7.1.3"><mo id="S4.E24.7.7.7.1.3.1" stretchy="false">(</mo><mi id="S4.E24.7.7.7.1.1" xref="S4.E24.7.7.7.1.1.cmml">k</mi><mo id="S4.E24.7.7.7.1.3.2" stretchy="false">)</mo></mrow></msup><mo id="S4.E24.25.25.25.1.1.1.4" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E24.25.25.25.1.1.1.3.3"><mo id="S4.E24.8.8.8" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E24.25.25.25.1.1.1.1.1.1"><msubsup id="S4.E24.25.25.25.1.1.1.1.1.1.3"><mover accent="true" id="S4.E24.9.9.9" xref="S4.E24.9.9.9.cmml"><mi id="S4.E24.9.9.9.2" xref="S4.E24.9.9.9.2.cmml">𝑺</mi><mo id="S4.E24.9.9.9.1" xref="S4.E24.9.9.9.1.cmml">~</mo></mover><mi id="S4.E24.10.10.10.1" xref="S4.E24.10.10.10.1.cmml">n</mi><mrow id="S4.E24.11.11.11.1.3"><mo id="S4.E24.11.11.11.1.3.1" stretchy="false">(</mo><mi id="S4.E24.11.11.11.1.1" xref="S4.E24.11.11.11.1.1.cmml">k</mi><mo id="S4.E24.11.11.11.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E24.25.25.25.1.1.1.1.1.1.2" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E24.25.25.25.1.1.1.1.1.1.1.1"><mo id="S4.E24.12.12.12" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E24.25.25.25.1.1.1.1.1.1.1.1.1"><mover accent="true" id="S4.E24.13.13.13" xref="S4.E24.13.13.13.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E24.13.13.13.2" mathvariant="bold-italic" xref="S4.E24.13.13.13.2.cmml">ϵ</mi><mo id="S4.E24.13.13.13.1" xref="S4.E24.13.13.13.1.cmml">¯</mo></mover><mi id="S4.E24.14.14.14.1" xref="S4.E24.14.14.14.1.cmml">n</mi></msub><mo id="S4.E24.15.15.15" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E24.16.16.16" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E24.25.25.25.1.1.1.2.2.2"><mi id="S4.E24.17.17.17" xref="S4.E24.17.17.17.cmml">𝚲</mi><mi id="S4.E24.18.18.18.1" xref="S4.E24.18.18.18.1.cmml">Z</mi></msub><mo id="S4.E24.19.19.19" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><msubsup id="S4.E24.25.25.25.1.1.1.3.3.3"><mi id="S4.E24.20.20.20" xref="S4.E24.20.20.20.cmml">𝒁</mi><mi id="S4.E24.21.21.21.1" xref="S4.E24.21.21.21.1.cmml">n</mi><mrow id="S4.E24.22.22.22.1.3"><mo id="S4.E24.22.22.22.1.3.1" stretchy="false">(</mo><mi id="S4.E24.22.22.22.1.1" xref="S4.E24.22.22.22.1.1.cmml">k</mi><mo id="S4.E24.22.22.22.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E24.23.23.23" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.E24.24.24.24" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E27.m1.1.1.2c"><mtd id="S4.E27.m1.1.1.2d"><mrow id="S4.E25.28.28"><mo id="S4.E25.28.28a" xref="S4.E27.m1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E25.28.28.28"><msup id="S4.E25.25.25.25.1"><mrow id="S4.E25.25.25.25.1.1.1"><mo id="S4.E25.2.2.2" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><msup id="S4.E25.25.25.25.1.1.1.1"><mover accent="true" id="S4.E25.3.3.3" xref="S4.E25.3.3.3.cmml"><mi id="S4.E25.3.3.3.2" xref="S4.E25.3.3.3.2.cmml">α</mi><mo id="S4.E25.3.3.3.1" xref="S4.E25.3.3.3.1.cmml">^</mo></mover><mrow id="S4.E25.4.4.4.1.3"><mo id="S4.E25.4.4.4.1.3.1" stretchy="false">(</mo><mi id="S4.E25.4.4.4.1.1" xref="S4.E25.4.4.4.1.1.cmml">k</mi><mo id="S4.E25.4.4.4.1.3.2" stretchy="false">)</mo></mrow></msup><mo id="S4.E25.5.5.5" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo></mrow><mn id="S4.E25.6.6.6.1" xref="S4.E25.6.6.6.1.cmml">2</mn></msup><mo id="S4.E25.28.28.28.5" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E25.28.28.28.4.3"><mo id="S4.E25.7.7.7" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E25.26.26.26.2.1.1"><msubsup id="S4.E25.26.26.26.2.1.1.2"><mover accent="true" id="S4.E25.8.8.8" xref="S4.E25.8.8.8.cmml"><mi id="S4.E25.8.8.8.2" xref="S4.E25.8.8.8.2.cmml">𝑺</mi><mo id="S4.E25.8.8.8.1" xref="S4.E25.8.8.8.1.cmml">~</mo></mover><mi id="S4.E25.9.9.9.1" xref="S4.E25.9.9.9.1.cmml">n</mi><mrow id="S4.E25.10.10.10.1.3"><mo id="S4.E25.10.10.10.1.3.1" stretchy="false">(</mo><mi id="S4.E25.10.10.10.1.1" xref="S4.E25.10.10.10.1.1.cmml">k</mi><mo id="S4.E25.10.10.10.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E25.26.26.26.2.1.1.1" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E25.26.26.26.2.1.1.3"><mo id="S4.E25.11.11.11" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><mover accent="true" id="S4.E25.12.12.12" xref="S4.E25.12.12.12.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E25.12.12.12.2" mathvariant="bold-italic" xref="S4.E25.12.12.12.2.cmml">ϵ</mi><mo id="S4.E25.12.12.12.1" xref="S4.E25.12.12.12.1.cmml">¯</mo></mover><mo id="S4.E25.13.13.13" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E25.14.14.14" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E25.27.27.27.3.2.2"><mi id="S4.E25.15.15.15" xref="S4.E25.15.15.15.cmml">𝚲</mi><mi id="S4.E25.16.16.16.1" xref="S4.E25.16.16.16.1.cmml">Z</mi></msub><mo id="S4.E25.17.17.17" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E25.28.28.28.4.3.3"><msubsup id="S4.E25.28.28.28.4.3.3.2"><mover accent="true" id="S4.E25.18.18.18" xref="S4.E25.18.18.18.cmml"><mi id="S4.E25.18.18.18.2" xref="S4.E25.18.18.18.2.cmml">𝑺</mi><mo id="S4.E25.18.18.18.1" xref="S4.E25.18.18.18.1.cmml">~</mo></mover><mi id="S4.E25.19.19.19.1" xref="S4.E25.19.19.19.1.cmml">n</mi><mrow id="S4.E25.20.20.20.1.3"><mo id="S4.E25.20.20.20.1.3.1" stretchy="false">(</mo><mi id="S4.E25.20.20.20.1.1" xref="S4.E25.20.20.20.1.1.cmml">k</mi><mo id="S4.E25.20.20.20.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E25.28.28.28.4.3.3.1" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E25.28.28.28.4.3.3.3"><mo id="S4.E25.21.21.21" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><mover accent="true" id="S4.E25.22.22.22" xref="S4.E25.22.22.22.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E25.22.22.22.2" mathvariant="bold-italic" xref="S4.E25.22.22.22.2.cmml">ϵ</mi><mo id="S4.E25.22.22.22.1" xref="S4.E25.22.22.22.1.cmml">¯</mo></mover><mo id="S4.E25.23.23.23" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E25.24.24.24" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">⟩</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E27.m1.1.1.2e"><mtd class="ltx_align_right" columnalign="right" id="S4.E27.m1.1.1.2f"><mrow id="S4.E26.41.41.41"><mrow id="S4.E26.41.41.41.1"><mrow id="S4.E26.41.41.41.1.2"><mo id="S4.E26.41.41.41.1.2a" xref="S4.E27.m1.1.1.1.1.1.cmml">+</mo><mrow id="S4.E26.41.41.41.1.2.2"><msup id="S4.E26.41.41.41.1.1.1.1"><mrow id="S4.E26.41.41.41.1.1.1.1.1.1"><mo id="S4.E26.2.2.2" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><msup id="S4.E26.41.41.41.1.1.1.1.1.1.1"><mover accent="true" id="S4.E26.3.3.3" xref="S4.E26.3.3.3.cmml"><mi id="S4.E26.3.3.3.2" xref="S4.E26.3.3.3.2.cmml">α</mi><mo id="S4.E26.3.3.3.1" xref="S4.E26.3.3.3.1.cmml">^</mo></mover><mrow id="S4.E26.4.4.4.1.3"><mo id="S4.E26.4.4.4.1.3.1" stretchy="false">(</mo><mi id="S4.E26.4.4.4.1.1" xref="S4.E26.4.4.4.1.1.cmml">k</mi><mo id="S4.E26.4.4.4.1.3.2" stretchy="false">)</mo></mrow></msup><mo id="S4.E26.5.5.5" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo></mrow><mn id="S4.E26.6.6.6.1" xref="S4.E26.6.6.6.1.cmml">2</mn></msup><mo id="S4.E26.41.41.41.1.2.2.3" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mpadded width="1.032em"><mtext id="S4.E26.7.7.7" xref="S4.E26.7.7.7a.cmml">Tr</mtext></mpadded><mo id="S4.E26.41.41.41.1.2.2.3a" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1"><mo id="S4.E26.8.8.8" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1.1"><mover accent="true" id="S4.E26.9.9.9" xref="S4.E26.9.9.9.cmml"><mover accent="true" id="S4.E26.9.9.9.2" xref="S4.E26.9.9.9.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E26.9.9.9.2.2" mathvariant="bold-italic" xref="S4.E26.9.9.9.2.2.cmml">ϵ</mi><mo id="S4.E26.9.9.9.2.1" xref="S4.E26.9.9.9.2.1.cmml">¯</mo></mover><mo id="S4.E26.9.9.9.1" xref="S4.E26.9.9.9.1.cmml">¯</mo></mover><mo id="S4.E26.41.41.41.1.2.2.2.1.1.4" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.3.3"><mo id="S4.E26.10.10.10" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">⟨</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.1.1.1"><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.1.1.1.2"><msub id="S4.E26.41.41.41.1.2.2.2.1.1.1.1.1.2.1"><mo id="S4.E26.11.11.11" rspace="0.167em" xref="S4.E26.11.11.11.cmml">∇</mo><mi id="S4.E26.12.12.12.1" xref="S4.E26.12.12.12.1.cmml">ϕ</mi></msub><msubsup id="S4.E26.41.41.41.1.2.2.2.1.1.1.1.1.2.2"><mover accent="true" id="S4.E26.13.13.13" xref="S4.E26.13.13.13.cmml"><mi id="S4.E26.13.13.13.2" xref="S4.E26.13.13.13.2.cmml">𝑺</mi><mo id="S4.E26.13.13.13.1" xref="S4.E26.13.13.13.1.cmml">~</mo></mover><mi id="S4.E26.14.14.14.1" xref="S4.E26.14.14.14.1.cmml">n</mi><mrow id="S4.E26.15.15.15.1.3"><mo id="S4.E26.15.15.15.1.3.1" stretchy="false">(</mo><mi id="S4.E26.15.15.15.1.1" xref="S4.E26.15.15.15.1.1.cmml">k</mi><mo id="S4.E26.15.15.15.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E26.41.41.41.1.2.2.2.1.1.1.1.1.1" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.1.1.1.3"><mo id="S4.E26.16.16.16" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><mover accent="true" id="S4.E26.17.17.17" xref="S4.E26.17.17.17.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E26.17.17.17.2" mathvariant="bold-italic" xref="S4.E26.17.17.17.2.cmml">ϵ</mi><mo id="S4.E26.17.17.17.1" xref="S4.E26.17.17.17.1.cmml">¯</mo></mover><mo id="S4.E26.18.18.18" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E26.19.19.19" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><msub id="S4.E26.41.41.41.1.2.2.2.1.1.2.2.2"><mi id="S4.E26.20.20.20" xref="S4.E26.20.20.20.cmml">𝚲</mi><mi id="S4.E26.21.21.21.1" xref="S4.E26.21.21.21.1.cmml">Z</mi></msub><mo id="S4.E26.22.22.22" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">|</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.3.3.3"><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.3.3.3.2"><msub id="S4.E26.41.41.41.1.2.2.2.1.1.3.3.3.2.1"><mo id="S4.E26.23.23.23" rspace="0.167em" xref="S4.E26.23.23.23.cmml">∇</mo><mi id="S4.E26.24.24.24.1" xref="S4.E26.24.24.24.1.cmml">ϕ</mi></msub><msubsup id="S4.E26.41.41.41.1.2.2.2.1.1.3.3.3.2.2"><mover accent="true" id="S4.E26.25.25.25" xref="S4.E26.25.25.25.cmml"><mi id="S4.E26.25.25.25.2" xref="S4.E26.25.25.25.2.cmml">𝑺</mi><mo id="S4.E26.25.25.25.1" xref="S4.E26.25.25.25.1.cmml">~</mo></mover><mi id="S4.E26.26.26.26.1" xref="S4.E26.26.26.26.1.cmml">n</mi><mrow id="S4.E26.27.27.27.1.3"><mo id="S4.E26.27.27.27.1.3.1" stretchy="false">(</mo><mi id="S4.E26.27.27.27.1.1" xref="S4.E26.27.27.27.1.1.cmml">k</mi><mo id="S4.E26.27.27.27.1.3.2" stretchy="false">)</mo></mrow></msubsup></mrow><mo id="S4.E26.41.41.41.1.2.2.2.1.1.3.3.3.1" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E26.41.41.41.1.2.2.2.1.1.3.3.3.3"><mo id="S4.E26.28.28.28" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><mover accent="true" id="S4.E26.29.29.29" xref="S4.E26.29.29.29.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E26.29.29.29.2" mathvariant="bold-italic" xref="S4.E26.29.29.29.2.cmml">ϵ</mi><mo id="S4.E26.29.29.29.1" xref="S4.E26.29.29.29.1.cmml">¯</mo></mover><mo id="S4.E26.30.30.30" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mpadded width="0.219em"><mo id="S4.E26.31.31.31" maxsize="120%" minsize="120%" xref="S4.E27.m1.1.1.1.1.1.cmml">⟩</mo></mpadded></mrow></mrow><mo id="S4.E26.32.32.32" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E26.33.33.33" xref="S4.E26.33.33.33.cmml">−</mo><mrow id="S4.E26.41.41.41.1.3"><mi id="S4.E26.34.34.34" xref="S4.E26.34.34.34.cmml">ζ</mi><mo id="S4.E26.41.41.41.1.3.1" xref="S4.E27.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E26.41.41.41.1.3.2"><mo id="S4.E26.35.35.35" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">(</mo><mover accent="true" id="S4.E26.36.36.36" xref="S4.E26.36.36.36.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E26.36.36.36.2" mathvariant="bold-italic" xref="S4.E26.36.36.36.2.cmml">ϵ</mi><mo id="S4.E26.36.36.36.1" xref="S4.E26.36.36.36.1.cmml">¯</mo></mover><mo id="S4.E26.37.37.37" xref="S4.E27.m1.1.1.1.1.1.cmml">,</mo><mover accent="true" id="S4.E26.38.38.38" xref="S4.E26.38.38.38.cmml"><mover accent="true" id="S4.E26.38.38.38.2" xref="S4.E26.38.38.38.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E26.38.38.38.2.2" mathvariant="bold-italic" xref="S4.E26.38.38.38.2.2.cmml">ϵ</mi><mo id="S4.E26.38.38.38.2.1" xref="S4.E26.38.38.38.2.1.cmml">¯</mo></mover><mo id="S4.E26.38.38.38.1" xref="S4.E26.38.38.38.1.cmml">¯</mo></mover><mo id="S4.E26.39.39.39" stretchy="false" xref="S4.E27.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E26.40.40.40" lspace="0em" xref="S4.E27.m1.1.1.1.1.1.cmml">.</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E27.m1.1b"><apply id="S4.E27.m1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="latexml" id="S4.E24.3.3.3.cmml" xref="S4.E24.3.3.3">proportional-to</csymbol><apply id="S4.E27.m1.1.1.1.1.1.9.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.9.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E24.1.1.1.cmml" xref="S4.E24.1.1.1">𝐷</ci><apply id="S4.E24.2.2.2.1.cmml" xref="S4.E24.2.2.2.1"><times id="S4.E24.2.2.2.1.1.cmml" xref="S4.E24.2.2.2.1.1"></times><ci id="S4.E24.2.2.2.1.2.cmml" xref="S4.E24.2.2.2.1.2">𝐾</ci><ci id="S4.E24.2.2.2.1.3.cmml" xref="S4.E24.2.2.2.1.3">𝐿</ci></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.7.cmml" xref="S4.E24.25.25.25a"><minus id="S4.E26.33.33.33.cmml" xref="S4.E26.33.33.33"></minus><apply id="S4.E27.m1.1.1.1.1.1.7.7.cmml" xref="S4.E24.25.25.25a"><plus id="S4.E25.1.1.1.cmml" xref="S4.E24.25.25.25a"></plus><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><minus id="S4.E24.4.4.4.cmml" xref="S4.E24.25.25.25a"></minus><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E24.25.25.25a"><abs id="S4.E27.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E24.25.25.25a"></abs><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.5.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.5.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E24.6.6.6.cmml" xref="S4.E24.6.6.6"><ci id="S4.E24.6.6.6.1.cmml" xref="S4.E24.6.6.6.1">^</ci><ci id="S4.E24.6.6.6.2.cmml" xref="S4.E24.6.6.6.2">𝛼</ci></apply><ci id="S4.E24.7.7.7.1.1.cmml" xref="S4.E24.7.7.7.1.1">𝑘</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.3.4.cmml" xref="S4.E24.25.25.25a"><csymbol cd="latexml" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.3.4.1.cmml" xref="S4.E24.25.25.25a">quantum-operator-product</csymbol><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><apply id="S4.E24.9.9.9.cmml" xref="S4.E24.9.9.9"><ci id="S4.E24.9.9.9.1.cmml" xref="S4.E24.9.9.9.1">~</ci><ci id="S4.E24.9.9.9.2.cmml" xref="S4.E24.9.9.9.2">𝑺</ci></apply><ci id="S4.E24.10.10.10.1.cmml" xref="S4.E24.10.10.10.1">𝑛</ci></apply><ci id="S4.E24.11.11.11.1.1.cmml" xref="S4.E24.11.11.11.1.1">𝑘</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><apply id="S4.E24.13.13.13.cmml" xref="S4.E24.13.13.13"><ci id="S4.E24.13.13.13.1.cmml" xref="S4.E24.13.13.13.1">¯</ci><ci id="S4.E24.13.13.13.2.cmml" xref="S4.E24.13.13.13.2">bold-italic-ϵ</ci></apply><ci id="S4.E24.14.14.14.1.cmml" xref="S4.E24.14.14.14.1">𝑛</ci></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E24.17.17.17.cmml" xref="S4.E24.17.17.17">𝚲</ci><ci id="S4.E24.18.18.18.1.cmml" xref="S4.E24.18.18.18.1">𝑍</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.3.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.3.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.3.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E24.20.20.20.cmml" xref="S4.E24.20.20.20">𝒁</ci><ci id="S4.E24.21.21.21.1.cmml" xref="S4.E24.21.21.21.1">𝑛</ci></apply><ci id="S4.E24.22.22.22.1.1.cmml" xref="S4.E24.22.22.22.1.1">𝑘</ci></apply></apply></apply></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.5.5.5.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.5.5.5.5.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.2.2.2.1.2.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.2.2.2.1.1.2.cmml" xref="S4.E24.25.25.25a"><abs id="S4.E27.m1.1.1.1.1.1.2.2.2.1.1.2.1.cmml" xref="S4.E24.25.25.25a"></abs><apply id="S4.E27.m1.1.1.1.1.1.2.2.2.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.2.2.2.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E25.3.3.3.cmml" xref="S4.E25.3.3.3"><ci id="S4.E25.3.3.3.1.cmml" xref="S4.E25.3.3.3.1">^</ci><ci id="S4.E25.3.3.3.2.cmml" xref="S4.E25.3.3.3.2">𝛼</ci></apply><ci id="S4.E25.4.4.4.1.1.cmml" xref="S4.E25.4.4.4.1.1">𝑘</ci></apply></apply><cn id="S4.E25.6.6.6.1.cmml" type="integer" xref="S4.E25.6.6.6.1">2</cn></apply><apply id="S4.E27.m1.1.1.1.1.1.5.5.5.4.4.cmml" xref="S4.E24.25.25.25a"><csymbol cd="latexml" id="S4.E27.m1.1.1.1.1.1.5.5.5.4.4.1.cmml" xref="S4.E24.25.25.25a">quantum-operator-product</csymbol><apply id="S4.E27.m1.1.1.1.1.1.3.3.3.2.1.1.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.3.3.3.2.1.1.1.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.3.3.3.2.1.1.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.3.3.3.2.1.1.2.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.3.3.3.2.1.1.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.3.3.3.2.1.1.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><apply id="S4.E25.8.8.8.cmml" xref="S4.E25.8.8.8"><ci id="S4.E25.8.8.8.1.cmml" xref="S4.E25.8.8.8.1">~</ci><ci id="S4.E25.8.8.8.2.cmml" xref="S4.E25.8.8.8.2">𝑺</ci></apply><ci id="S4.E25.9.9.9.1.cmml" xref="S4.E25.9.9.9.1">𝑛</ci></apply><ci id="S4.E25.10.10.10.1.1.cmml" xref="S4.E25.10.10.10.1.1">𝑘</ci></apply><apply id="S4.E25.12.12.12.cmml" xref="S4.E25.12.12.12"><ci id="S4.E25.12.12.12.1.cmml" xref="S4.E25.12.12.12.1">¯</ci><ci id="S4.E25.12.12.12.2.cmml" xref="S4.E25.12.12.12.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.4.4.4.3.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.4.4.4.3.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E25.15.15.15.cmml" xref="S4.E25.15.15.15">𝚲</ci><ci id="S4.E25.16.16.16.1.cmml" xref="S4.E25.16.16.16.1">𝑍</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.5.5.5.4.3.3.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.5.5.5.4.3.3.1.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.5.5.5.4.3.3.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.5.5.5.4.3.3.2.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.5.5.5.4.3.3.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.5.5.5.4.3.3.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><apply id="S4.E25.18.18.18.cmml" xref="S4.E25.18.18.18"><ci id="S4.E25.18.18.18.1.cmml" xref="S4.E25.18.18.18.1">~</ci><ci id="S4.E25.18.18.18.2.cmml" xref="S4.E25.18.18.18.2">𝑺</ci></apply><ci id="S4.E25.19.19.19.1.cmml" xref="S4.E25.19.19.19.1">𝑛</ci></apply><ci id="S4.E25.20.20.20.1.1.cmml" xref="S4.E25.20.20.20.1.1">𝑘</ci></apply><apply id="S4.E25.22.22.22.cmml" xref="S4.E25.22.22.22"><ci id="S4.E25.22.22.22.1.cmml" xref="S4.E25.22.22.22.1">¯</ci><ci id="S4.E25.22.22.22.2.cmml" xref="S4.E25.22.22.22.2">bold-italic-ϵ</ci></apply></apply></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.7.7.7.3.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.6.6.6.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.6.6.6.1.2.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.6.6.6.1.1.2.cmml" xref="S4.E24.25.25.25a"><abs id="S4.E27.m1.1.1.1.1.1.6.6.6.1.1.2.1.cmml" xref="S4.E24.25.25.25a"></abs><apply id="S4.E27.m1.1.1.1.1.1.6.6.6.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.6.6.6.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E26.3.3.3.cmml" xref="S4.E26.3.3.3"><ci id="S4.E26.3.3.3.1.cmml" xref="S4.E26.3.3.3.1">^</ci><ci id="S4.E26.3.3.3.2.cmml" xref="S4.E26.3.3.3.2">𝛼</ci></apply><ci id="S4.E26.4.4.4.1.1.cmml" xref="S4.E26.4.4.4.1.1">𝑘</ci></apply></apply><cn id="S4.E26.6.6.6.1.cmml" type="integer" xref="S4.E26.6.6.6.1">2</cn></apply><ci id="S4.E26.7.7.7a.cmml" xref="S4.E26.7.7.7"><mtext id="S4.E26.7.7.7.cmml" xref="S4.E26.7.7.7">Tr</mtext></ci><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.4.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E26.9.9.9.cmml" xref="S4.E26.9.9.9"><ci id="S4.E26.9.9.9.1.cmml" xref="S4.E26.9.9.9.1">¯</ci><apply id="S4.E26.9.9.9.2.cmml" xref="S4.E26.9.9.9.2"><ci id="S4.E26.9.9.9.2.1.cmml" xref="S4.E26.9.9.9.2.1">¯</ci><ci id="S4.E26.9.9.9.2.2.cmml" xref="S4.E26.9.9.9.2.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.4.cmml" xref="S4.E24.25.25.25a"><csymbol cd="latexml" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.4.1.cmml" xref="S4.E24.25.25.25a">quantum-operator-product</csymbol><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.1.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.cmml" xref="S4.E24.25.25.25a"><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.1.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E26.11.11.11.cmml" xref="S4.E26.11.11.11">∇</ci><ci id="S4.E26.12.12.12.1.cmml" xref="S4.E26.12.12.12.1">italic-ϕ</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.2.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><apply id="S4.E26.13.13.13.cmml" xref="S4.E26.13.13.13"><ci id="S4.E26.13.13.13.1.cmml" xref="S4.E26.13.13.13.1">~</ci><ci id="S4.E26.13.13.13.2.cmml" xref="S4.E26.13.13.13.2">𝑺</ci></apply><ci id="S4.E26.14.14.14.1.cmml" xref="S4.E26.14.14.14.1">𝑛</ci></apply><ci id="S4.E26.15.15.15.1.1.cmml" xref="S4.E26.15.15.15.1.1">𝑘</ci></apply></apply><apply id="S4.E26.17.17.17.cmml" xref="S4.E26.17.17.17"><ci id="S4.E26.17.17.17.1.cmml" xref="S4.E26.17.17.17.1">¯</ci><ci id="S4.E26.17.17.17.2.cmml" xref="S4.E26.17.17.17.2">bold-italic-ϵ</ci></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.2.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.2.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E26.20.20.20.cmml" xref="S4.E26.20.20.20">𝚲</ci><ci id="S4.E26.21.21.21.1.cmml" xref="S4.E26.21.21.21.1">𝑍</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.1.cmml" xref="S4.E24.25.25.25a"></times><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.cmml" xref="S4.E24.25.25.25a"><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.1.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.1.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><ci id="S4.E26.23.23.23.cmml" xref="S4.E26.23.23.23">∇</ci><ci id="S4.E26.24.24.24.1.cmml" xref="S4.E26.24.24.24.1">italic-ϕ</ci></apply><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.2.1.cmml" xref="S4.E24.25.25.25a">superscript</csymbol><apply id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.2.2.cmml" xref="S4.E24.25.25.25a"><csymbol cd="ambiguous" id="S4.E27.m1.1.1.1.1.1.7.7.7.2.1.1.3.3.3.2.2.2.1.cmml" xref="S4.E24.25.25.25a">subscript</csymbol><apply id="S4.E26.25.25.25.cmml" xref="S4.E26.25.25.25"><ci id="S4.E26.25.25.25.1.cmml" xref="S4.E26.25.25.25.1">~</ci><ci id="S4.E26.25.25.25.2.cmml" xref="S4.E26.25.25.25.2">𝑺</ci></apply><ci id="S4.E26.26.26.26.1.cmml" xref="S4.E26.26.26.26.1">𝑛</ci></apply><ci id="S4.E26.27.27.27.1.1.cmml" xref="S4.E26.27.27.27.1.1">𝑘</ci></apply></apply><apply id="S4.E26.29.29.29.cmml" xref="S4.E26.29.29.29"><ci id="S4.E26.29.29.29.1.cmml" xref="S4.E26.29.29.29.1">¯</ci><ci id="S4.E26.29.29.29.2.cmml" xref="S4.E26.29.29.29.2">bold-italic-ϵ</ci></apply></apply></apply></apply></apply></apply><apply id="S4.E27.m1.1.1.1.1.1.7.9.cmml" xref="S4.E24.25.25.25a"><times id="S4.E27.m1.1.1.1.1.1.7.9.1.cmml" xref="S4.E24.25.25.25a"></times><ci id="S4.E26.34.34.34.cmml" xref="S4.E26.34.34.34">𝜁</ci><interval closure="open" id="S4.E27.m1.1.1.1.1.1.7.9.3.cmml" xref="S4.E24.25.25.25a"><apply id="S4.E26.36.36.36.cmml" xref="S4.E26.36.36.36"><ci id="S4.E26.36.36.36.1.cmml" xref="S4.E26.36.36.36.1">¯</ci><ci id="S4.E26.36.36.36.2.cmml" xref="S4.E26.36.36.36.2">bold-italic-ϵ</ci></apply><apply id="S4.E26.38.38.38.cmml" xref="S4.E26.38.38.38"><ci id="S4.E26.38.38.38.1.cmml" xref="S4.E26.38.38.38.1">¯</ci><apply id="S4.E26.38.38.38.2.cmml" xref="S4.E26.38.38.38.2"><ci id="S4.E26.38.38.38.2.1.cmml" xref="S4.E26.38.38.38.2.1">¯</ci><ci id="S4.E26.38.38.38.2.2.cmml" xref="S4.E26.38.38.38.2.2">bold-italic-ϵ</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E27.m1.1c">\displaystyle\begin{multlined}D_{KL}\propto-\left|\hat{\alpha}^{(k)}\big{% \langle}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}% _{Z}\big{|}\bm{Z}_{n}^{(k)}\big{\rangle}\right|\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\big{\langle}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{|}\bm{\Lambda}_{Z}\big{|}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{\rangle}\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\text{Tr}\!\left(\bar{\bar{\bm{\epsilon}}% }\big{\langle}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}})\big% {|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{% \epsilon}})\big{\rangle}\!\right)-\zeta(\bar{\bm{\epsilon}},\bar{\bar{\bm{% \epsilon}}}).\end{multlined}D_{KL}\propto-\left|\hat{\alpha}^{(k)}\big{\langle% }\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}}_{n})\big{|}\bm{\Lambda}_{Z}% \big{|}\bm{Z}_{n}^{(k)}\big{\rangle}\right|\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\big{\langle}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{|}\bm{\Lambda}_{Z}\big{|}\tilde{\bm{S}}_{n}^{(k)}\!(% \bar{\bm{\epsilon}})\big{\rangle}\\ +\big{|}\hat{\alpha}^{(k)}\big{|}^{2}\text{Tr}\!\left(\bar{\bar{\bm{\epsilon}}% }\big{\langle}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{\epsilon}})\big% {|}\bm{\Lambda}_{Z}\big{|}\nabla_{\phi}\tilde{\bm{S}}_{n}^{(k)}\!(\bar{\bm{% \epsilon}})\big{\rangle}\!\right)-\zeta(\bar{\bm{\epsilon}},\bar{\bar{\bm{% \epsilon}}}).</annotation><annotation encoding="application/x-llamapun" id="S4.E27.m1.1d">start_ROW start_CELL italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ∝ - | over^ start_ARG italic_α end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ⟨ over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ⟩ | end_CELL end_ROW start_ROW start_CELL + | over^ start_ARG italic_α end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟨ over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG ) ⟩ end_CELL end_ROW start_ROW start_CELL + | over^ start_ARG italic_α end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT Tr ( over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG ⟨ ∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG ) | bold_Λ start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT | ∇ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT over~ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϵ end_ARG ) ⟩ ) - italic_ζ ( over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG ) . 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">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p3.6">Noting that <math alttext="\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}" class="ltx_Math" display="inline" id="S4.p3.5.m1.2"><semantics id="S4.p3.5.m1.2a"><mrow id="S4.p3.5.m1.2.2" xref="S4.p3.5.m1.2.2.cmml"><msub id="S4.p3.5.m1.2.2.4" xref="S4.p3.5.m1.2.2.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p3.5.m1.2.2.4.2" mathvariant="bold-italic" xref="S4.p3.5.m1.2.2.4.2.cmml">ϕ</mi><mi id="S4.p3.5.m1.2.2.4.3" xref="S4.p3.5.m1.2.2.4.3.cmml">n</mi></msub><mo fence="false" id="S4.p3.5.m1.2.2.3" xref="S4.p3.5.m1.2.2.3.cmml">|</mo><mrow id="S4.p3.5.m1.2.2.2.2" xref="S4.p3.5.m1.2.2.2.3.cmml"><msub id="S4.p3.5.m1.1.1.1.1.1" xref="S4.p3.5.m1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p3.5.m1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.p3.5.m1.1.1.1.1.1.2.cmml">ϕ</mi><mrow id="S4.p3.5.m1.1.1.1.1.1.3" xref="S4.p3.5.m1.1.1.1.1.1.3.cmml"><mi id="S4.p3.5.m1.1.1.1.1.1.3.2" xref="S4.p3.5.m1.1.1.1.1.1.3.2.cmml">n</mi><mo id="S4.p3.5.m1.1.1.1.1.1.3.1" xref="S4.p3.5.m1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S4.p3.5.m1.1.1.1.1.1.3.3" xref="S4.p3.5.m1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.p3.5.m1.2.2.2.2.3" xref="S4.p3.5.m1.2.2.2.3.cmml">,</mo><msub id="S4.p3.5.m1.2.2.2.2.2" xref="S4.p3.5.m1.2.2.2.2.2.cmml"><mi id="S4.p3.5.m1.2.2.2.2.2.2" xref="S4.p3.5.m1.2.2.2.2.2.2.cmml">𝚲</mi><mi id="S4.p3.5.m1.2.2.2.2.2.3" xref="S4.p3.5.m1.2.2.2.2.2.3.cmml">a</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.5.m1.2b"><apply id="S4.p3.5.m1.2.2.cmml" xref="S4.p3.5.m1.2.2"><csymbol cd="latexml" id="S4.p3.5.m1.2.2.3.cmml" xref="S4.p3.5.m1.2.2.3">conditional</csymbol><apply id="S4.p3.5.m1.2.2.4.cmml" xref="S4.p3.5.m1.2.2.4"><csymbol cd="ambiguous" id="S4.p3.5.m1.2.2.4.1.cmml" xref="S4.p3.5.m1.2.2.4">subscript</csymbol><ci id="S4.p3.5.m1.2.2.4.2.cmml" xref="S4.p3.5.m1.2.2.4.2">bold-italic-ϕ</ci><ci id="S4.p3.5.m1.2.2.4.3.cmml" xref="S4.p3.5.m1.2.2.4.3">𝑛</ci></apply><list id="S4.p3.5.m1.2.2.2.3.cmml" xref="S4.p3.5.m1.2.2.2.2"><apply id="S4.p3.5.m1.1.1.1.1.1.cmml" xref="S4.p3.5.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p3.5.m1.1.1.1.1.1.1.cmml" xref="S4.p3.5.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.p3.5.m1.1.1.1.1.1.2.cmml" xref="S4.p3.5.m1.1.1.1.1.1.2">bold-italic-ϕ</ci><apply id="S4.p3.5.m1.1.1.1.1.1.3.cmml" xref="S4.p3.5.m1.1.1.1.1.1.3"><minus id="S4.p3.5.m1.1.1.1.1.1.3.1.cmml" xref="S4.p3.5.m1.1.1.1.1.1.3.1"></minus><ci id="S4.p3.5.m1.1.1.1.1.1.3.2.cmml" xref="S4.p3.5.m1.1.1.1.1.1.3.2">𝑛</ci><cn id="S4.p3.5.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.p3.5.m1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.p3.5.m1.2.2.2.2.2.cmml" xref="S4.p3.5.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.p3.5.m1.2.2.2.2.2.1.cmml" xref="S4.p3.5.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.p3.5.m1.2.2.2.2.2.2.cmml" xref="S4.p3.5.m1.2.2.2.2.2.2">𝚲</ci><ci id="S4.p3.5.m1.2.2.2.2.2.3.cmml" xref="S4.p3.5.m1.2.2.2.2.2.3">𝑎</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m1.2c">\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m1.2d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is Gaussian, the message <math alttext="\epsilon^{(\bm{\phi}_{n-1}\to\bm{\phi}_{n})}" class="ltx_Math" display="inline" id="S4.p3.6.m2.1"><semantics id="S4.p3.6.m2.1a"><msup id="S4.p3.6.m2.1.2" xref="S4.p3.6.m2.1.2.cmml"><mi id="S4.p3.6.m2.1.2.2" xref="S4.p3.6.m2.1.2.2.cmml">ϵ</mi><mrow id="S4.p3.6.m2.1.1.1.1" xref="S4.p3.6.m2.1.1.1.1.1.cmml"><mo id="S4.p3.6.m2.1.1.1.1.2" stretchy="false" xref="S4.p3.6.m2.1.1.1.1.1.cmml">(</mo><mrow id="S4.p3.6.m2.1.1.1.1.1" xref="S4.p3.6.m2.1.1.1.1.1.cmml"><msub id="S4.p3.6.m2.1.1.1.1.1.2" xref="S4.p3.6.m2.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p3.6.m2.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S4.p3.6.m2.1.1.1.1.1.2.2.cmml">ϕ</mi><mrow id="S4.p3.6.m2.1.1.1.1.1.2.3" xref="S4.p3.6.m2.1.1.1.1.1.2.3.cmml"><mi id="S4.p3.6.m2.1.1.1.1.1.2.3.2" xref="S4.p3.6.m2.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S4.p3.6.m2.1.1.1.1.1.2.3.1" xref="S4.p3.6.m2.1.1.1.1.1.2.3.1.cmml">−</mo><mn id="S4.p3.6.m2.1.1.1.1.1.2.3.3" xref="S4.p3.6.m2.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.p3.6.m2.1.1.1.1.1.1" stretchy="false" xref="S4.p3.6.m2.1.1.1.1.1.1.cmml">→</mo><msub id="S4.p3.6.m2.1.1.1.1.1.3" xref="S4.p3.6.m2.1.1.1.1.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p3.6.m2.1.1.1.1.1.3.2" mathvariant="bold-italic" xref="S4.p3.6.m2.1.1.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.p3.6.m2.1.1.1.1.1.3.3" xref="S4.p3.6.m2.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.p3.6.m2.1.1.1.1.3" stretchy="false" xref="S4.p3.6.m2.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p3.6.m2.1b"><apply id="S4.p3.6.m2.1.2.cmml" xref="S4.p3.6.m2.1.2"><csymbol cd="ambiguous" id="S4.p3.6.m2.1.2.1.cmml" xref="S4.p3.6.m2.1.2">superscript</csymbol><ci id="S4.p3.6.m2.1.2.2.cmml" xref="S4.p3.6.m2.1.2.2">italic-ϵ</ci><apply id="S4.p3.6.m2.1.1.1.1.1.cmml" xref="S4.p3.6.m2.1.1.1.1"><ci id="S4.p3.6.m2.1.1.1.1.1.1.cmml" xref="S4.p3.6.m2.1.1.1.1.1.1">→</ci><apply id="S4.p3.6.m2.1.1.1.1.1.2.cmml" xref="S4.p3.6.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.p3.6.m2.1.1.1.1.1.2.1.cmml" xref="S4.p3.6.m2.1.1.1.1.1.2">subscript</csymbol><ci id="S4.p3.6.m2.1.1.1.1.1.2.2.cmml" xref="S4.p3.6.m2.1.1.1.1.1.2.2">bold-italic-ϕ</ci><apply id="S4.p3.6.m2.1.1.1.1.1.2.3.cmml" xref="S4.p3.6.m2.1.1.1.1.1.2.3"><minus id="S4.p3.6.m2.1.1.1.1.1.2.3.1.cmml" xref="S4.p3.6.m2.1.1.1.1.1.2.3.1"></minus><ci id="S4.p3.6.m2.1.1.1.1.1.2.3.2.cmml" xref="S4.p3.6.m2.1.1.1.1.1.2.3.2">𝑛</ci><cn id="S4.p3.6.m2.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S4.p3.6.m2.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.p3.6.m2.1.1.1.1.1.3.cmml" xref="S4.p3.6.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.p3.6.m2.1.1.1.1.1.3.1.cmml" xref="S4.p3.6.m2.1.1.1.1.1.3">subscript</csymbol><ci id="S4.p3.6.m2.1.1.1.1.1.3.2.cmml" xref="S4.p3.6.m2.1.1.1.1.1.3.2">bold-italic-ϕ</ci><ci id="S4.p3.6.m2.1.1.1.1.1.3.3.cmml" xref="S4.p3.6.m2.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.6.m2.1c">\epsilon^{(\bm{\phi}_{n-1}\to\bm{\phi}_{n})}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.6.m2.1d">italic_ϵ start_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math> can be calculated as</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="\ln\big{(}\epsilon^{(\bm{\phi}_{n-1}\to\bm{\phi}_{n})}\big{)}\propto\\ -\frac{1}{2}\big{\langle}\bm{\phi}_{n}-\bm{T}\overline{\bm{\phi}}_{n-1}\big{|}% \bm{G}^{-\top}\overline{\bm{\Lambda}}_{a}\bm{G}^{-1}\big{|}\bm{\phi}_{n}-\bm{T% }\overline{\bm{\phi}}_{n-1}\big{\rangle}," class="ltx_Math" display="block" id="S4.E28.m1.34"><semantics id="S4.E28.m1.34a"><mtable displaystyle="true" id="S4.E28.m1.34.34.3" rowspacing="0pt"><mtr id="S4.E28.m1.34.34.3a"><mtd class="ltx_align_left" columnalign="left" id="S4.E28.m1.34.34.3b"><mrow id="S4.E28.m1.33.33.2.32.7.7"><mrow id="S4.E28.m1.33.33.2.32.7.7.7.1"><mi id="S4.E28.m1.1.1.1.1.1.1" xref="S4.E28.m1.1.1.1.1.1.1.cmml">ln</mi><mo id="S4.E28.m1.33.33.2.32.7.7.7.1a" xref="S4.E28.m1.32.32.1.1.1.cmml">⁡</mo><mrow id="S4.E28.m1.33.33.2.32.7.7.7.1.1"><mo id="S4.E28.m1.2.2.2.2.2.2" maxsize="120%" minsize="120%" xref="S4.E28.m1.32.32.1.1.1.cmml">(</mo><msup id="S4.E28.m1.33.33.2.32.7.7.7.1.1.1"><mi id="S4.E28.m1.3.3.3.3.3.3" xref="S4.E28.m1.3.3.3.3.3.3.cmml">ϵ</mi><mrow id="S4.E28.m1.4.4.4.4.4.4.1.1" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.cmml"><mo id="S4.E28.m1.4.4.4.4.4.4.1.1.2" stretchy="false" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.cmml">(</mo><mrow id="S4.E28.m1.4.4.4.4.4.4.1.1.1" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.cmml"><msub id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.2" mathvariant="bold-italic" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.2.cmml">ϕ</mi><mrow id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.cmml"><mi id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.2" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.2.cmml">n</mi><mo id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.1" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.1.cmml">−</mo><mn id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.3" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.E28.m1.4.4.4.4.4.4.1.1.1.1" stretchy="false" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.1.cmml">→</mo><msub id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.2" mathvariant="bold-italic" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.3" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.E28.m1.4.4.4.4.4.4.1.1.3" stretchy="false" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.E28.m1.5.5.5.5.5.5" maxsize="120%" minsize="120%" xref="S4.E28.m1.32.32.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E28.m1.6.6.6.6.6.6" xref="S4.E28.m1.6.6.6.6.6.6.cmml">∝</mo><mi id="S4.E28.m1.33.33.2.32.7.7.8" xref="S4.E28.m1.32.32.1.1.1.cmml"></mi></mrow></mtd></mtr><mtr id="S4.E28.m1.34.34.3c"><mtd class="ltx_align_right" columnalign="right" id="S4.E28.m1.34.34.3d"><mrow id="S4.E28.m1.34.34.3.33.26.26.26"><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1"><mo id="S4.E28.m1.34.34.3.33.26.26.26.1a" xref="S4.E28.m1.32.32.1.1.1.cmml">−</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.3"><mfrac id="S4.E28.m1.8.8.8.2.2.2" xref="S4.E28.m1.8.8.8.2.2.2.cmml"><mn id="S4.E28.m1.8.8.8.2.2.2.2" xref="S4.E28.m1.8.8.8.2.2.2.2.cmml">1</mn><mn id="S4.E28.m1.8.8.8.2.2.2.3" xref="S4.E28.m1.8.8.8.2.2.2.3.cmml">2</mn></mfrac><mo id="S4.E28.m1.34.34.3.33.26.26.26.1.3.4" xref="S4.E28.m1.32.32.1.1.1.cmml">⁢</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.3.3.3"><mo id="S4.E28.m1.9.9.9.3.3.3" maxsize="120%" minsize="120%" xref="S4.E28.m1.32.32.1.1.1.cmml">⟨</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.1.1.1.1"><msub id="S4.E28.m1.34.34.3.33.26.26.26.1.1.1.1.1.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E28.m1.10.10.10.4.4.4" mathvariant="bold-italic" xref="S4.E28.m1.10.10.10.4.4.4.cmml">ϕ</mi><mi id="S4.E28.m1.11.11.11.5.5.5.1" xref="S4.E28.m1.11.11.11.5.5.5.1.cmml">n</mi></msub><mo id="S4.E28.m1.12.12.12.6.6.6" xref="S4.E28.m1.12.12.12.6.6.6.cmml">−</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.1.1.1.1.2"><mi id="S4.E28.m1.13.13.13.7.7.7" xref="S4.E28.m1.13.13.13.7.7.7.cmml">𝑻</mi><mo id="S4.E28.m1.34.34.3.33.26.26.26.1.1.1.1.1.2.1" xref="S4.E28.m1.32.32.1.1.1.cmml">⁢</mo><msub id="S4.E28.m1.34.34.3.33.26.26.26.1.1.1.1.1.2.2"><mover accent="true" id="S4.E28.m1.14.14.14.8.8.8" xref="S4.E28.m1.14.14.14.8.8.8.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E28.m1.14.14.14.8.8.8.2" mathvariant="bold-italic" xref="S4.E28.m1.14.14.14.8.8.8.2.cmml">ϕ</mi><mo id="S4.E28.m1.14.14.14.8.8.8.1" xref="S4.E28.m1.14.14.14.8.8.8.1.cmml">¯</mo></mover><mrow id="S4.E28.m1.15.15.15.9.9.9.1" xref="S4.E28.m1.15.15.15.9.9.9.1.cmml"><mi id="S4.E28.m1.15.15.15.9.9.9.1.2" xref="S4.E28.m1.15.15.15.9.9.9.1.2.cmml">n</mi><mo id="S4.E28.m1.15.15.15.9.9.9.1.1" xref="S4.E28.m1.15.15.15.9.9.9.1.1.cmml">−</mo><mn id="S4.E28.m1.15.15.15.9.9.9.1.3" xref="S4.E28.m1.15.15.15.9.9.9.1.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S4.E28.m1.16.16.16.10.10.10" maxsize="120%" minsize="120%" xref="S4.E28.m1.32.32.1.1.1.cmml">|</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.2.2.2.2"><msup id="S4.E28.m1.34.34.3.33.26.26.26.1.2.2.2.2.2"><mi id="S4.E28.m1.17.17.17.11.11.11" xref="S4.E28.m1.17.17.17.11.11.11.cmml">𝑮</mi><mrow id="S4.E28.m1.18.18.18.12.12.12.1" xref="S4.E28.m1.18.18.18.12.12.12.1.cmml"><mi id="S4.E28.m1.18.18.18.12.12.12.1.2" xref="S4.E28.m1.18.18.18.12.12.12.1.2.cmml"></mi><mo id="S4.E28.m1.18.18.18.12.12.12.1.1" rspace="0em" xref="S4.E28.m1.18.18.18.12.12.12.1.1.cmml">−</mo><mo id="S4.E28.m1.18.18.18.12.12.12.1.3" lspace="0em" xref="S4.E28.m1.18.18.18.12.12.12.1.3.cmml">⊤</mo></mrow></msup><mo id="S4.E28.m1.34.34.3.33.26.26.26.1.2.2.2.2.1" xref="S4.E28.m1.32.32.1.1.1.cmml">⁢</mo><msub id="S4.E28.m1.34.34.3.33.26.26.26.1.2.2.2.2.3"><mover accent="true" id="S4.E28.m1.19.19.19.13.13.13" xref="S4.E28.m1.19.19.19.13.13.13.cmml"><mi id="S4.E28.m1.19.19.19.13.13.13.2" xref="S4.E28.m1.19.19.19.13.13.13.2.cmml">𝚲</mi><mo id="S4.E28.m1.19.19.19.13.13.13.1" xref="S4.E28.m1.19.19.19.13.13.13.1.cmml">¯</mo></mover><mi id="S4.E28.m1.20.20.20.14.14.14.1" xref="S4.E28.m1.20.20.20.14.14.14.1.cmml">a</mi></msub><mo id="S4.E28.m1.34.34.3.33.26.26.26.1.2.2.2.2.1a" xref="S4.E28.m1.32.32.1.1.1.cmml">⁢</mo><msup id="S4.E28.m1.34.34.3.33.26.26.26.1.2.2.2.2.4"><mi id="S4.E28.m1.21.21.21.15.15.15" xref="S4.E28.m1.21.21.21.15.15.15.cmml">𝑮</mi><mrow id="S4.E28.m1.22.22.22.16.16.16.1" xref="S4.E28.m1.22.22.22.16.16.16.1.cmml"><mo id="S4.E28.m1.22.22.22.16.16.16.1a" xref="S4.E28.m1.22.22.22.16.16.16.1.cmml">−</mo><mn id="S4.E28.m1.22.22.22.16.16.16.1.2" xref="S4.E28.m1.22.22.22.16.16.16.1.2.cmml">1</mn></mrow></msup></mrow><mo id="S4.E28.m1.23.23.23.17.17.17" maxsize="120%" minsize="120%" xref="S4.E28.m1.32.32.1.1.1.cmml">|</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.3.3.3.3"><msub id="S4.E28.m1.34.34.3.33.26.26.26.1.3.3.3.3.1"><mi class="ltx_mathvariant_bold-italic" id="S4.E28.m1.24.24.24.18.18.18" mathvariant="bold-italic" xref="S4.E28.m1.24.24.24.18.18.18.cmml">ϕ</mi><mi id="S4.E28.m1.25.25.25.19.19.19.1" xref="S4.E28.m1.25.25.25.19.19.19.1.cmml">n</mi></msub><mo id="S4.E28.m1.26.26.26.20.20.20" xref="S4.E28.m1.26.26.26.20.20.20.cmml">−</mo><mrow id="S4.E28.m1.34.34.3.33.26.26.26.1.3.3.3.3.2"><mi id="S4.E28.m1.27.27.27.21.21.21" xref="S4.E28.m1.27.27.27.21.21.21.cmml">𝑻</mi><mo id="S4.E28.m1.34.34.3.33.26.26.26.1.3.3.3.3.2.1" xref="S4.E28.m1.32.32.1.1.1.cmml">⁢</mo><msub id="S4.E28.m1.34.34.3.33.26.26.26.1.3.3.3.3.2.2"><mover accent="true" id="S4.E28.m1.28.28.28.22.22.22" xref="S4.E28.m1.28.28.28.22.22.22.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E28.m1.28.28.28.22.22.22.2" mathvariant="bold-italic" xref="S4.E28.m1.28.28.28.22.22.22.2.cmml">ϕ</mi><mo id="S4.E28.m1.28.28.28.22.22.22.1" xref="S4.E28.m1.28.28.28.22.22.22.1.cmml">¯</mo></mover><mrow id="S4.E28.m1.29.29.29.23.23.23.1" xref="S4.E28.m1.29.29.29.23.23.23.1.cmml"><mi id="S4.E28.m1.29.29.29.23.23.23.1.2" xref="S4.E28.m1.29.29.29.23.23.23.1.2.cmml">n</mi><mo id="S4.E28.m1.29.29.29.23.23.23.1.1" xref="S4.E28.m1.29.29.29.23.23.23.1.1.cmml">−</mo><mn id="S4.E28.m1.29.29.29.23.23.23.1.3" xref="S4.E28.m1.29.29.29.23.23.23.1.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S4.E28.m1.30.30.30.24.24.24" maxsize="120%" minsize="120%" xref="S4.E28.m1.32.32.1.1.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S4.E28.m1.31.31.31.25.25.25" xref="S4.E28.m1.32.32.1.1.1.cmml">,</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E28.m1.34b"><apply id="S4.E28.m1.32.32.1.1.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="latexml" id="S4.E28.m1.6.6.6.6.6.6.cmml" xref="S4.E28.m1.6.6.6.6.6.6">proportional-to</csymbol><apply id="S4.E28.m1.32.32.1.1.1.1.2.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><ln id="S4.E28.m1.1.1.1.1.1.1.cmml" xref="S4.E28.m1.1.1.1.1.1.1"></ln><apply id="S4.E28.m1.32.32.1.1.1.1.1.1.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.1.1.1.1.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">superscript</csymbol><ci id="S4.E28.m1.3.3.3.3.3.3.cmml" xref="S4.E28.m1.3.3.3.3.3.3">italic-ϵ</ci><apply id="S4.E28.m1.4.4.4.4.4.4.1.1.1.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1"><ci id="S4.E28.m1.4.4.4.4.4.4.1.1.1.1.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.1">→</ci><apply id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.1.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2">subscript</csymbol><ci id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.2.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.2">bold-italic-ϕ</ci><apply id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3"><minus id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.1.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.1"></minus><ci id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.2.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.2">𝑛</ci><cn id="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.3.cmml" type="integer" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.1.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3">subscript</csymbol><ci id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.2.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.2">bold-italic-ϕ</ci><ci id="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.3.cmml" xref="S4.E28.m1.4.4.4.4.4.4.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply><apply id="S4.E28.m1.32.32.1.1.1.4.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><minus id="S4.E28.m1.7.7.7.1.1.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"></minus><apply id="S4.E28.m1.32.32.1.1.1.4.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><times id="S4.E28.m1.32.32.1.1.1.4.3.4.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"></times><apply id="S4.E28.m1.8.8.8.2.2.2.cmml" xref="S4.E28.m1.8.8.8.2.2.2"><divide id="S4.E28.m1.8.8.8.2.2.2.1.cmml" xref="S4.E28.m1.8.8.8.2.2.2"></divide><cn id="S4.E28.m1.8.8.8.2.2.2.2.cmml" type="integer" xref="S4.E28.m1.8.8.8.2.2.2.2">1</cn><cn id="S4.E28.m1.8.8.8.2.2.2.3.cmml" type="integer" xref="S4.E28.m1.8.8.8.2.2.2.3">2</cn></apply><apply id="S4.E28.m1.32.32.1.1.1.4.3.3.4.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="latexml" id="S4.E28.m1.32.32.1.1.1.4.3.3.4.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">quantum-operator-product</csymbol><apply id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><minus id="S4.E28.m1.12.12.12.6.6.6.cmml" xref="S4.E28.m1.12.12.12.6.6.6"></minus><apply id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.2.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">subscript</csymbol><ci id="S4.E28.m1.10.10.10.4.4.4.cmml" xref="S4.E28.m1.10.10.10.4.4.4">bold-italic-ϕ</ci><ci id="S4.E28.m1.11.11.11.5.5.5.1.cmml" xref="S4.E28.m1.11.11.11.5.5.5.1">𝑛</ci></apply><apply id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><times id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.3.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"></times><ci id="S4.E28.m1.13.13.13.7.7.7.cmml" xref="S4.E28.m1.13.13.13.7.7.7">𝑻</ci><apply id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.3.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.2.1.1.1.1.3.3.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">subscript</csymbol><apply id="S4.E28.m1.14.14.14.8.8.8.cmml" xref="S4.E28.m1.14.14.14.8.8.8"><ci id="S4.E28.m1.14.14.14.8.8.8.1.cmml" xref="S4.E28.m1.14.14.14.8.8.8.1">¯</ci><ci id="S4.E28.m1.14.14.14.8.8.8.2.cmml" xref="S4.E28.m1.14.14.14.8.8.8.2">bold-italic-ϕ</ci></apply><apply id="S4.E28.m1.15.15.15.9.9.9.1.cmml" xref="S4.E28.m1.15.15.15.9.9.9.1"><minus id="S4.E28.m1.15.15.15.9.9.9.1.1.cmml" xref="S4.E28.m1.15.15.15.9.9.9.1.1"></minus><ci id="S4.E28.m1.15.15.15.9.9.9.1.2.cmml" xref="S4.E28.m1.15.15.15.9.9.9.1.2">𝑛</ci><cn id="S4.E28.m1.15.15.15.9.9.9.1.3.cmml" type="integer" xref="S4.E28.m1.15.15.15.9.9.9.1.3">1</cn></apply></apply></apply></apply><apply id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><times id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"></times><apply id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.2.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.2.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">superscript</csymbol><ci id="S4.E28.m1.17.17.17.11.11.11.cmml" xref="S4.E28.m1.17.17.17.11.11.11">𝑮</ci><apply id="S4.E28.m1.18.18.18.12.12.12.1.cmml" xref="S4.E28.m1.18.18.18.12.12.12.1"><minus id="S4.E28.m1.18.18.18.12.12.12.1.1.cmml" xref="S4.E28.m1.18.18.18.12.12.12.1.1"></minus><csymbol cd="latexml" id="S4.E28.m1.18.18.18.12.12.12.1.2.cmml" xref="S4.E28.m1.18.18.18.12.12.12.1.2">absent</csymbol><csymbol cd="latexml" id="S4.E28.m1.18.18.18.12.12.12.1.3.cmml" xref="S4.E28.m1.18.18.18.12.12.12.1.3">top</csymbol></apply></apply><apply id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.3.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">subscript</csymbol><apply id="S4.E28.m1.19.19.19.13.13.13.cmml" xref="S4.E28.m1.19.19.19.13.13.13"><ci id="S4.E28.m1.19.19.19.13.13.13.1.cmml" xref="S4.E28.m1.19.19.19.13.13.13.1">¯</ci><ci id="S4.E28.m1.19.19.19.13.13.13.2.cmml" xref="S4.E28.m1.19.19.19.13.13.13.2">𝚲</ci></apply><ci id="S4.E28.m1.20.20.20.14.14.14.1.cmml" xref="S4.E28.m1.20.20.20.14.14.14.1">𝑎</ci></apply><apply id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.4.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.3.2.2.2.2.4.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">superscript</csymbol><ci id="S4.E28.m1.21.21.21.15.15.15.cmml" xref="S4.E28.m1.21.21.21.15.15.15">𝑮</ci><apply id="S4.E28.m1.22.22.22.16.16.16.1.cmml" xref="S4.E28.m1.22.22.22.16.16.16.1"><minus id="S4.E28.m1.22.22.22.16.16.16.1.1.cmml" xref="S4.E28.m1.22.22.22.16.16.16.1"></minus><cn id="S4.E28.m1.22.22.22.16.16.16.1.2.cmml" type="integer" xref="S4.E28.m1.22.22.22.16.16.16.1.2">1</cn></apply></apply></apply><apply id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><minus id="S4.E28.m1.26.26.26.20.20.20.cmml" xref="S4.E28.m1.26.26.26.20.20.20"></minus><apply id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.2.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.2.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">subscript</csymbol><ci id="S4.E28.m1.24.24.24.18.18.18.cmml" xref="S4.E28.m1.24.24.24.18.18.18">bold-italic-ϕ</ci><ci id="S4.E28.m1.25.25.25.19.19.19.1.cmml" xref="S4.E28.m1.25.25.25.19.19.19.1">𝑛</ci></apply><apply id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><times id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.3.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"></times><ci id="S4.E28.m1.27.27.27.21.21.21.cmml" xref="S4.E28.m1.27.27.27.21.21.21">𝑻</ci><apply id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.3.3.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a"><csymbol cd="ambiguous" id="S4.E28.m1.32.32.1.1.1.4.3.3.3.3.3.3.1.cmml" xref="S4.E28.m1.33.33.2.32.7.7.7.1a">subscript</csymbol><apply id="S4.E28.m1.28.28.28.22.22.22.cmml" xref="S4.E28.m1.28.28.28.22.22.22"><ci id="S4.E28.m1.28.28.28.22.22.22.1.cmml" xref="S4.E28.m1.28.28.28.22.22.22.1">¯</ci><ci id="S4.E28.m1.28.28.28.22.22.22.2.cmml" xref="S4.E28.m1.28.28.28.22.22.22.2">bold-italic-ϕ</ci></apply><apply id="S4.E28.m1.29.29.29.23.23.23.1.cmml" xref="S4.E28.m1.29.29.29.23.23.23.1"><minus id="S4.E28.m1.29.29.29.23.23.23.1.1.cmml" xref="S4.E28.m1.29.29.29.23.23.23.1.1"></minus><ci id="S4.E28.m1.29.29.29.23.23.23.1.2.cmml" xref="S4.E28.m1.29.29.29.23.23.23.1.2">𝑛</ci><cn id="S4.E28.m1.29.29.29.23.23.23.1.3.cmml" type="integer" xref="S4.E28.m1.29.29.29.23.23.23.1.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E28.m1.34c">\ln\big{(}\epsilon^{(\bm{\phi}_{n-1}\to\bm{\phi}_{n})}\big{)}\propto\\ -\frac{1}{2}\big{\langle}\bm{\phi}_{n}-\bm{T}\overline{\bm{\phi}}_{n-1}\big{|}% \bm{G}^{-\top}\overline{\bm{\Lambda}}_{a}\bm{G}^{-1}\big{|}\bm{\phi}_{n}-\bm{T% }\overline{\bm{\phi}}_{n-1}\big{\rangle},</annotation><annotation encoding="application/x-llamapun" id="S4.E28.m1.34d">start_ROW start_CELL roman_ln ( italic_ϵ start_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) ∝ end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ⟨ bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - bold_italic_T over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT | bold_italic_G start_POSTSUPERSCRIPT - ⊤ end_POSTSUPERSCRIPT over¯ start_ARG bold_Λ end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT bold_italic_G start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - bold_italic_T over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ⟩ , 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">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p3.7">leading to the following functional form for the message</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx7"> <tbody id="S4.E29"><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\epsilon^{(\bm{\phi}_{n-1}\to\bm{\phi}_{n})}=\mathcal{N}\!\big{(}% \bm{\phi}_{n};\bm{T}\overline{\bm{\phi}}_{n-1},\bm{G}^{-\top}\overline{\bm{% \Lambda}}_{a}\bm{G}\big{)}." class="ltx_Math" display="inline" id="S4.E29.m1.2"><semantics id="S4.E29.m1.2a"><mrow id="S4.E29.m1.2.2.1" xref="S4.E29.m1.2.2.1.1.cmml"><mrow id="S4.E29.m1.2.2.1.1" xref="S4.E29.m1.2.2.1.1.cmml"><msup id="S4.E29.m1.2.2.1.1.5" xref="S4.E29.m1.2.2.1.1.5.cmml"><mi id="S4.E29.m1.2.2.1.1.5.2" xref="S4.E29.m1.2.2.1.1.5.2.cmml">ϵ</mi><mrow id="S4.E29.m1.1.1.1.1" xref="S4.E29.m1.1.1.1.1.1.cmml"><mo id="S4.E29.m1.1.1.1.1.2" stretchy="false" xref="S4.E29.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E29.m1.1.1.1.1.1" xref="S4.E29.m1.1.1.1.1.1.cmml"><msub id="S4.E29.m1.1.1.1.1.1.2" xref="S4.E29.m1.1.1.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E29.m1.1.1.1.1.1.2.2" mathvariant="bold-italic" xref="S4.E29.m1.1.1.1.1.1.2.2.cmml">ϕ</mi><mrow id="S4.E29.m1.1.1.1.1.1.2.3" xref="S4.E29.m1.1.1.1.1.1.2.3.cmml"><mi id="S4.E29.m1.1.1.1.1.1.2.3.2" xref="S4.E29.m1.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S4.E29.m1.1.1.1.1.1.2.3.1" xref="S4.E29.m1.1.1.1.1.1.2.3.1.cmml">−</mo><mn id="S4.E29.m1.1.1.1.1.1.2.3.3" xref="S4.E29.m1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.E29.m1.1.1.1.1.1.1" stretchy="false" xref="S4.E29.m1.1.1.1.1.1.1.cmml">→</mo><msub id="S4.E29.m1.1.1.1.1.1.3" xref="S4.E29.m1.1.1.1.1.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E29.m1.1.1.1.1.1.3.2" mathvariant="bold-italic" xref="S4.E29.m1.1.1.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.E29.m1.1.1.1.1.1.3.3" xref="S4.E29.m1.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.E29.m1.1.1.1.1.3" stretchy="false" xref="S4.E29.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.E29.m1.2.2.1.1.4" xref="S4.E29.m1.2.2.1.1.4.cmml">=</mo><mrow id="S4.E29.m1.2.2.1.1.3" xref="S4.E29.m1.2.2.1.1.3.cmml"><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E29.m1.2.2.1.1.3.5" xref="S4.E29.m1.2.2.1.1.3.5.cmml">𝒩</mi></mpadded><mo id="S4.E29.m1.2.2.1.1.3.4" xref="S4.E29.m1.2.2.1.1.3.4.cmml">⁢</mo><mrow id="S4.E29.m1.2.2.1.1.3.3.3" xref="S4.E29.m1.2.2.1.1.3.3.4.cmml"><mo id="S4.E29.m1.2.2.1.1.3.3.3.4" maxsize="120%" minsize="120%" xref="S4.E29.m1.2.2.1.1.3.3.4.cmml">(</mo><msub id="S4.E29.m1.2.2.1.1.1.1.1.1" xref="S4.E29.m1.2.2.1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E29.m1.2.2.1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.E29.m1.2.2.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.E29.m1.2.2.1.1.1.1.1.1.3" xref="S4.E29.m1.2.2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.E29.m1.2.2.1.1.3.3.3.5" xref="S4.E29.m1.2.2.1.1.3.3.4.cmml">;</mo><mrow id="S4.E29.m1.2.2.1.1.2.2.2.2" xref="S4.E29.m1.2.2.1.1.2.2.2.2.cmml"><mi id="S4.E29.m1.2.2.1.1.2.2.2.2.2" xref="S4.E29.m1.2.2.1.1.2.2.2.2.2.cmml">𝑻</mi><mo id="S4.E29.m1.2.2.1.1.2.2.2.2.1" xref="S4.E29.m1.2.2.1.1.2.2.2.2.1.cmml">⁢</mo><msub id="S4.E29.m1.2.2.1.1.2.2.2.2.3" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.cmml"><mover accent="true" id="S4.E29.m1.2.2.1.1.2.2.2.2.3.2" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.2" mathvariant="bold-italic" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.2.cmml">ϕ</mi><mo id="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.1" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.1.cmml">¯</mo></mover><mrow id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.cmml"><mi id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.2" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.2.cmml">n</mi><mo id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.1" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.1.cmml">−</mo><mn id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.3" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S4.E29.m1.2.2.1.1.3.3.3.6" xref="S4.E29.m1.2.2.1.1.3.3.4.cmml">,</mo><mrow id="S4.E29.m1.2.2.1.1.3.3.3.3" xref="S4.E29.m1.2.2.1.1.3.3.3.3.cmml"><msup id="S4.E29.m1.2.2.1.1.3.3.3.3.2" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.cmml"><mi id="S4.E29.m1.2.2.1.1.3.3.3.3.2.2" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.2.cmml">𝑮</mi><mrow id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.cmml"><mi id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.2" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.2.cmml"></mi><mo id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.1" rspace="0em" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.1.cmml">−</mo><mo id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.3" lspace="0em" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.3.cmml">⊤</mo></mrow></msup><mo id="S4.E29.m1.2.2.1.1.3.3.3.3.1" xref="S4.E29.m1.2.2.1.1.3.3.3.3.1.cmml">⁢</mo><msub id="S4.E29.m1.2.2.1.1.3.3.3.3.3" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.cmml"><mover accent="true" id="S4.E29.m1.2.2.1.1.3.3.3.3.3.2" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.cmml"><mi id="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.2" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.2.cmml">𝚲</mi><mo id="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.1" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.1.cmml">¯</mo></mover><mi id="S4.E29.m1.2.2.1.1.3.3.3.3.3.3" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.3.cmml">a</mi></msub><mo id="S4.E29.m1.2.2.1.1.3.3.3.3.1a" xref="S4.E29.m1.2.2.1.1.3.3.3.3.1.cmml">⁢</mo><mi id="S4.E29.m1.2.2.1.1.3.3.3.3.4" xref="S4.E29.m1.2.2.1.1.3.3.3.3.4.cmml">𝑮</mi></mrow><mo id="S4.E29.m1.2.2.1.1.3.3.3.7" maxsize="120%" minsize="120%" xref="S4.E29.m1.2.2.1.1.3.3.4.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E29.m1.2.2.1.2" lspace="0em" xref="S4.E29.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E29.m1.2b"><apply id="S4.E29.m1.2.2.1.1.cmml" xref="S4.E29.m1.2.2.1"><eq id="S4.E29.m1.2.2.1.1.4.cmml" xref="S4.E29.m1.2.2.1.1.4"></eq><apply id="S4.E29.m1.2.2.1.1.5.cmml" xref="S4.E29.m1.2.2.1.1.5"><csymbol cd="ambiguous" id="S4.E29.m1.2.2.1.1.5.1.cmml" xref="S4.E29.m1.2.2.1.1.5">superscript</csymbol><ci id="S4.E29.m1.2.2.1.1.5.2.cmml" xref="S4.E29.m1.2.2.1.1.5.2">italic-ϵ</ci><apply id="S4.E29.m1.1.1.1.1.1.cmml" xref="S4.E29.m1.1.1.1.1"><ci id="S4.E29.m1.1.1.1.1.1.1.cmml" xref="S4.E29.m1.1.1.1.1.1.1">→</ci><apply id="S4.E29.m1.1.1.1.1.1.2.cmml" xref="S4.E29.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E29.m1.1.1.1.1.1.2.1.cmml" xref="S4.E29.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.E29.m1.1.1.1.1.1.2.2.cmml" xref="S4.E29.m1.1.1.1.1.1.2.2">bold-italic-ϕ</ci><apply id="S4.E29.m1.1.1.1.1.1.2.3.cmml" xref="S4.E29.m1.1.1.1.1.1.2.3"><minus id="S4.E29.m1.1.1.1.1.1.2.3.1.cmml" xref="S4.E29.m1.1.1.1.1.1.2.3.1"></minus><ci id="S4.E29.m1.1.1.1.1.1.2.3.2.cmml" xref="S4.E29.m1.1.1.1.1.1.2.3.2">𝑛</ci><cn id="S4.E29.m1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S4.E29.m1.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.E29.m1.1.1.1.1.1.3.cmml" xref="S4.E29.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E29.m1.1.1.1.1.1.3.1.cmml" xref="S4.E29.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.E29.m1.1.1.1.1.1.3.2.cmml" xref="S4.E29.m1.1.1.1.1.1.3.2">bold-italic-ϕ</ci><ci id="S4.E29.m1.1.1.1.1.1.3.3.cmml" xref="S4.E29.m1.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E29.m1.2.2.1.1.3.cmml" xref="S4.E29.m1.2.2.1.1.3"><times id="S4.E29.m1.2.2.1.1.3.4.cmml" xref="S4.E29.m1.2.2.1.1.3.4"></times><ci id="S4.E29.m1.2.2.1.1.3.5.cmml" xref="S4.E29.m1.2.2.1.1.3.5">𝒩</ci><list id="S4.E29.m1.2.2.1.1.3.3.4.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3"><apply id="S4.E29.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E29.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E29.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E29.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E29.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E29.m1.2.2.1.1.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.E29.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S4.E29.m1.2.2.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S4.E29.m1.2.2.1.1.2.2.2.2.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2"><times id="S4.E29.m1.2.2.1.1.2.2.2.2.1.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.1"></times><ci id="S4.E29.m1.2.2.1.1.2.2.2.2.2.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.2">𝑻</ci><apply id="S4.E29.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E29.m1.2.2.1.1.2.2.2.2.3.1.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3">subscript</csymbol><apply id="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.2"><ci id="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.1.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.1">¯</ci><ci id="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.2.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.2.2">bold-italic-ϕ</ci></apply><apply id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3"><minus id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.1.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.1"></minus><ci id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.2.cmml" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.2">𝑛</ci><cn id="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.3.cmml" type="integer" xref="S4.E29.m1.2.2.1.1.2.2.2.2.3.3.3">1</cn></apply></apply></apply><apply id="S4.E29.m1.2.2.1.1.3.3.3.3.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3"><times id="S4.E29.m1.2.2.1.1.3.3.3.3.1.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.1"></times><apply id="S4.E29.m1.2.2.1.1.3.3.3.3.2.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2"><csymbol cd="ambiguous" id="S4.E29.m1.2.2.1.1.3.3.3.3.2.1.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2">superscript</csymbol><ci id="S4.E29.m1.2.2.1.1.3.3.3.3.2.2.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.2">𝑮</ci><apply id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3"><minus id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.1.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.1"></minus><csymbol cd="latexml" id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.2.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.2">absent</csymbol><csymbol cd="latexml" id="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.3.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.2.3.3">top</csymbol></apply></apply><apply id="S4.E29.m1.2.2.1.1.3.3.3.3.3.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3"><csymbol cd="ambiguous" id="S4.E29.m1.2.2.1.1.3.3.3.3.3.1.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3">subscript</csymbol><apply id="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.2"><ci id="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.1.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.1">¯</ci><ci id="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.2.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.2.2">𝚲</ci></apply><ci id="S4.E29.m1.2.2.1.1.3.3.3.3.3.3.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.3.3">𝑎</ci></apply><ci id="S4.E29.m1.2.2.1.1.3.3.3.3.4.cmml" xref="S4.E29.m1.2.2.1.1.3.3.3.3.4">𝑮</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E29.m1.2c">\displaystyle\epsilon^{(\bm{\phi}_{n-1}\to\bm{\phi}_{n})}=\mathcal{N}\!\big{(}% \bm{\phi}_{n};\bm{T}\overline{\bm{\phi}}_{n-1},\bm{G}^{-\top}\overline{\bm{% \Lambda}}_{a}\bm{G}\big{)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E29.m1.2d">italic_ϵ start_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT = caligraphic_N ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; bold_italic_T over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_italic_G start_POSTSUPERSCRIPT - ⊤ end_POSTSUPERSCRIPT over¯ start_ARG bold_Λ end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT bold_italic_G ) .</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="S4.p3.8">Similarly,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E30"> <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="\epsilon^{(\phi_{n+1}\rightarrow\phi_{n})}=\mathcal{N}\!\big{(}\phi_{n};\bm{T}% ^{-1}\bar{\phi}_{n+1},\bm{T}^{\top}\bm{G}^{-\top}\overline{\bm{\Lambda}}_{a}% \bm{G}^{-1}\bm{T}\big{)}." class="ltx_Math" display="block" id="S4.E30.m1.2"><semantics id="S4.E30.m1.2a"><mrow id="S4.E30.m1.2.2.1" xref="S4.E30.m1.2.2.1.1.cmml"><mrow id="S4.E30.m1.2.2.1.1" xref="S4.E30.m1.2.2.1.1.cmml"><msup id="S4.E30.m1.2.2.1.1.5" xref="S4.E30.m1.2.2.1.1.5.cmml"><mi id="S4.E30.m1.2.2.1.1.5.2" xref="S4.E30.m1.2.2.1.1.5.2.cmml">ϵ</mi><mrow id="S4.E30.m1.1.1.1.1" xref="S4.E30.m1.1.1.1.1.1.cmml"><mo id="S4.E30.m1.1.1.1.1.2" stretchy="false" xref="S4.E30.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E30.m1.1.1.1.1.1" xref="S4.E30.m1.1.1.1.1.1.cmml"><msub id="S4.E30.m1.1.1.1.1.1.2" xref="S4.E30.m1.1.1.1.1.1.2.cmml"><mi id="S4.E30.m1.1.1.1.1.1.2.2" xref="S4.E30.m1.1.1.1.1.1.2.2.cmml">ϕ</mi><mrow id="S4.E30.m1.1.1.1.1.1.2.3" xref="S4.E30.m1.1.1.1.1.1.2.3.cmml"><mi id="S4.E30.m1.1.1.1.1.1.2.3.2" xref="S4.E30.m1.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S4.E30.m1.1.1.1.1.1.2.3.1" xref="S4.E30.m1.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="S4.E30.m1.1.1.1.1.1.2.3.3" xref="S4.E30.m1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.E30.m1.1.1.1.1.1.1" stretchy="false" xref="S4.E30.m1.1.1.1.1.1.1.cmml">→</mo><msub id="S4.E30.m1.1.1.1.1.1.3" xref="S4.E30.m1.1.1.1.1.1.3.cmml"><mi id="S4.E30.m1.1.1.1.1.1.3.2" xref="S4.E30.m1.1.1.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.E30.m1.1.1.1.1.1.3.3" xref="S4.E30.m1.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.E30.m1.1.1.1.1.3" stretchy="false" xref="S4.E30.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.E30.m1.2.2.1.1.4" xref="S4.E30.m1.2.2.1.1.4.cmml">=</mo><mrow id="S4.E30.m1.2.2.1.1.3" xref="S4.E30.m1.2.2.1.1.3.cmml"><mpadded width="0.798em"><mi class="ltx_font_mathcaligraphic" id="S4.E30.m1.2.2.1.1.3.5" xref="S4.E30.m1.2.2.1.1.3.5.cmml">𝒩</mi></mpadded><mo id="S4.E30.m1.2.2.1.1.3.4" xref="S4.E30.m1.2.2.1.1.3.4.cmml">⁢</mo><mrow id="S4.E30.m1.2.2.1.1.3.3.3" xref="S4.E30.m1.2.2.1.1.3.3.4.cmml"><mo id="S4.E30.m1.2.2.1.1.3.3.3.4" maxsize="120%" minsize="120%" xref="S4.E30.m1.2.2.1.1.3.3.4.cmml">(</mo><msub id="S4.E30.m1.2.2.1.1.1.1.1.1" xref="S4.E30.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S4.E30.m1.2.2.1.1.1.1.1.1.2" xref="S4.E30.m1.2.2.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.E30.m1.2.2.1.1.1.1.1.1.3" xref="S4.E30.m1.2.2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.E30.m1.2.2.1.1.3.3.3.5" xref="S4.E30.m1.2.2.1.1.3.3.4.cmml">;</mo><mrow id="S4.E30.m1.2.2.1.1.2.2.2.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.cmml"><msup id="S4.E30.m1.2.2.1.1.2.2.2.2.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.cmml"><mi id="S4.E30.m1.2.2.1.1.2.2.2.2.2.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.2.cmml">𝑻</mi><mrow id="S4.E30.m1.2.2.1.1.2.2.2.2.2.3" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.cmml"><mo id="S4.E30.m1.2.2.1.1.2.2.2.2.2.3a" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.cmml">−</mo><mn id="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.E30.m1.2.2.1.1.2.2.2.2.1" xref="S4.E30.m1.2.2.1.1.2.2.2.2.1.cmml">⁢</mo><msub id="S4.E30.m1.2.2.1.1.2.2.2.2.3" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.cmml"><mover accent="true" id="S4.E30.m1.2.2.1.1.2.2.2.2.3.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.cmml"><mi id="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.2.cmml">ϕ</mi><mo id="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.1" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.1.cmml">¯</mo></mover><mrow id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.cmml"><mi id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.2" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.2.cmml">n</mi><mo id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.1" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.1.cmml">+</mo><mn id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.3" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S4.E30.m1.2.2.1.1.3.3.3.6" xref="S4.E30.m1.2.2.1.1.3.3.4.cmml">,</mo><mrow id="S4.E30.m1.2.2.1.1.3.3.3.3" xref="S4.E30.m1.2.2.1.1.3.3.3.3.cmml"><msup id="S4.E30.m1.2.2.1.1.3.3.3.3.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2.cmml"><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.2.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2.2.cmml">𝑻</mi><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.2.3" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2.3.cmml">⊤</mo></msup><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.1" xref="S4.E30.m1.2.2.1.1.3.3.3.3.1.cmml">⁢</mo><msup id="S4.E30.m1.2.2.1.1.3.3.3.3.3" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.cmml"><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.3.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.2.cmml">𝑮</mi><mrow id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.cmml"><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.2.cmml"></mi><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.1" rspace="0em" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.1.cmml">−</mo><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.3" lspace="0em" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.3.cmml">⊤</mo></mrow></msup><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.1a" xref="S4.E30.m1.2.2.1.1.3.3.3.3.1.cmml">⁢</mo><msub id="S4.E30.m1.2.2.1.1.3.3.3.3.4" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.cmml"><mover accent="true" id="S4.E30.m1.2.2.1.1.3.3.3.3.4.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.cmml"><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.2.cmml">𝚲</mi><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.1" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.1.cmml">¯</mo></mover><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.4.3" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.3.cmml">a</mi></msub><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.1b" xref="S4.E30.m1.2.2.1.1.3.3.3.3.1.cmml">⁢</mo><msup id="S4.E30.m1.2.2.1.1.3.3.3.3.5" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.cmml"><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.5.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.2.cmml">𝑮</mi><mrow id="S4.E30.m1.2.2.1.1.3.3.3.3.5.3" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.cmml"><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.5.3a" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.cmml">−</mo><mn id="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.2" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.2.cmml">1</mn></mrow></msup><mo id="S4.E30.m1.2.2.1.1.3.3.3.3.1c" xref="S4.E30.m1.2.2.1.1.3.3.3.3.1.cmml">⁢</mo><mi id="S4.E30.m1.2.2.1.1.3.3.3.3.6" xref="S4.E30.m1.2.2.1.1.3.3.3.3.6.cmml">𝑻</mi></mrow><mo id="S4.E30.m1.2.2.1.1.3.3.3.7" maxsize="120%" minsize="120%" xref="S4.E30.m1.2.2.1.1.3.3.4.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E30.m1.2.2.1.2" lspace="0em" xref="S4.E30.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E30.m1.2b"><apply id="S4.E30.m1.2.2.1.1.cmml" xref="S4.E30.m1.2.2.1"><eq id="S4.E30.m1.2.2.1.1.4.cmml" xref="S4.E30.m1.2.2.1.1.4"></eq><apply id="S4.E30.m1.2.2.1.1.5.cmml" xref="S4.E30.m1.2.2.1.1.5"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.5.1.cmml" xref="S4.E30.m1.2.2.1.1.5">superscript</csymbol><ci id="S4.E30.m1.2.2.1.1.5.2.cmml" xref="S4.E30.m1.2.2.1.1.5.2">italic-ϵ</ci><apply id="S4.E30.m1.1.1.1.1.1.cmml" xref="S4.E30.m1.1.1.1.1"><ci id="S4.E30.m1.1.1.1.1.1.1.cmml" xref="S4.E30.m1.1.1.1.1.1.1">→</ci><apply id="S4.E30.m1.1.1.1.1.1.2.cmml" xref="S4.E30.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E30.m1.1.1.1.1.1.2.1.cmml" xref="S4.E30.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.E30.m1.1.1.1.1.1.2.2.cmml" xref="S4.E30.m1.1.1.1.1.1.2.2">italic-ϕ</ci><apply id="S4.E30.m1.1.1.1.1.1.2.3.cmml" xref="S4.E30.m1.1.1.1.1.1.2.3"><plus id="S4.E30.m1.1.1.1.1.1.2.3.1.cmml" xref="S4.E30.m1.1.1.1.1.1.2.3.1"></plus><ci id="S4.E30.m1.1.1.1.1.1.2.3.2.cmml" xref="S4.E30.m1.1.1.1.1.1.2.3.2">𝑛</ci><cn id="S4.E30.m1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S4.E30.m1.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.E30.m1.1.1.1.1.1.3.cmml" xref="S4.E30.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E30.m1.1.1.1.1.1.3.1.cmml" xref="S4.E30.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.E30.m1.1.1.1.1.1.3.2.cmml" xref="S4.E30.m1.1.1.1.1.1.3.2">italic-ϕ</ci><ci id="S4.E30.m1.1.1.1.1.1.3.3.cmml" xref="S4.E30.m1.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply><apply id="S4.E30.m1.2.2.1.1.3.cmml" xref="S4.E30.m1.2.2.1.1.3"><times id="S4.E30.m1.2.2.1.1.3.4.cmml" xref="S4.E30.m1.2.2.1.1.3.4"></times><ci id="S4.E30.m1.2.2.1.1.3.5.cmml" xref="S4.E30.m1.2.2.1.1.3.5">𝒩</ci><list id="S4.E30.m1.2.2.1.1.3.3.4.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3"><apply id="S4.E30.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E30.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E30.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E30.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E30.m1.2.2.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="S4.E30.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S4.E30.m1.2.2.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S4.E30.m1.2.2.1.1.2.2.2.2.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2"><times id="S4.E30.m1.2.2.1.1.2.2.2.2.1.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.1"></times><apply id="S4.E30.m1.2.2.1.1.2.2.2.2.2.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.2.2.2.2.2.1.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2">superscript</csymbol><ci id="S4.E30.m1.2.2.1.1.2.2.2.2.2.2.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.2">𝑻</ci><apply id="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.3"><minus id="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.1.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.3"></minus><cn id="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.2.cmml" type="integer" xref="S4.E30.m1.2.2.1.1.2.2.2.2.2.3.2">1</cn></apply></apply><apply id="S4.E30.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.2.2.2.2.3.1.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3">subscript</csymbol><apply id="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.2"><ci id="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.1.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.1">¯</ci><ci id="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.2.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.2.2">italic-ϕ</ci></apply><apply id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3"><plus id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.1.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.1"></plus><ci id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.2.cmml" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.2">𝑛</ci><cn id="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.3.cmml" type="integer" xref="S4.E30.m1.2.2.1.1.2.2.2.2.3.3.3">1</cn></apply></apply></apply><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3"><times id="S4.E30.m1.2.2.1.1.3.3.3.3.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.1"></times><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.3.3.3.3.2.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2">superscript</csymbol><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.2.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2.2">𝑻</ci><csymbol cd="latexml" id="S4.E30.m1.2.2.1.1.3.3.3.3.2.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.2.3">top</csymbol></apply><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.3.3.3.3.3.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3">superscript</csymbol><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.3.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.2">𝑮</ci><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3"><minus id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.1"></minus><csymbol cd="latexml" id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.2">absent</csymbol><csymbol cd="latexml" id="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.3.3.3">top</csymbol></apply></apply><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.4.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.3.3.3.3.4.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4">subscript</csymbol><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.2"><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.1">¯</ci><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.2.2">𝚲</ci></apply><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.4.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.4.3">𝑎</ci></apply><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.5.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5"><csymbol cd="ambiguous" id="S4.E30.m1.2.2.1.1.3.3.3.3.5.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5">superscript</csymbol><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.5.2.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.2">𝑮</ci><apply id="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.3"><minus id="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.1.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.3"></minus><cn id="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.2.cmml" type="integer" xref="S4.E30.m1.2.2.1.1.3.3.3.3.5.3.2">1</cn></apply></apply><ci id="S4.E30.m1.2.2.1.1.3.3.3.3.6.cmml" xref="S4.E30.m1.2.2.1.1.3.3.3.3.6">𝑻</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E30.m1.2c">\epsilon^{(\phi_{n+1}\rightarrow\phi_{n})}=\mathcal{N}\!\big{(}\phi_{n};\bm{T}% ^{-1}\bar{\phi}_{n+1},\bm{T}^{\top}\bm{G}^{-\top}\overline{\bm{\Lambda}}_{a}% \bm{G}^{-1}\bm{T}\big{)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E30.m1.2d">italic_ϵ start_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT → italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT = caligraphic_N ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ; bold_italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over¯ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , bold_italic_T start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_italic_G start_POSTSUPERSCRIPT - ⊤ end_POSTSUPERSCRIPT over¯ start_ARG bold_Λ end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT bold_italic_G start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_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">(30)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">It is now possible to calculate the surrogate <math alttext="q(\bm{\phi}_{n})" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mrow id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml"><mi id="S4.p4.1.m1.1.1.3" xref="S4.p4.1.m1.1.1.3.cmml">q</mi><mo id="S4.p4.1.m1.1.1.2" xref="S4.p4.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.p4.1.m1.1.1.1.1" xref="S4.p4.1.m1.1.1.1.1.1.cmml"><mo id="S4.p4.1.m1.1.1.1.1.2" stretchy="false" xref="S4.p4.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.p4.1.m1.1.1.1.1.1" xref="S4.p4.1.m1.1.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p4.1.m1.1.1.1.1.1.2" mathvariant="bold-italic" xref="S4.p4.1.m1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.p4.1.m1.1.1.1.1.1.3" xref="S4.p4.1.m1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.p4.1.m1.1.1.1.1.3" stretchy="false" xref="S4.p4.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><apply id="S4.p4.1.m1.1.1.cmml" xref="S4.p4.1.m1.1.1"><times id="S4.p4.1.m1.1.1.2.cmml" xref="S4.p4.1.m1.1.1.2"></times><ci id="S4.p4.1.m1.1.1.3.cmml" xref="S4.p4.1.m1.1.1.3">𝑞</ci><apply id="S4.p4.1.m1.1.1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p4.1.m1.1.1.1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.p4.1.m1.1.1.1.1.1.2.cmml" xref="S4.p4.1.m1.1.1.1.1.1.2">bold-italic-ϕ</ci><ci id="S4.p4.1.m1.1.1.1.1.1.3.cmml" xref="S4.p4.1.m1.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">q(\bm{\phi}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">italic_q ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> since all messages are Gaussians. The surrogate is a product of Gaussians which can be derived using the standard result</p> <table class="ltx_equation ltx_eqn_table" id="S4.E31"> <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="\mathcal{N}(\bm{\mu}_{\text{total}},\bm{\Lambda}_{\text{total}})=\prod_{n=0}^{% N}\mathcal{N}(\bm{\mu}_{n},\bm{\Lambda}_{n})," class="ltx_Math" display="block" id="S4.E31.m1.1"><semantics id="S4.E31.m1.1a"><mrow id="S4.E31.m1.1.1.1" xref="S4.E31.m1.1.1.1.1.cmml"><mrow id="S4.E31.m1.1.1.1.1" xref="S4.E31.m1.1.1.1.1.cmml"><mrow id="S4.E31.m1.1.1.1.1.2" xref="S4.E31.m1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E31.m1.1.1.1.1.2.4" xref="S4.E31.m1.1.1.1.1.2.4.cmml">𝒩</mi><mo id="S4.E31.m1.1.1.1.1.2.3" xref="S4.E31.m1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S4.E31.m1.1.1.1.1.2.2.2" xref="S4.E31.m1.1.1.1.1.2.2.3.cmml"><mo id="S4.E31.m1.1.1.1.1.2.2.2.3" stretchy="false" xref="S4.E31.m1.1.1.1.1.2.2.3.cmml">(</mo><msub id="S4.E31.m1.1.1.1.1.1.1.1.1" xref="S4.E31.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E31.m1.1.1.1.1.1.1.1.1.2" xref="S4.E31.m1.1.1.1.1.1.1.1.1.2.cmml">𝝁</mi><mtext id="S4.E31.m1.1.1.1.1.1.1.1.1.3" xref="S4.E31.m1.1.1.1.1.1.1.1.1.3a.cmml">total</mtext></msub><mo id="S4.E31.m1.1.1.1.1.2.2.2.4" xref="S4.E31.m1.1.1.1.1.2.2.3.cmml">,</mo><msub id="S4.E31.m1.1.1.1.1.2.2.2.2" xref="S4.E31.m1.1.1.1.1.2.2.2.2.cmml"><mi id="S4.E31.m1.1.1.1.1.2.2.2.2.2" xref="S4.E31.m1.1.1.1.1.2.2.2.2.2.cmml">𝚲</mi><mtext id="S4.E31.m1.1.1.1.1.2.2.2.2.3" xref="S4.E31.m1.1.1.1.1.2.2.2.2.3a.cmml">total</mtext></msub><mo id="S4.E31.m1.1.1.1.1.2.2.2.5" stretchy="false" xref="S4.E31.m1.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.E31.m1.1.1.1.1.5" rspace="0.111em" xref="S4.E31.m1.1.1.1.1.5.cmml">=</mo><mrow id="S4.E31.m1.1.1.1.1.4" xref="S4.E31.m1.1.1.1.1.4.cmml"><munderover id="S4.E31.m1.1.1.1.1.4.3" xref="S4.E31.m1.1.1.1.1.4.3.cmml"><mo id="S4.E31.m1.1.1.1.1.4.3.2.2" movablelimits="false" xref="S4.E31.m1.1.1.1.1.4.3.2.2.cmml">∏</mo><mrow id="S4.E31.m1.1.1.1.1.4.3.2.3" xref="S4.E31.m1.1.1.1.1.4.3.2.3.cmml"><mi id="S4.E31.m1.1.1.1.1.4.3.2.3.2" xref="S4.E31.m1.1.1.1.1.4.3.2.3.2.cmml">n</mi><mo id="S4.E31.m1.1.1.1.1.4.3.2.3.1" xref="S4.E31.m1.1.1.1.1.4.3.2.3.1.cmml">=</mo><mn id="S4.E31.m1.1.1.1.1.4.3.2.3.3" xref="S4.E31.m1.1.1.1.1.4.3.2.3.3.cmml">0</mn></mrow><mi id="S4.E31.m1.1.1.1.1.4.3.3" xref="S4.E31.m1.1.1.1.1.4.3.3.cmml">N</mi></munderover><mrow id="S4.E31.m1.1.1.1.1.4.2" xref="S4.E31.m1.1.1.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E31.m1.1.1.1.1.4.2.4" xref="S4.E31.m1.1.1.1.1.4.2.4.cmml">𝒩</mi><mo id="S4.E31.m1.1.1.1.1.4.2.3" xref="S4.E31.m1.1.1.1.1.4.2.3.cmml">⁢</mo><mrow id="S4.E31.m1.1.1.1.1.4.2.2.2" xref="S4.E31.m1.1.1.1.1.4.2.2.3.cmml"><mo id="S4.E31.m1.1.1.1.1.4.2.2.2.3" stretchy="false" xref="S4.E31.m1.1.1.1.1.4.2.2.3.cmml">(</mo><msub id="S4.E31.m1.1.1.1.1.3.1.1.1.1" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1.cmml"><mi id="S4.E31.m1.1.1.1.1.3.1.1.1.1.2" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1.2.cmml">𝝁</mi><mi id="S4.E31.m1.1.1.1.1.3.1.1.1.1.3" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.E31.m1.1.1.1.1.4.2.2.2.4" xref="S4.E31.m1.1.1.1.1.4.2.2.3.cmml">,</mo><msub id="S4.E31.m1.1.1.1.1.4.2.2.2.2" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2.cmml"><mi id="S4.E31.m1.1.1.1.1.4.2.2.2.2.2" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2.2.cmml">𝚲</mi><mi id="S4.E31.m1.1.1.1.1.4.2.2.2.2.3" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2.3.cmml">n</mi></msub><mo id="S4.E31.m1.1.1.1.1.4.2.2.2.5" stretchy="false" xref="S4.E31.m1.1.1.1.1.4.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E31.m1.1.1.1.2" xref="S4.E31.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E31.m1.1b"><apply id="S4.E31.m1.1.1.1.1.cmml" xref="S4.E31.m1.1.1.1"><eq id="S4.E31.m1.1.1.1.1.5.cmml" xref="S4.E31.m1.1.1.1.1.5"></eq><apply id="S4.E31.m1.1.1.1.1.2.cmml" xref="S4.E31.m1.1.1.1.1.2"><times id="S4.E31.m1.1.1.1.1.2.3.cmml" xref="S4.E31.m1.1.1.1.1.2.3"></times><ci id="S4.E31.m1.1.1.1.1.2.4.cmml" xref="S4.E31.m1.1.1.1.1.2.4">𝒩</ci><interval closure="open" id="S4.E31.m1.1.1.1.1.2.2.3.cmml" xref="S4.E31.m1.1.1.1.1.2.2.2"><apply id="S4.E31.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E31.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E31.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E31.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E31.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E31.m1.1.1.1.1.1.1.1.1.2">𝝁</ci><ci id="S4.E31.m1.1.1.1.1.1.1.1.1.3a.cmml" xref="S4.E31.m1.1.1.1.1.1.1.1.1.3"><mtext id="S4.E31.m1.1.1.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S4.E31.m1.1.1.1.1.1.1.1.1.3">total</mtext></ci></apply><apply id="S4.E31.m1.1.1.1.1.2.2.2.2.cmml" xref="S4.E31.m1.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S4.E31.m1.1.1.1.1.2.2.2.2.1.cmml" xref="S4.E31.m1.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S4.E31.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S4.E31.m1.1.1.1.1.2.2.2.2.2">𝚲</ci><ci id="S4.E31.m1.1.1.1.1.2.2.2.2.3a.cmml" xref="S4.E31.m1.1.1.1.1.2.2.2.2.3"><mtext id="S4.E31.m1.1.1.1.1.2.2.2.2.3.cmml" mathsize="70%" xref="S4.E31.m1.1.1.1.1.2.2.2.2.3">total</mtext></ci></apply></interval></apply><apply id="S4.E31.m1.1.1.1.1.4.cmml" xref="S4.E31.m1.1.1.1.1.4"><apply id="S4.E31.m1.1.1.1.1.4.3.cmml" xref="S4.E31.m1.1.1.1.1.4.3"><csymbol cd="ambiguous" id="S4.E31.m1.1.1.1.1.4.3.1.cmml" xref="S4.E31.m1.1.1.1.1.4.3">superscript</csymbol><apply id="S4.E31.m1.1.1.1.1.4.3.2.cmml" xref="S4.E31.m1.1.1.1.1.4.3"><csymbol cd="ambiguous" id="S4.E31.m1.1.1.1.1.4.3.2.1.cmml" xref="S4.E31.m1.1.1.1.1.4.3">subscript</csymbol><csymbol cd="latexml" id="S4.E31.m1.1.1.1.1.4.3.2.2.cmml" xref="S4.E31.m1.1.1.1.1.4.3.2.2">product</csymbol><apply id="S4.E31.m1.1.1.1.1.4.3.2.3.cmml" xref="S4.E31.m1.1.1.1.1.4.3.2.3"><eq id="S4.E31.m1.1.1.1.1.4.3.2.3.1.cmml" xref="S4.E31.m1.1.1.1.1.4.3.2.3.1"></eq><ci id="S4.E31.m1.1.1.1.1.4.3.2.3.2.cmml" xref="S4.E31.m1.1.1.1.1.4.3.2.3.2">𝑛</ci><cn id="S4.E31.m1.1.1.1.1.4.3.2.3.3.cmml" type="integer" xref="S4.E31.m1.1.1.1.1.4.3.2.3.3">0</cn></apply></apply><ci id="S4.E31.m1.1.1.1.1.4.3.3.cmml" xref="S4.E31.m1.1.1.1.1.4.3.3">𝑁</ci></apply><apply id="S4.E31.m1.1.1.1.1.4.2.cmml" xref="S4.E31.m1.1.1.1.1.4.2"><times id="S4.E31.m1.1.1.1.1.4.2.3.cmml" xref="S4.E31.m1.1.1.1.1.4.2.3"></times><ci id="S4.E31.m1.1.1.1.1.4.2.4.cmml" xref="S4.E31.m1.1.1.1.1.4.2.4">𝒩</ci><interval closure="open" id="S4.E31.m1.1.1.1.1.4.2.2.3.cmml" xref="S4.E31.m1.1.1.1.1.4.2.2.2"><apply id="S4.E31.m1.1.1.1.1.3.1.1.1.1.cmml" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.E31.m1.1.1.1.1.3.1.1.1.1.1.cmml" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1">subscript</csymbol><ci id="S4.E31.m1.1.1.1.1.3.1.1.1.1.2.cmml" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1.2">𝝁</ci><ci id="S4.E31.m1.1.1.1.1.3.1.1.1.1.3.cmml" xref="S4.E31.m1.1.1.1.1.3.1.1.1.1.3">𝑛</ci></apply><apply id="S4.E31.m1.1.1.1.1.4.2.2.2.2.cmml" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.E31.m1.1.1.1.1.4.2.2.2.2.1.cmml" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2">subscript</csymbol><ci id="S4.E31.m1.1.1.1.1.4.2.2.2.2.2.cmml" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2.2">𝚲</ci><ci id="S4.E31.m1.1.1.1.1.4.2.2.2.2.3.cmml" xref="S4.E31.m1.1.1.1.1.4.2.2.2.2.3">𝑛</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E31.m1.1c">\mathcal{N}(\bm{\mu}_{\text{total}},\bm{\Lambda}_{\text{total}})=\prod_{n=0}^{% N}\mathcal{N}(\bm{\mu}_{n},\bm{\Lambda}_{n}),</annotation><annotation encoding="application/x-llamapun" id="S4.E31.m1.1d">caligraphic_N ( bold_italic_μ start_POSTSUBSCRIPT total end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT total end_POSTSUBSCRIPT ) = ∏ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT caligraphic_N ( bold_italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_n 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">(31)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p4.2">with</p> <table class="ltx_equation ltx_eqn_table" id="S4.E32"> <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{\Lambda}_{\text{total}}=\sum_{n=0}^{N}\bm{\Lambda}_{n},\phantom{mmm}\bm{% \mu}_{\text{total}}=\bm{\Lambda}_{\text{total}}^{-1}\sum_{n=0}^{N}\bm{\Lambda}% _{n}\bm{\mu}_{n}." class="ltx_Math" display="block" id="S4.E32.m1.1"><semantics id="S4.E32.m1.1a"><mrow id="S4.E32.m1.1.1.1"><mrow id="S4.E32.m1.1.1.1.1.2" xref="S4.E32.m1.1.1.1.1.3.cmml"><mrow id="S4.E32.m1.1.1.1.1.1.1" xref="S4.E32.m1.1.1.1.1.1.1.cmml"><msub id="S4.E32.m1.1.1.1.1.1.1.2" xref="S4.E32.m1.1.1.1.1.1.1.2.cmml"><mi id="S4.E32.m1.1.1.1.1.1.1.2.2" xref="S4.E32.m1.1.1.1.1.1.1.2.2.cmml">𝚲</mi><mtext id="S4.E32.m1.1.1.1.1.1.1.2.3" xref="S4.E32.m1.1.1.1.1.1.1.2.3a.cmml">total</mtext></msub><mo id="S4.E32.m1.1.1.1.1.1.1.1" rspace="0.111em" xref="S4.E32.m1.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E32.m1.1.1.1.1.1.1.3" xref="S4.E32.m1.1.1.1.1.1.1.3.cmml"><munderover id="S4.E32.m1.1.1.1.1.1.1.3.1" xref="S4.E32.m1.1.1.1.1.1.1.3.1.cmml"><mo id="S4.E32.m1.1.1.1.1.1.1.3.1.2.2" movablelimits="false" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.2.cmml">∑</mo><mrow id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.cmml"><mi id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.2" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.2.cmml">n</mi><mo id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.1" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.3" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.3.cmml">0</mn></mrow><mi id="S4.E32.m1.1.1.1.1.1.1.3.1.3" xref="S4.E32.m1.1.1.1.1.1.1.3.1.3.cmml">N</mi></munderover><msub id="S4.E32.m1.1.1.1.1.1.1.3.2" xref="S4.E32.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S4.E32.m1.1.1.1.1.1.1.3.2.2" xref="S4.E32.m1.1.1.1.1.1.1.3.2.2.cmml">𝚲</mi><mi id="S4.E32.m1.1.1.1.1.1.1.3.2.3" xref="S4.E32.m1.1.1.1.1.1.1.3.2.3.cmml">n</mi></msub></mrow></mrow><mo id="S4.E32.m1.1.1.1.1.2.3" rspace="2.797em" xref="S4.E32.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S4.E32.m1.1.1.1.1.2.2" xref="S4.E32.m1.1.1.1.1.2.2.cmml"><msub id="S4.E32.m1.1.1.1.1.2.2.2" xref="S4.E32.m1.1.1.1.1.2.2.2.cmml"><mi id="S4.E32.m1.1.1.1.1.2.2.2.2" xref="S4.E32.m1.1.1.1.1.2.2.2.2.cmml">𝝁</mi><mtext id="S4.E32.m1.1.1.1.1.2.2.2.3" xref="S4.E32.m1.1.1.1.1.2.2.2.3a.cmml">total</mtext></msub><mo id="S4.E32.m1.1.1.1.1.2.2.1" xref="S4.E32.m1.1.1.1.1.2.2.1.cmml">=</mo><mrow id="S4.E32.m1.1.1.1.1.2.2.3" xref="S4.E32.m1.1.1.1.1.2.2.3.cmml"><msubsup id="S4.E32.m1.1.1.1.1.2.2.3.2" xref="S4.E32.m1.1.1.1.1.2.2.3.2.cmml"><mi id="S4.E32.m1.1.1.1.1.2.2.3.2.2.2" xref="S4.E32.m1.1.1.1.1.2.2.3.2.2.2.cmml">𝚲</mi><mtext id="S4.E32.m1.1.1.1.1.2.2.3.2.2.3" xref="S4.E32.m1.1.1.1.1.2.2.3.2.2.3a.cmml">total</mtext><mrow id="S4.E32.m1.1.1.1.1.2.2.3.2.3" xref="S4.E32.m1.1.1.1.1.2.2.3.2.3.cmml"><mo id="S4.E32.m1.1.1.1.1.2.2.3.2.3a" xref="S4.E32.m1.1.1.1.1.2.2.3.2.3.cmml">−</mo><mn id="S4.E32.m1.1.1.1.1.2.2.3.2.3.2" xref="S4.E32.m1.1.1.1.1.2.2.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.E32.m1.1.1.1.1.2.2.3.1" xref="S4.E32.m1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mrow id="S4.E32.m1.1.1.1.1.2.2.3.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.cmml"><munderover id="S4.E32.m1.1.1.1.1.2.2.3.3.1" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.cmml"><mo id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.2" movablelimits="false" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.2.cmml">∑</mo><mrow id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.cmml"><mi id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.2" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.2.cmml">n</mi><mo id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.1" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.1.cmml">=</mo><mn id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.3.cmml">0</mn></mrow><mi id="S4.E32.m1.1.1.1.1.2.2.3.3.1.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.3.cmml">N</mi></munderover><mrow id="S4.E32.m1.1.1.1.1.2.2.3.3.2" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.cmml"><msub id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.cmml"><mi id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.2" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.2.cmml">𝚲</mi><mi id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.3.cmml">n</mi></msub><mo id="S4.E32.m1.1.1.1.1.2.2.3.3.2.1" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.1.cmml">⁢</mo><msub id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.cmml"><mi id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.2" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.2.cmml">𝝁</mi><mi id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.3" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.3.cmml">n</mi></msub></mrow></mrow></mrow></mrow></mrow><mo id="S4.E32.m1.1.1.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E32.m1.1b"><apply id="S4.E32.m1.1.1.1.1.3.cmml" xref="S4.E32.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.3a.cmml" xref="S4.E32.m1.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S4.E32.m1.1.1.1.1.1.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1"><eq id="S4.E32.m1.1.1.1.1.1.1.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.1"></eq><apply id="S4.E32.m1.1.1.1.1.1.1.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.1.1.2.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.E32.m1.1.1.1.1.1.1.2.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.2.2">𝚲</ci><ci id="S4.E32.m1.1.1.1.1.1.1.2.3a.cmml" xref="S4.E32.m1.1.1.1.1.1.1.2.3"><mtext id="S4.E32.m1.1.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S4.E32.m1.1.1.1.1.1.1.2.3">total</mtext></ci></apply><apply id="S4.E32.m1.1.1.1.1.1.1.3.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3"><apply id="S4.E32.m1.1.1.1.1.1.1.3.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.1.1.3.1.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1">superscript</csymbol><apply id="S4.E32.m1.1.1.1.1.1.1.3.1.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.1.1.3.1.2.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1">subscript</csymbol><sum id="S4.E32.m1.1.1.1.1.1.1.3.1.2.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.2"></sum><apply id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3"><eq id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.1"></eq><ci id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.2">𝑛</ci><cn id="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.E32.m1.1.1.1.1.1.1.3.1.2.3.3">0</cn></apply></apply><ci id="S4.E32.m1.1.1.1.1.1.1.3.1.3.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.1.3">𝑁</ci></apply><apply id="S4.E32.m1.1.1.1.1.1.1.3.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="S4.E32.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.2.2">𝚲</ci><ci id="S4.E32.m1.1.1.1.1.1.1.3.2.3.cmml" xref="S4.E32.m1.1.1.1.1.1.1.3.2.3">𝑛</ci></apply></apply></apply><apply id="S4.E32.m1.1.1.1.1.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2"><eq id="S4.E32.m1.1.1.1.1.2.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.1"></eq><apply id="S4.E32.m1.1.1.1.1.2.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.E32.m1.1.1.1.1.2.2.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.2.2">𝝁</ci><ci id="S4.E32.m1.1.1.1.1.2.2.2.3a.cmml" xref="S4.E32.m1.1.1.1.1.2.2.2.3"><mtext id="S4.E32.m1.1.1.1.1.2.2.2.3.cmml" mathsize="70%" xref="S4.E32.m1.1.1.1.1.2.2.2.3">total</mtext></ci></apply><apply id="S4.E32.m1.1.1.1.1.2.2.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3"><times id="S4.E32.m1.1.1.1.1.2.2.3.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.1"></times><apply id="S4.E32.m1.1.1.1.1.2.2.3.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.3.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2">superscript</csymbol><apply id="S4.E32.m1.1.1.1.1.2.2.3.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.3.2.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2">subscript</csymbol><ci id="S4.E32.m1.1.1.1.1.2.2.3.2.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2.2.2">𝚲</ci><ci id="S4.E32.m1.1.1.1.1.2.2.3.2.2.3a.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2.2.3"><mtext id="S4.E32.m1.1.1.1.1.2.2.3.2.2.3.cmml" mathsize="70%" xref="S4.E32.m1.1.1.1.1.2.2.3.2.2.3">total</mtext></ci></apply><apply id="S4.E32.m1.1.1.1.1.2.2.3.2.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2.3"><minus id="S4.E32.m1.1.1.1.1.2.2.3.2.3.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.2.3"></minus><cn id="S4.E32.m1.1.1.1.1.2.2.3.2.3.2.cmml" type="integer" xref="S4.E32.m1.1.1.1.1.2.2.3.2.3.2">1</cn></apply></apply><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3"><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.3.3.1.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1">superscript</csymbol><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1">subscript</csymbol><sum id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.2"></sum><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3"><eq id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.1"></eq><ci id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.2">𝑛</ci><cn id="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.3.cmml" type="integer" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.2.3.3">0</cn></apply></apply><ci id="S4.E32.m1.1.1.1.1.2.2.3.3.1.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.1.3">𝑁</ci></apply><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2"><times id="S4.E32.m1.1.1.1.1.2.2.3.3.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.1"></times><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2">subscript</csymbol><ci id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.2">𝚲</ci><ci id="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.2.3">𝑛</ci></apply><apply id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3"><csymbol cd="ambiguous" id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.1.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3">subscript</csymbol><ci id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.2.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.2">𝝁</ci><ci id="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.3.cmml" xref="S4.E32.m1.1.1.1.1.2.2.3.3.2.3.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E32.m1.1c">\bm{\Lambda}_{\text{total}}=\sum_{n=0}^{N}\bm{\Lambda}_{n},\phantom{mmm}\bm{% \mu}_{\text{total}}=\bm{\Lambda}_{\text{total}}^{-1}\sum_{n=0}^{N}\bm{\Lambda}% _{n}\bm{\mu}_{n}.</annotation><annotation encoding="application/x-llamapun" id="S4.E32.m1.1d">bold_Λ start_POSTSUBSCRIPT total end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT bold_Λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_italic_μ start_POSTSUBSCRIPT total end_POSTSUBSCRIPT = bold_Λ start_POSTSUBSCRIPT total end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT bold_Λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT bold_italic_μ start_POSTSUBSCRIPT italic_n 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">(32)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.5">We now derive the surrogate <math alttext="q(\bm{\Lambda}_{a})" class="ltx_Math" display="inline" id="S4.p5.1.m1.1"><semantics id="S4.p5.1.m1.1a"><mrow id="S4.p5.1.m1.1.1" xref="S4.p5.1.m1.1.1.cmml"><mi id="S4.p5.1.m1.1.1.3" xref="S4.p5.1.m1.1.1.3.cmml">q</mi><mo id="S4.p5.1.m1.1.1.2" xref="S4.p5.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.p5.1.m1.1.1.1.1" xref="S4.p5.1.m1.1.1.1.1.1.cmml"><mo id="S4.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S4.p5.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.p5.1.m1.1.1.1.1.1" xref="S4.p5.1.m1.1.1.1.1.1.cmml"><mi id="S4.p5.1.m1.1.1.1.1.1.2" xref="S4.p5.1.m1.1.1.1.1.1.2.cmml">𝚲</mi><mi id="S4.p5.1.m1.1.1.1.1.1.3" xref="S4.p5.1.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S4.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.1.m1.1b"><apply id="S4.p5.1.m1.1.1.cmml" xref="S4.p5.1.m1.1.1"><times id="S4.p5.1.m1.1.1.2.cmml" xref="S4.p5.1.m1.1.1.2"></times><ci id="S4.p5.1.m1.1.1.3.cmml" xref="S4.p5.1.m1.1.1.3">𝑞</ci><apply id="S4.p5.1.m1.1.1.1.1.1.cmml" xref="S4.p5.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p5.1.m1.1.1.1.1.1.1.cmml" xref="S4.p5.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.p5.1.m1.1.1.1.1.1.2.cmml" xref="S4.p5.1.m1.1.1.1.1.1.2">𝚲</ci><ci id="S4.p5.1.m1.1.1.1.1.1.3.cmml" xref="S4.p5.1.m1.1.1.1.1.1.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.1.m1.1c">q(\bm{\Lambda}_{a})</annotation><annotation encoding="application/x-llamapun" id="S4.p5.1.m1.1d">italic_q ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT )</annotation></semantics></math>. To simplify calculations we will impose a diagonal gamma prior on <math alttext="\bm{\Lambda}_{a}" class="ltx_Math" display="inline" id="S4.p5.2.m2.1"><semantics id="S4.p5.2.m2.1a"><msub id="S4.p5.2.m2.1.1" xref="S4.p5.2.m2.1.1.cmml"><mi id="S4.p5.2.m2.1.1.2" xref="S4.p5.2.m2.1.1.2.cmml">𝚲</mi><mi id="S4.p5.2.m2.1.1.3" xref="S4.p5.2.m2.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p5.2.m2.1b"><apply id="S4.p5.2.m2.1.1.cmml" xref="S4.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p5.2.m2.1.1.1.cmml" xref="S4.p5.2.m2.1.1">subscript</csymbol><ci id="S4.p5.2.m2.1.1.2.cmml" xref="S4.p5.2.m2.1.1.2">𝚲</ci><ci id="S4.p5.2.m2.1.1.3.cmml" xref="S4.p5.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.2.m2.1c">\bm{\Lambda}_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.2.m2.1d">bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>, i.e., <math alttext="p(\bm{\Lambda}_{a,{i\neq j}})=0" class="ltx_Math" display="inline" id="S4.p5.3.m3.3"><semantics id="S4.p5.3.m3.3a"><mrow id="S4.p5.3.m3.3.3" xref="S4.p5.3.m3.3.3.cmml"><mrow id="S4.p5.3.m3.3.3.1" xref="S4.p5.3.m3.3.3.1.cmml"><mi id="S4.p5.3.m3.3.3.1.3" xref="S4.p5.3.m3.3.3.1.3.cmml">p</mi><mo id="S4.p5.3.m3.3.3.1.2" xref="S4.p5.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="S4.p5.3.m3.3.3.1.1.1" xref="S4.p5.3.m3.3.3.1.1.1.1.cmml"><mo id="S4.p5.3.m3.3.3.1.1.1.2" stretchy="false" xref="S4.p5.3.m3.3.3.1.1.1.1.cmml">(</mo><msub id="S4.p5.3.m3.3.3.1.1.1.1" xref="S4.p5.3.m3.3.3.1.1.1.1.cmml"><mi id="S4.p5.3.m3.3.3.1.1.1.1.2" xref="S4.p5.3.m3.3.3.1.1.1.1.2.cmml">𝚲</mi><mrow id="S4.p5.3.m3.2.2.2" xref="S4.p5.3.m3.2.2.2.cmml"><mrow id="S4.p5.3.m3.2.2.2.4.2" xref="S4.p5.3.m3.2.2.2.4.1.cmml"><mi id="S4.p5.3.m3.1.1.1.1" xref="S4.p5.3.m3.1.1.1.1.cmml">a</mi><mo id="S4.p5.3.m3.2.2.2.4.2.1" xref="S4.p5.3.m3.2.2.2.4.1.cmml">,</mo><mi id="S4.p5.3.m3.2.2.2.2" xref="S4.p5.3.m3.2.2.2.2.cmml">i</mi></mrow><mo id="S4.p5.3.m3.2.2.2.3" xref="S4.p5.3.m3.2.2.2.3.cmml">≠</mo><mi id="S4.p5.3.m3.2.2.2.5" xref="S4.p5.3.m3.2.2.2.5.cmml">j</mi></mrow></msub><mo id="S4.p5.3.m3.3.3.1.1.1.3" stretchy="false" xref="S4.p5.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.p5.3.m3.3.3.2" xref="S4.p5.3.m3.3.3.2.cmml">=</mo><mn id="S4.p5.3.m3.3.3.3" xref="S4.p5.3.m3.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.3.m3.3b"><apply id="S4.p5.3.m3.3.3.cmml" xref="S4.p5.3.m3.3.3"><eq id="S4.p5.3.m3.3.3.2.cmml" xref="S4.p5.3.m3.3.3.2"></eq><apply id="S4.p5.3.m3.3.3.1.cmml" xref="S4.p5.3.m3.3.3.1"><times id="S4.p5.3.m3.3.3.1.2.cmml" xref="S4.p5.3.m3.3.3.1.2"></times><ci id="S4.p5.3.m3.3.3.1.3.cmml" xref="S4.p5.3.m3.3.3.1.3">𝑝</ci><apply id="S4.p5.3.m3.3.3.1.1.1.1.cmml" xref="S4.p5.3.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.p5.3.m3.3.3.1.1.1.1.1.cmml" xref="S4.p5.3.m3.3.3.1.1.1">subscript</csymbol><ci id="S4.p5.3.m3.3.3.1.1.1.1.2.cmml" xref="S4.p5.3.m3.3.3.1.1.1.1.2">𝚲</ci><apply id="S4.p5.3.m3.2.2.2.cmml" xref="S4.p5.3.m3.2.2.2"><neq id="S4.p5.3.m3.2.2.2.3.cmml" xref="S4.p5.3.m3.2.2.2.3"></neq><list id="S4.p5.3.m3.2.2.2.4.1.cmml" xref="S4.p5.3.m3.2.2.2.4.2"><ci id="S4.p5.3.m3.1.1.1.1.cmml" xref="S4.p5.3.m3.1.1.1.1">𝑎</ci><ci id="S4.p5.3.m3.2.2.2.2.cmml" xref="S4.p5.3.m3.2.2.2.2">𝑖</ci></list><ci id="S4.p5.3.m3.2.2.2.5.cmml" xref="S4.p5.3.m3.2.2.2.5">𝑗</ci></apply></apply></apply><cn id="S4.p5.3.m3.3.3.3.cmml" type="integer" xref="S4.p5.3.m3.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.3.m3.3c">p(\bm{\Lambda}_{a,{i\neq j}})=0</annotation><annotation encoding="application/x-llamapun" id="S4.p5.3.m3.3d">italic_p ( bold_Λ start_POSTSUBSCRIPT italic_a , italic_i ≠ italic_j end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>, with shape parameter <math alttext="\zeta/2" class="ltx_Math" display="inline" id="S4.p5.4.m4.1"><semantics id="S4.p5.4.m4.1a"><mrow id="S4.p5.4.m4.1.1" xref="S4.p5.4.m4.1.1.cmml"><mi id="S4.p5.4.m4.1.1.2" xref="S4.p5.4.m4.1.1.2.cmml">ζ</mi><mo id="S4.p5.4.m4.1.1.1" xref="S4.p5.4.m4.1.1.1.cmml">/</mo><mn id="S4.p5.4.m4.1.1.3" xref="S4.p5.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.4.m4.1b"><apply id="S4.p5.4.m4.1.1.cmml" xref="S4.p5.4.m4.1.1"><divide id="S4.p5.4.m4.1.1.1.cmml" xref="S4.p5.4.m4.1.1.1"></divide><ci id="S4.p5.4.m4.1.1.2.cmml" xref="S4.p5.4.m4.1.1.2">𝜁</ci><cn id="S4.p5.4.m4.1.1.3.cmml" type="integer" xref="S4.p5.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.4.m4.1c">\zeta/2</annotation><annotation encoding="application/x-llamapun" id="S4.p5.4.m4.1d">italic_ζ / 2</annotation></semantics></math> and rate parameter <math alttext="\chi/2" class="ltx_Math" display="inline" id="S4.p5.5.m5.1"><semantics id="S4.p5.5.m5.1a"><mrow id="S4.p5.5.m5.1.1" xref="S4.p5.5.m5.1.1.cmml"><mi id="S4.p5.5.m5.1.1.2" xref="S4.p5.5.m5.1.1.2.cmml">χ</mi><mo id="S4.p5.5.m5.1.1.1" xref="S4.p5.5.m5.1.1.1.cmml">/</mo><mn id="S4.p5.5.m5.1.1.3" xref="S4.p5.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.5.m5.1b"><apply id="S4.p5.5.m5.1.1.cmml" xref="S4.p5.5.m5.1.1"><divide id="S4.p5.5.m5.1.1.1.cmml" xref="S4.p5.5.m5.1.1.1"></divide><ci id="S4.p5.5.m5.1.1.2.cmml" xref="S4.p5.5.m5.1.1.2">𝜒</ci><cn id="S4.p5.5.m5.1.1.3.cmml" type="integer" xref="S4.p5.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.5.m5.1c">\chi/2</annotation><annotation encoding="application/x-llamapun" id="S4.p5.5.m5.1d">italic_χ / 2</annotation></semantics></math>. With this, the log-likelihood can be expressed as,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E33"> <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="\ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}))\propto\frac{1}{2}\ln(|% \bm{\Lambda}_{a}|)\\ -\frac{1}{2}\big{\langle}\bm{G}^{-1}(\phi_{n}-\bm{T}\phi_{n-1})\big{|}\bm{% \Lambda}_{a}\big{|}\bm{G}^{-1}(\bm{\phi}_{n}-\bm{T}\bm{\phi}_{n-1})\big{\rangle}" class="ltx_Math" display="block" id="S4.E33.m1.61"><semantics id="S4.E33.m1.61a"><mtable displaystyle="true" id="S4.E33.m1.61.61.10" rowspacing="0pt" xref="S4.E33.m1.56.56.5.cmml"><mtr id="S4.E33.m1.61.61.10a" xref="S4.E33.m1.56.56.5.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E33.m1.61.61.10b" xref="S4.E33.m1.56.56.5.cmml"><mrow id="S4.E33.m1.58.58.7.53.25.25" xref="S4.E33.m1.56.56.5.cmml"><mrow id="S4.E33.m1.57.57.6.52.24.24.24.1" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.1.1.1.1.1.1" xref="S4.E33.m1.1.1.1.1.1.1.cmml">ln</mi><mo id="S4.E33.m1.57.57.6.52.24.24.24.1a" xref="S4.E33.m1.56.56.5.cmml">⁡</mo><mrow id="S4.E33.m1.57.57.6.52.24.24.24.1.1" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.2.2.2.2.2.2" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">(</mo><mrow id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.3.3.3.3.3.3" xref="S4.E33.m1.3.3.3.3.3.3.cmml">p</mi><mo id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.2" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><mrow id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.4.4.4.4.4.4" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">(</mo><mrow id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><msub id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.1.1.1.3" xref="S4.E33.m1.56.56.5.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E33.m1.5.5.5.5.5.5" mathvariant="bold-italic" xref="S4.E33.m1.5.5.5.5.5.5.cmml">ϕ</mi><mi id="S4.E33.m1.6.6.6.6.6.6.1" xref="S4.E33.m1.6.6.6.6.6.6.1.cmml">n</mi></msub><mo fence="false" id="S4.E33.m1.7.7.7.7.7.7" xref="S4.E33.m1.7.7.7.7.7.7.cmml">|</mo><mrow id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.1.1.1.2.2" xref="S4.E33.m1.56.56.5.cmml"><msub id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.1.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E33.m1.8.8.8.8.8.8" mathvariant="bold-italic" xref="S4.E33.m1.8.8.8.8.8.8.cmml">ϕ</mi><mrow id="S4.E33.m1.9.9.9.9.9.9.1" xref="S4.E33.m1.9.9.9.9.9.9.1.cmml"><mi id="S4.E33.m1.9.9.9.9.9.9.1.2" xref="S4.E33.m1.9.9.9.9.9.9.1.2.cmml">n</mi><mo id="S4.E33.m1.9.9.9.9.9.9.1.1" xref="S4.E33.m1.9.9.9.9.9.9.1.1.cmml">−</mo><mn id="S4.E33.m1.9.9.9.9.9.9.1.3" xref="S4.E33.m1.9.9.9.9.9.9.1.3.cmml">1</mn></mrow></msub><mo id="S4.E33.m1.10.10.10.10.10.10" xref="S4.E33.m1.56.56.5.cmml">,</mo><msub id="S4.E33.m1.57.57.6.52.24.24.24.1.1.1.1.1.1.2.2.2" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.11.11.11.11.11.11" xref="S4.E33.m1.11.11.11.11.11.11.cmml">𝚲</mi><mi id="S4.E33.m1.12.12.12.12.12.12.1" xref="S4.E33.m1.12.12.12.12.12.12.1.cmml">a</mi></msub></mrow></mrow><mo id="S4.E33.m1.13.13.13.13.13.13" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">)</mo></mrow></mrow><mo id="S4.E33.m1.14.14.14.14.14.14" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">)</mo></mrow></mrow><mo id="S4.E33.m1.15.15.15.15.15.15" xref="S4.E33.m1.15.15.15.15.15.15.cmml">∝</mo><mrow id="S4.E33.m1.58.58.7.53.25.25.25" xref="S4.E33.m1.56.56.5.cmml"><mfrac id="S4.E33.m1.16.16.16.16.16.16" xref="S4.E33.m1.16.16.16.16.16.16.cmml"><mn id="S4.E33.m1.16.16.16.16.16.16.2" xref="S4.E33.m1.16.16.16.16.16.16.2.cmml">1</mn><mn id="S4.E33.m1.16.16.16.16.16.16.3" xref="S4.E33.m1.16.16.16.16.16.16.3.cmml">2</mn></mfrac><mo id="S4.E33.m1.58.58.7.53.25.25.25.2" lspace="0.167em" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><mrow id="S4.E33.m1.58.58.7.53.25.25.25.1.1" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.17.17.17.17.17.17" xref="S4.E33.m1.17.17.17.17.17.17.cmml">ln</mi><mo id="S4.E33.m1.58.58.7.53.25.25.25.1.1a" xref="S4.E33.m1.56.56.5.cmml">⁡</mo><mrow id="S4.E33.m1.58.58.7.53.25.25.25.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.18.18.18.18.18.18" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">(</mo><mrow id="S4.E33.m1.58.58.7.53.25.25.25.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.19.19.19.19.19.19" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">|</mo><msub id="S4.E33.m1.58.58.7.53.25.25.25.1.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.20.20.20.20.20.20" xref="S4.E33.m1.20.20.20.20.20.20.cmml">𝚲</mi><mi id="S4.E33.m1.21.21.21.21.21.21.1" xref="S4.E33.m1.21.21.21.21.21.21.1.cmml">a</mi></msub><mo id="S4.E33.m1.22.22.22.22.22.22" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">|</mo></mrow><mo id="S4.E33.m1.23.23.23.23.23.23" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E33.m1.61.61.10c" xref="S4.E33.m1.56.56.5.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E33.m1.61.61.10d" xref="S4.E33.m1.56.56.5.cmml"><mrow id="S4.E33.m1.61.61.10.56.31.31" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.61.61.10.56.31.31a" xref="S4.E33.m1.56.56.5.cmml">−</mo><mrow id="S4.E33.m1.61.61.10.56.31.31.31" xref="S4.E33.m1.56.56.5.cmml"><mfrac id="S4.E33.m1.25.25.25.2.2.2" xref="S4.E33.m1.25.25.25.2.2.2.cmml"><mn id="S4.E33.m1.25.25.25.2.2.2.2" xref="S4.E33.m1.25.25.25.2.2.2.2.cmml">1</mn><mn id="S4.E33.m1.25.25.25.2.2.2.3" xref="S4.E33.m1.25.25.25.2.2.2.3.cmml">2</mn></mfrac><mo id="S4.E33.m1.61.61.10.56.31.31.31.4" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><mrow id="S4.E33.m1.61.61.10.56.31.31.31.3.3" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.26.26.26.3.3.3" maxsize="120%" minsize="120%" xref="S4.E33.m1.56.56.5.cmml">⟨</mo><mrow id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><msup id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.3" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.27.27.27.4.4.4" xref="S4.E33.m1.27.27.27.4.4.4.cmml">𝑮</mi><mrow id="S4.E33.m1.28.28.28.5.5.5.1" xref="S4.E33.m1.28.28.28.5.5.5.1.cmml"><mo id="S4.E33.m1.28.28.28.5.5.5.1a" xref="S4.E33.m1.28.28.28.5.5.5.1.cmml">−</mo><mn id="S4.E33.m1.28.28.28.5.5.5.1.2" xref="S4.E33.m1.28.28.28.5.5.5.1.2.cmml">1</mn></mrow></msup><mo id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.2" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><mrow id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.29.29.29.6.6.6" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">(</mo><mrow id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><msub id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.30.30.30.7.7.7" xref="S4.E33.m1.30.30.30.7.7.7.cmml">ϕ</mi><mi id="S4.E33.m1.31.31.31.8.8.8.1" xref="S4.E33.m1.31.31.31.8.8.8.1.cmml">n</mi></msub><mo id="S4.E33.m1.32.32.32.9.9.9" xref="S4.E33.m1.32.32.32.9.9.9.cmml">−</mo><mrow id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.1.1.1.2" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.33.33.33.10.10.10" xref="S4.E33.m1.33.33.33.10.10.10.cmml">𝑻</mi><mo id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.1.1.1.2.1" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><msub id="S4.E33.m1.59.59.8.54.29.29.29.1.1.1.1.1.1.2.2" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.34.34.34.11.11.11" xref="S4.E33.m1.34.34.34.11.11.11.cmml">ϕ</mi><mrow id="S4.E33.m1.35.35.35.12.12.12.1" xref="S4.E33.m1.35.35.35.12.12.12.1.cmml"><mi id="S4.E33.m1.35.35.35.12.12.12.1.2" xref="S4.E33.m1.35.35.35.12.12.12.1.2.cmml">n</mi><mo id="S4.E33.m1.35.35.35.12.12.12.1.1" xref="S4.E33.m1.35.35.35.12.12.12.1.1.cmml">−</mo><mn id="S4.E33.m1.35.35.35.12.12.12.1.3" xref="S4.E33.m1.35.35.35.12.12.12.1.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S4.E33.m1.36.36.36.13.13.13" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">)</mo></mrow></mrow><mo id="S4.E33.m1.37.37.37.14.14.14" maxsize="120%" minsize="120%" xref="S4.E33.m1.56.56.5.cmml">|</mo><msub id="S4.E33.m1.60.60.9.55.30.30.30.2.2.2" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.38.38.38.15.15.15" xref="S4.E33.m1.38.38.38.15.15.15.cmml">𝚲</mi><mi id="S4.E33.m1.39.39.39.16.16.16.1" xref="S4.E33.m1.39.39.39.16.16.16.1.cmml">a</mi></msub><mo id="S4.E33.m1.40.40.40.17.17.17" maxsize="120%" minsize="120%" xref="S4.E33.m1.56.56.5.cmml">|</mo><mrow id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3" xref="S4.E33.m1.56.56.5.cmml"><msup id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.3" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.41.41.41.18.18.18" xref="S4.E33.m1.41.41.41.18.18.18.cmml">𝑮</mi><mrow id="S4.E33.m1.42.42.42.19.19.19.1" xref="S4.E33.m1.42.42.42.19.19.19.1.cmml"><mo id="S4.E33.m1.42.42.42.19.19.19.1a" xref="S4.E33.m1.42.42.42.19.19.19.1.cmml">−</mo><mn id="S4.E33.m1.42.42.42.19.19.19.1.2" xref="S4.E33.m1.42.42.42.19.19.19.1.2.cmml">1</mn></mrow></msup><mo id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.2" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><mrow id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.1.1" xref="S4.E33.m1.56.56.5.cmml"><mo id="S4.E33.m1.43.43.43.20.20.20" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">(</mo><mrow id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><msub id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.1.1.1.1" xref="S4.E33.m1.56.56.5.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E33.m1.44.44.44.21.21.21" mathvariant="bold-italic" xref="S4.E33.m1.44.44.44.21.21.21.cmml">ϕ</mi><mi id="S4.E33.m1.45.45.45.22.22.22.1" xref="S4.E33.m1.45.45.45.22.22.22.1.cmml">n</mi></msub><mo id="S4.E33.m1.46.46.46.23.23.23" xref="S4.E33.m1.46.46.46.23.23.23.cmml">−</mo><mrow id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.1.1.1.2" xref="S4.E33.m1.56.56.5.cmml"><mi id="S4.E33.m1.47.47.47.24.24.24" xref="S4.E33.m1.47.47.47.24.24.24.cmml">𝑻</mi><mo id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.1.1.1.2.1" xref="S4.E33.m1.56.56.5.cmml">⁢</mo><msub id="S4.E33.m1.61.61.10.56.31.31.31.3.3.3.1.1.1.2.2" xref="S4.E33.m1.56.56.5.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E33.m1.48.48.48.25.25.25" mathvariant="bold-italic" xref="S4.E33.m1.48.48.48.25.25.25.cmml">ϕ</mi><mrow id="S4.E33.m1.49.49.49.26.26.26.1" xref="S4.E33.m1.49.49.49.26.26.26.1.cmml"><mi id="S4.E33.m1.49.49.49.26.26.26.1.2" xref="S4.E33.m1.49.49.49.26.26.26.1.2.cmml">n</mi><mo id="S4.E33.m1.49.49.49.26.26.26.1.1" xref="S4.E33.m1.49.49.49.26.26.26.1.1.cmml">−</mo><mn id="S4.E33.m1.49.49.49.26.26.26.1.3" xref="S4.E33.m1.49.49.49.26.26.26.1.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S4.E33.m1.50.50.50.27.27.27" stretchy="false" xref="S4.E33.m1.56.56.5.cmml">)</mo></mrow></mrow><mo id="S4.E33.m1.51.51.51.28.28.28" maxsize="120%" minsize="120%" xref="S4.E33.m1.56.56.5.cmml">⟩</mo></mrow></mrow></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E33.m1.61b"><apply id="S4.E33.m1.56.56.5.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="latexml" id="S4.E33.m1.15.15.15.15.15.15.cmml" xref="S4.E33.m1.15.15.15.15.15.15">proportional-to</csymbol><apply id="S4.E33.m1.52.52.1.1.2.cmml" xref="S4.E33.m1.61.61.10"><ln id="S4.E33.m1.1.1.1.1.1.1.cmml" xref="S4.E33.m1.1.1.1.1.1.1"></ln><apply id="S4.E33.m1.52.52.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.52.52.1.1.1.1.1.2.cmml" xref="S4.E33.m1.61.61.10"></times><ci id="S4.E33.m1.3.3.3.3.3.3.cmml" xref="S4.E33.m1.3.3.3.3.3.3">𝑝</ci><apply id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="latexml" id="S4.E33.m1.7.7.7.7.7.7.cmml" xref="S4.E33.m1.7.7.7.7.7.7">conditional</csymbol><apply id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.5.5.5.5.5.5.cmml" xref="S4.E33.m1.5.5.5.5.5.5">bold-italic-ϕ</ci><ci id="S4.E33.m1.6.6.6.6.6.6.1.cmml" xref="S4.E33.m1.6.6.6.6.6.6.1">𝑛</ci></apply><list id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E33.m1.61.61.10"><apply id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.8.8.8.8.8.8.cmml" xref="S4.E33.m1.8.8.8.8.8.8">bold-italic-ϕ</ci><apply id="S4.E33.m1.9.9.9.9.9.9.1.cmml" xref="S4.E33.m1.9.9.9.9.9.9.1"><minus id="S4.E33.m1.9.9.9.9.9.9.1.1.cmml" xref="S4.E33.m1.9.9.9.9.9.9.1.1"></minus><ci id="S4.E33.m1.9.9.9.9.9.9.1.2.cmml" xref="S4.E33.m1.9.9.9.9.9.9.1.2">𝑛</ci><cn id="S4.E33.m1.9.9.9.9.9.9.1.3.cmml" type="integer" xref="S4.E33.m1.9.9.9.9.9.9.1.3">1</cn></apply></apply><apply id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.52.52.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.11.11.11.11.11.11.cmml" xref="S4.E33.m1.11.11.11.11.11.11">𝚲</ci><ci id="S4.E33.m1.12.12.12.12.12.12.1.cmml" xref="S4.E33.m1.12.12.12.12.12.12.1">𝑎</ci></apply></list></apply></apply></apply><apply id="S4.E33.m1.56.56.5.5.cmml" xref="S4.E33.m1.61.61.10"><minus id="S4.E33.m1.24.24.24.1.1.1.cmml" xref="S4.E33.m1.61.61.10"></minus><apply id="S4.E33.m1.53.53.2.2.1.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.53.53.2.2.1.2.cmml" xref="S4.E33.m1.61.61.10"></times><apply id="S4.E33.m1.16.16.16.16.16.16.cmml" xref="S4.E33.m1.16.16.16.16.16.16"><divide id="S4.E33.m1.16.16.16.16.16.16.1.cmml" xref="S4.E33.m1.16.16.16.16.16.16"></divide><cn id="S4.E33.m1.16.16.16.16.16.16.2.cmml" type="integer" xref="S4.E33.m1.16.16.16.16.16.16.2">1</cn><cn id="S4.E33.m1.16.16.16.16.16.16.3.cmml" type="integer" xref="S4.E33.m1.16.16.16.16.16.16.3">2</cn></apply><apply id="S4.E33.m1.53.53.2.2.1.1.2.cmml" xref="S4.E33.m1.61.61.10"><ln id="S4.E33.m1.17.17.17.17.17.17.cmml" xref="S4.E33.m1.17.17.17.17.17.17"></ln><apply id="S4.E33.m1.53.53.2.2.1.1.1.1.1.2.cmml" xref="S4.E33.m1.61.61.10"><abs id="S4.E33.m1.53.53.2.2.1.1.1.1.1.2.1.cmml" xref="S4.E33.m1.61.61.10"></abs><apply id="S4.E33.m1.53.53.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.53.53.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.20.20.20.20.20.20.cmml" xref="S4.E33.m1.20.20.20.20.20.20">𝚲</ci><ci id="S4.E33.m1.21.21.21.21.21.21.1.cmml" xref="S4.E33.m1.21.21.21.21.21.21.1">𝑎</ci></apply></apply></apply></apply><apply id="S4.E33.m1.56.56.5.5.4.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.56.56.5.5.4.4.cmml" xref="S4.E33.m1.61.61.10"></times><apply id="S4.E33.m1.25.25.25.2.2.2.cmml" xref="S4.E33.m1.25.25.25.2.2.2"><divide id="S4.E33.m1.25.25.25.2.2.2.1.cmml" xref="S4.E33.m1.25.25.25.2.2.2"></divide><cn id="S4.E33.m1.25.25.25.2.2.2.2.cmml" type="integer" xref="S4.E33.m1.25.25.25.2.2.2.2">1</cn><cn id="S4.E33.m1.25.25.25.2.2.2.3.cmml" type="integer" xref="S4.E33.m1.25.25.25.2.2.2.3">2</cn></apply><apply id="S4.E33.m1.56.56.5.5.4.3.4.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="latexml" id="S4.E33.m1.56.56.5.5.4.3.4.1.cmml" xref="S4.E33.m1.61.61.10">quantum-operator-product</csymbol><apply id="S4.E33.m1.54.54.3.3.2.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.54.54.3.3.2.1.1.1.2.cmml" xref="S4.E33.m1.61.61.10"></times><apply id="S4.E33.m1.54.54.3.3.2.1.1.1.3.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.54.54.3.3.2.1.1.1.3.1.cmml" xref="S4.E33.m1.61.61.10">superscript</csymbol><ci id="S4.E33.m1.27.27.27.4.4.4.cmml" xref="S4.E33.m1.27.27.27.4.4.4">𝑮</ci><apply id="S4.E33.m1.28.28.28.5.5.5.1.cmml" xref="S4.E33.m1.28.28.28.5.5.5.1"><minus id="S4.E33.m1.28.28.28.5.5.5.1.1.cmml" xref="S4.E33.m1.28.28.28.5.5.5.1"></minus><cn id="S4.E33.m1.28.28.28.5.5.5.1.2.cmml" type="integer" xref="S4.E33.m1.28.28.28.5.5.5.1.2">1</cn></apply></apply><apply id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><minus id="S4.E33.m1.32.32.32.9.9.9.cmml" xref="S4.E33.m1.32.32.32.9.9.9"></minus><apply id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.2.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.2.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.30.30.30.7.7.7.cmml" xref="S4.E33.m1.30.30.30.7.7.7">italic-ϕ</ci><ci id="S4.E33.m1.31.31.31.8.8.8.1.cmml" xref="S4.E33.m1.31.31.31.8.8.8.1">𝑛</ci></apply><apply id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.3.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.3.1.cmml" xref="S4.E33.m1.61.61.10"></times><ci id="S4.E33.m1.33.33.33.10.10.10.cmml" xref="S4.E33.m1.33.33.33.10.10.10">𝑻</ci><apply id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.3.3.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.54.54.3.3.2.1.1.1.1.1.1.3.3.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.34.34.34.11.11.11.cmml" xref="S4.E33.m1.34.34.34.11.11.11">italic-ϕ</ci><apply id="S4.E33.m1.35.35.35.12.12.12.1.cmml" xref="S4.E33.m1.35.35.35.12.12.12.1"><minus id="S4.E33.m1.35.35.35.12.12.12.1.1.cmml" xref="S4.E33.m1.35.35.35.12.12.12.1.1"></minus><ci id="S4.E33.m1.35.35.35.12.12.12.1.2.cmml" xref="S4.E33.m1.35.35.35.12.12.12.1.2">𝑛</ci><cn id="S4.E33.m1.35.35.35.12.12.12.1.3.cmml" type="integer" xref="S4.E33.m1.35.35.35.12.12.12.1.3">1</cn></apply></apply></apply></apply></apply><apply id="S4.E33.m1.55.55.4.4.3.2.2.2.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.55.55.4.4.3.2.2.2.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.38.38.38.15.15.15.cmml" xref="S4.E33.m1.38.38.38.15.15.15">𝚲</ci><ci id="S4.E33.m1.39.39.39.16.16.16.1.cmml" xref="S4.E33.m1.39.39.39.16.16.16.1">𝑎</ci></apply><apply id="S4.E33.m1.56.56.5.5.4.3.3.3.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.56.56.5.5.4.3.3.3.2.cmml" xref="S4.E33.m1.61.61.10"></times><apply id="S4.E33.m1.56.56.5.5.4.3.3.3.3.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.56.56.5.5.4.3.3.3.3.1.cmml" xref="S4.E33.m1.61.61.10">superscript</csymbol><ci id="S4.E33.m1.41.41.41.18.18.18.cmml" xref="S4.E33.m1.41.41.41.18.18.18">𝑮</ci><apply id="S4.E33.m1.42.42.42.19.19.19.1.cmml" xref="S4.E33.m1.42.42.42.19.19.19.1"><minus id="S4.E33.m1.42.42.42.19.19.19.1.1.cmml" xref="S4.E33.m1.42.42.42.19.19.19.1"></minus><cn id="S4.E33.m1.42.42.42.19.19.19.1.2.cmml" type="integer" xref="S4.E33.m1.42.42.42.19.19.19.1.2">1</cn></apply></apply><apply id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.cmml" xref="S4.E33.m1.61.61.10"><minus id="S4.E33.m1.46.46.46.23.23.23.cmml" xref="S4.E33.m1.46.46.46.23.23.23"></minus><apply id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.2.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.2.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.44.44.44.21.21.21.cmml" xref="S4.E33.m1.44.44.44.21.21.21">bold-italic-ϕ</ci><ci id="S4.E33.m1.45.45.45.22.22.22.1.cmml" xref="S4.E33.m1.45.45.45.22.22.22.1">𝑛</ci></apply><apply id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.3.cmml" xref="S4.E33.m1.61.61.10"><times id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.3.1.cmml" xref="S4.E33.m1.61.61.10"></times><ci id="S4.E33.m1.47.47.47.24.24.24.cmml" xref="S4.E33.m1.47.47.47.24.24.24">𝑻</ci><apply id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.3.3.cmml" xref="S4.E33.m1.61.61.10"><csymbol cd="ambiguous" id="S4.E33.m1.56.56.5.5.4.3.3.3.1.1.1.3.3.1.cmml" xref="S4.E33.m1.61.61.10">subscript</csymbol><ci id="S4.E33.m1.48.48.48.25.25.25.cmml" xref="S4.E33.m1.48.48.48.25.25.25">bold-italic-ϕ</ci><apply id="S4.E33.m1.49.49.49.26.26.26.1.cmml" xref="S4.E33.m1.49.49.49.26.26.26.1"><minus id="S4.E33.m1.49.49.49.26.26.26.1.1.cmml" xref="S4.E33.m1.49.49.49.26.26.26.1.1"></minus><ci id="S4.E33.m1.49.49.49.26.26.26.1.2.cmml" xref="S4.E33.m1.49.49.49.26.26.26.1.2">𝑛</ci><cn id="S4.E33.m1.49.49.49.26.26.26.1.3.cmml" type="integer" xref="S4.E33.m1.49.49.49.26.26.26.1.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E33.m1.61c">\ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}))\propto\frac{1}{2}\ln(|% \bm{\Lambda}_{a}|)\\ -\frac{1}{2}\big{\langle}\bm{G}^{-1}(\phi_{n}-\bm{T}\phi_{n-1})\big{|}\bm{% \Lambda}_{a}\big{|}\bm{G}^{-1}(\bm{\phi}_{n}-\bm{T}\bm{\phi}_{n-1})\big{\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.E33.m1.61d">start_ROW start_CELL roman_ln ( italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) ∝ divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_ln ( | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | ) end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ⟨ bold_italic_G start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - bold_italic_T italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) | bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | bold_italic_G start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - bold_italic_T bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) ⟩ 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">(33)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p5.7">Taking the expectation w.r.t. <math alttext="\bm{\phi}_{n}" class="ltx_Math" display="inline" id="S4.p5.6.m1.1"><semantics id="S4.p5.6.m1.1a"><msub id="S4.p5.6.m1.1.1" xref="S4.p5.6.m1.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p5.6.m1.1.1.2" mathvariant="bold-italic" xref="S4.p5.6.m1.1.1.2.cmml">ϕ</mi><mi id="S4.p5.6.m1.1.1.3" xref="S4.p5.6.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p5.6.m1.1b"><apply id="S4.p5.6.m1.1.1.cmml" xref="S4.p5.6.m1.1.1"><csymbol cd="ambiguous" id="S4.p5.6.m1.1.1.1.cmml" xref="S4.p5.6.m1.1.1">subscript</csymbol><ci id="S4.p5.6.m1.1.1.2.cmml" xref="S4.p5.6.m1.1.1.2">bold-italic-ϕ</ci><ci id="S4.p5.6.m1.1.1.3.cmml" xref="S4.p5.6.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.6.m1.1c">\bm{\phi}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.6.m1.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bm{\phi}_{n-1}" class="ltx_Math" display="inline" id="S4.p5.7.m2.1"><semantics id="S4.p5.7.m2.1a"><msub id="S4.p5.7.m2.1.1" xref="S4.p5.7.m2.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.p5.7.m2.1.1.2" mathvariant="bold-italic" xref="S4.p5.7.m2.1.1.2.cmml">ϕ</mi><mrow id="S4.p5.7.m2.1.1.3" xref="S4.p5.7.m2.1.1.3.cmml"><mi id="S4.p5.7.m2.1.1.3.2" xref="S4.p5.7.m2.1.1.3.2.cmml">n</mi><mo id="S4.p5.7.m2.1.1.3.1" xref="S4.p5.7.m2.1.1.3.1.cmml">−</mo><mn id="S4.p5.7.m2.1.1.3.3" xref="S4.p5.7.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p5.7.m2.1b"><apply id="S4.p5.7.m2.1.1.cmml" xref="S4.p5.7.m2.1.1"><csymbol cd="ambiguous" id="S4.p5.7.m2.1.1.1.cmml" xref="S4.p5.7.m2.1.1">subscript</csymbol><ci id="S4.p5.7.m2.1.1.2.cmml" xref="S4.p5.7.m2.1.1.2">bold-italic-ϕ</ci><apply id="S4.p5.7.m2.1.1.3.cmml" xref="S4.p5.7.m2.1.1.3"><minus id="S4.p5.7.m2.1.1.3.1.cmml" xref="S4.p5.7.m2.1.1.3.1"></minus><ci id="S4.p5.7.m2.1.1.3.2.cmml" xref="S4.p5.7.m2.1.1.3.2">𝑛</ci><cn id="S4.p5.7.m2.1.1.3.3.cmml" type="integer" xref="S4.p5.7.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.7.m2.1c">\bm{\phi}_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.7.m2.1d">bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> yields</p> <table class="ltx_equation ltx_eqn_table" id="S4.E34"> <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="\mathbb{E}[\ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}))]\propto\frac% {1}{2}\sum_{k=1}^{4}\ln(|{\lambda}_{a,k}|)\\ -\frac{1}{2}\sum_{k=1}^{4}\mathbb{V}_{n,n-1,k}\lambda_{a,k}" class="ltx_Math" display="block" id="S4.E34.m1.42"><semantics id="S4.E34.m1.42a"><mtable displaystyle="true" id="S4.E34.m1.42.42.4" rowspacing="0pt" xref="S4.E34.m1.40.40.2.cmml"><mtr id="S4.E34.m1.42.42.4a" xref="S4.E34.m1.40.40.2.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E34.m1.42.42.4b" xref="S4.E34.m1.40.40.2.cmml"><mrow id="S4.E34.m1.42.42.4.40.31.31" xref="S4.E34.m1.40.40.2.cmml"><mrow id="S4.E34.m1.41.41.3.39.30.30.30" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.1.1.1.1.1.1" xref="S4.E34.m1.1.1.1.1.1.1.cmml">𝔼</mi><mo id="S4.E34.m1.41.41.3.39.30.30.30.2" xref="S4.E34.m1.40.40.2.cmml">⁢</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.2.2.2.2.2.2" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">[</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.3.3.3.3.3.3" xref="S4.E34.m1.3.3.3.3.3.3.cmml">ln</mi><mo id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1a" xref="S4.E34.m1.40.40.2.cmml">⁡</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.4.4.4.4.4.4" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">(</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.5.5.5.5.5.5" xref="S4.E34.m1.5.5.5.5.5.5.cmml">p</mi><mo id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.2" xref="S4.E34.m1.40.40.2.cmml">⁢</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.6.6.6.6.6.6" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">(</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><msub id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.1.1.1.3" xref="S4.E34.m1.40.40.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E34.m1.7.7.7.7.7.7" mathvariant="bold-italic" xref="S4.E34.m1.7.7.7.7.7.7.cmml">ϕ</mi><mi id="S4.E34.m1.8.8.8.8.8.8.1" xref="S4.E34.m1.8.8.8.8.8.8.1.cmml">n</mi></msub><mo fence="false" id="S4.E34.m1.9.9.9.9.9.9" xref="S4.E34.m1.9.9.9.9.9.9.cmml">|</mo><mrow id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E34.m1.40.40.2.cmml"><msub id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E34.m1.10.10.10.10.10.10" mathvariant="bold-italic" xref="S4.E34.m1.10.10.10.10.10.10.cmml">ϕ</mi><mrow id="S4.E34.m1.11.11.11.11.11.11.1" xref="S4.E34.m1.11.11.11.11.11.11.1.cmml"><mi id="S4.E34.m1.11.11.11.11.11.11.1.2" xref="S4.E34.m1.11.11.11.11.11.11.1.2.cmml">n</mi><mo id="S4.E34.m1.11.11.11.11.11.11.1.1" xref="S4.E34.m1.11.11.11.11.11.11.1.1.cmml">−</mo><mn id="S4.E34.m1.11.11.11.11.11.11.1.3" xref="S4.E34.m1.11.11.11.11.11.11.1.3.cmml">1</mn></mrow></msub><mo id="S4.E34.m1.12.12.12.12.12.12" xref="S4.E34.m1.40.40.2.cmml">,</mo><msub id="S4.E34.m1.41.41.3.39.30.30.30.1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.13.13.13.13.13.13" xref="S4.E34.m1.13.13.13.13.13.13.cmml">𝚲</mi><mi id="S4.E34.m1.14.14.14.14.14.14.1" xref="S4.E34.m1.14.14.14.14.14.14.1.cmml">a</mi></msub></mrow></mrow><mo id="S4.E34.m1.15.15.15.15.15.15" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">)</mo></mrow></mrow><mo id="S4.E34.m1.16.16.16.16.16.16" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">)</mo></mrow></mrow><mo id="S4.E34.m1.17.17.17.17.17.17" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">]</mo></mrow></mrow><mo id="S4.E34.m1.18.18.18.18.18.18" xref="S4.E34.m1.18.18.18.18.18.18.cmml">∝</mo><mrow id="S4.E34.m1.42.42.4.40.31.31.31" xref="S4.E34.m1.40.40.2.cmml"><mfrac id="S4.E34.m1.19.19.19.19.19.19" xref="S4.E34.m1.19.19.19.19.19.19.cmml"><mn id="S4.E34.m1.19.19.19.19.19.19.2" xref="S4.E34.m1.19.19.19.19.19.19.2.cmml">1</mn><mn id="S4.E34.m1.19.19.19.19.19.19.3" xref="S4.E34.m1.19.19.19.19.19.19.3.cmml">2</mn></mfrac><mo id="S4.E34.m1.42.42.4.40.31.31.31.2" xref="S4.E34.m1.40.40.2.cmml">⁢</mo><mrow id="S4.E34.m1.42.42.4.40.31.31.31.1" xref="S4.E34.m1.40.40.2.cmml"><munderover id="S4.E34.m1.42.42.4.40.31.31.31.1.2" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.20.20.20.20.20.20" movablelimits="false" xref="S4.E34.m1.20.20.20.20.20.20.cmml">∑</mo><mrow id="S4.E34.m1.21.21.21.21.21.21.1" xref="S4.E34.m1.21.21.21.21.21.21.1.cmml"><mi id="S4.E34.m1.21.21.21.21.21.21.1.2" xref="S4.E34.m1.21.21.21.21.21.21.1.2.cmml">k</mi><mo id="S4.E34.m1.21.21.21.21.21.21.1.1" xref="S4.E34.m1.21.21.21.21.21.21.1.1.cmml">=</mo><mn id="S4.E34.m1.21.21.21.21.21.21.1.3" xref="S4.E34.m1.21.21.21.21.21.21.1.3.cmml">1</mn></mrow><mn id="S4.E34.m1.22.22.22.22.22.22.1" xref="S4.E34.m1.22.22.22.22.22.22.1.cmml">4</mn></munderover><mrow id="S4.E34.m1.42.42.4.40.31.31.31.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.23.23.23.23.23.23" xref="S4.E34.m1.23.23.23.23.23.23.cmml">ln</mi><mo id="S4.E34.m1.42.42.4.40.31.31.31.1.1.1a" xref="S4.E34.m1.40.40.2.cmml">⁡</mo><mrow id="S4.E34.m1.42.42.4.40.31.31.31.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.24.24.24.24.24.24" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">(</mo><mrow id="S4.E34.m1.42.42.4.40.31.31.31.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.25.25.25.25.25.25" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">|</mo><msub id="S4.E34.m1.42.42.4.40.31.31.31.1.1.1.1.1.1.1" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.26.26.26.26.26.26" xref="S4.E34.m1.26.26.26.26.26.26.cmml">λ</mi><mrow id="S4.E34.m1.27.27.27.27.27.27.1.4" xref="S4.E34.m1.27.27.27.27.27.27.1.3.cmml"><mi id="S4.E34.m1.27.27.27.27.27.27.1.1" xref="S4.E34.m1.27.27.27.27.27.27.1.1.cmml">a</mi><mo id="S4.E34.m1.27.27.27.27.27.27.1.4.1" xref="S4.E34.m1.27.27.27.27.27.27.1.3.cmml">,</mo><mi id="S4.E34.m1.27.27.27.27.27.27.1.2" xref="S4.E34.m1.27.27.27.27.27.27.1.2.cmml">k</mi></mrow></msub><mo id="S4.E34.m1.28.28.28.28.28.28" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">|</mo></mrow><mo id="S4.E34.m1.29.29.29.29.29.29" stretchy="false" xref="S4.E34.m1.40.40.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E34.m1.42.42.4c" xref="S4.E34.m1.40.40.2.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E34.m1.42.42.4d" xref="S4.E34.m1.40.40.2.cmml"><mrow id="S4.E34.m1.38.38.38.9.9" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.38.38.38.9.9a" xref="S4.E34.m1.40.40.2.cmml">−</mo><mrow id="S4.E34.m1.38.38.38.9.9.10" xref="S4.E34.m1.40.40.2.cmml"><mfrac id="S4.E34.m1.31.31.31.2.2.2" xref="S4.E34.m1.31.31.31.2.2.2.cmml"><mn id="S4.E34.m1.31.31.31.2.2.2.2" xref="S4.E34.m1.31.31.31.2.2.2.2.cmml">1</mn><mn id="S4.E34.m1.31.31.31.2.2.2.3" xref="S4.E34.m1.31.31.31.2.2.2.3.cmml">2</mn></mfrac><mo id="S4.E34.m1.38.38.38.9.9.10.1" xref="S4.E34.m1.40.40.2.cmml">⁢</mo><mrow id="S4.E34.m1.38.38.38.9.9.10.2" xref="S4.E34.m1.40.40.2.cmml"><munderover id="S4.E34.m1.38.38.38.9.9.10.2.1" xref="S4.E34.m1.40.40.2.cmml"><mo id="S4.E34.m1.32.32.32.3.3.3" movablelimits="false" xref="S4.E34.m1.32.32.32.3.3.3.cmml">∑</mo><mrow id="S4.E34.m1.33.33.33.4.4.4.1" xref="S4.E34.m1.33.33.33.4.4.4.1.cmml"><mi id="S4.E34.m1.33.33.33.4.4.4.1.2" xref="S4.E34.m1.33.33.33.4.4.4.1.2.cmml">k</mi><mo id="S4.E34.m1.33.33.33.4.4.4.1.1" xref="S4.E34.m1.33.33.33.4.4.4.1.1.cmml">=</mo><mn id="S4.E34.m1.33.33.33.4.4.4.1.3" xref="S4.E34.m1.33.33.33.4.4.4.1.3.cmml">1</mn></mrow><mn id="S4.E34.m1.34.34.34.5.5.5.1" xref="S4.E34.m1.34.34.34.5.5.5.1.cmml">4</mn></munderover><mrow id="S4.E34.m1.38.38.38.9.9.10.2.2" xref="S4.E34.m1.40.40.2.cmml"><msub id="S4.E34.m1.38.38.38.9.9.10.2.2.2" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.35.35.35.6.6.6" xref="S4.E34.m1.35.35.35.6.6.6.cmml">𝕍</mi><mrow id="S4.E34.m1.36.36.36.7.7.7.1.3" xref="S4.E34.m1.36.36.36.7.7.7.1.4.cmml"><mi id="S4.E34.m1.36.36.36.7.7.7.1.1" xref="S4.E34.m1.36.36.36.7.7.7.1.1.cmml">n</mi><mo id="S4.E34.m1.36.36.36.7.7.7.1.3.2" xref="S4.E34.m1.36.36.36.7.7.7.1.4.cmml">,</mo><mrow id="S4.E34.m1.36.36.36.7.7.7.1.3.1" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.cmml"><mi id="S4.E34.m1.36.36.36.7.7.7.1.3.1.2" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.2.cmml">n</mi><mo id="S4.E34.m1.36.36.36.7.7.7.1.3.1.1" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.1.cmml">−</mo><mn id="S4.E34.m1.36.36.36.7.7.7.1.3.1.3" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.3.cmml">1</mn></mrow><mo id="S4.E34.m1.36.36.36.7.7.7.1.3.3" xref="S4.E34.m1.36.36.36.7.7.7.1.4.cmml">,</mo><mi id="S4.E34.m1.36.36.36.7.7.7.1.2" xref="S4.E34.m1.36.36.36.7.7.7.1.2.cmml">k</mi></mrow></msub><mo id="S4.E34.m1.38.38.38.9.9.10.2.2.1" xref="S4.E34.m1.40.40.2.cmml">⁢</mo><msub id="S4.E34.m1.38.38.38.9.9.10.2.2.3" xref="S4.E34.m1.40.40.2.cmml"><mi id="S4.E34.m1.37.37.37.8.8.8" xref="S4.E34.m1.37.37.37.8.8.8.cmml">λ</mi><mrow id="S4.E34.m1.38.38.38.9.9.9.1.4" xref="S4.E34.m1.38.38.38.9.9.9.1.3.cmml"><mi id="S4.E34.m1.38.38.38.9.9.9.1.1" xref="S4.E34.m1.38.38.38.9.9.9.1.1.cmml">a</mi><mo id="S4.E34.m1.38.38.38.9.9.9.1.4.1" xref="S4.E34.m1.38.38.38.9.9.9.1.3.cmml">,</mo><mi id="S4.E34.m1.38.38.38.9.9.9.1.2" xref="S4.E34.m1.38.38.38.9.9.9.1.2.cmml">k</mi></mrow></msub></mrow></mrow></mrow></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E34.m1.42b"><apply id="S4.E34.m1.40.40.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="latexml" id="S4.E34.m1.18.18.18.18.18.18.cmml" xref="S4.E34.m1.18.18.18.18.18.18">proportional-to</csymbol><apply id="S4.E34.m1.39.39.1.1.cmml" xref="S4.E34.m1.42.42.4"><times id="S4.E34.m1.39.39.1.1.2.cmml" xref="S4.E34.m1.42.42.4"></times><ci id="S4.E34.m1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.1.1.1.1.1.1">𝔼</ci><apply id="S4.E34.m1.39.39.1.1.1.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="latexml" id="S4.E34.m1.39.39.1.1.1.2.1.cmml" xref="S4.E34.m1.42.42.4">delimited-[]</csymbol><apply id="S4.E34.m1.39.39.1.1.1.1.1.2.cmml" xref="S4.E34.m1.42.42.4"><ln id="S4.E34.m1.3.3.3.3.3.3.cmml" xref="S4.E34.m1.3.3.3.3.3.3"></ln><apply id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.42.42.4"><times id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E34.m1.42.42.4"></times><ci id="S4.E34.m1.5.5.5.5.5.5.cmml" xref="S4.E34.m1.5.5.5.5.5.5">𝑝</ci><apply id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="latexml" id="S4.E34.m1.9.9.9.9.9.9.cmml" xref="S4.E34.m1.9.9.9.9.9.9">conditional</csymbol><apply id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><ci id="S4.E34.m1.7.7.7.7.7.7.cmml" xref="S4.E34.m1.7.7.7.7.7.7">bold-italic-ϕ</ci><ci id="S4.E34.m1.8.8.8.8.8.8.1.cmml" xref="S4.E34.m1.8.8.8.8.8.8.1">𝑛</ci></apply><list id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E34.m1.42.42.4"><apply id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><ci id="S4.E34.m1.10.10.10.10.10.10.cmml" xref="S4.E34.m1.10.10.10.10.10.10">bold-italic-ϕ</ci><apply id="S4.E34.m1.11.11.11.11.11.11.1.cmml" xref="S4.E34.m1.11.11.11.11.11.11.1"><minus id="S4.E34.m1.11.11.11.11.11.11.1.1.cmml" xref="S4.E34.m1.11.11.11.11.11.11.1.1"></minus><ci id="S4.E34.m1.11.11.11.11.11.11.1.2.cmml" xref="S4.E34.m1.11.11.11.11.11.11.1.2">𝑛</ci><cn id="S4.E34.m1.11.11.11.11.11.11.1.3.cmml" type="integer" xref="S4.E34.m1.11.11.11.11.11.11.1.3">1</cn></apply></apply><apply id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.39.39.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><ci id="S4.E34.m1.13.13.13.13.13.13.cmml" xref="S4.E34.m1.13.13.13.13.13.13">𝚲</ci><ci id="S4.E34.m1.14.14.14.14.14.14.1.cmml" xref="S4.E34.m1.14.14.14.14.14.14.1">𝑎</ci></apply></list></apply></apply></apply></apply></apply><apply id="S4.E34.m1.40.40.2.2.cmml" xref="S4.E34.m1.42.42.4"><minus id="S4.E34.m1.30.30.30.1.1.1.cmml" xref="S4.E34.m1.42.42.4"></minus><apply id="S4.E34.m1.40.40.2.2.1.cmml" xref="S4.E34.m1.42.42.4"><times id="S4.E34.m1.40.40.2.2.1.2.cmml" xref="S4.E34.m1.42.42.4"></times><apply id="S4.E34.m1.19.19.19.19.19.19.cmml" xref="S4.E34.m1.19.19.19.19.19.19"><divide id="S4.E34.m1.19.19.19.19.19.19.1.cmml" xref="S4.E34.m1.19.19.19.19.19.19"></divide><cn id="S4.E34.m1.19.19.19.19.19.19.2.cmml" type="integer" xref="S4.E34.m1.19.19.19.19.19.19.2">1</cn><cn id="S4.E34.m1.19.19.19.19.19.19.3.cmml" type="integer" xref="S4.E34.m1.19.19.19.19.19.19.3">2</cn></apply><apply id="S4.E34.m1.40.40.2.2.1.1.cmml" xref="S4.E34.m1.42.42.4"><apply id="S4.E34.m1.40.40.2.2.1.1.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.1.1.2.1.cmml" xref="S4.E34.m1.42.42.4">superscript</csymbol><apply id="S4.E34.m1.40.40.2.2.1.1.2.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.1.1.2.2.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><sum id="S4.E34.m1.20.20.20.20.20.20.cmml" xref="S4.E34.m1.20.20.20.20.20.20"></sum><apply id="S4.E34.m1.21.21.21.21.21.21.1.cmml" xref="S4.E34.m1.21.21.21.21.21.21.1"><eq id="S4.E34.m1.21.21.21.21.21.21.1.1.cmml" xref="S4.E34.m1.21.21.21.21.21.21.1.1"></eq><ci id="S4.E34.m1.21.21.21.21.21.21.1.2.cmml" xref="S4.E34.m1.21.21.21.21.21.21.1.2">𝑘</ci><cn id="S4.E34.m1.21.21.21.21.21.21.1.3.cmml" type="integer" xref="S4.E34.m1.21.21.21.21.21.21.1.3">1</cn></apply></apply><cn id="S4.E34.m1.22.22.22.22.22.22.1.cmml" type="integer" xref="S4.E34.m1.22.22.22.22.22.22.1">4</cn></apply><apply id="S4.E34.m1.40.40.2.2.1.1.1.2.cmml" xref="S4.E34.m1.42.42.4"><ln id="S4.E34.m1.23.23.23.23.23.23.cmml" xref="S4.E34.m1.23.23.23.23.23.23"></ln><apply id="S4.E34.m1.40.40.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E34.m1.42.42.4"><abs id="S4.E34.m1.40.40.2.2.1.1.1.1.1.1.2.1.cmml" xref="S4.E34.m1.42.42.4"></abs><apply id="S4.E34.m1.40.40.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><ci id="S4.E34.m1.26.26.26.26.26.26.cmml" xref="S4.E34.m1.26.26.26.26.26.26">𝜆</ci><list id="S4.E34.m1.27.27.27.27.27.27.1.3.cmml" xref="S4.E34.m1.27.27.27.27.27.27.1.4"><ci id="S4.E34.m1.27.27.27.27.27.27.1.1.cmml" xref="S4.E34.m1.27.27.27.27.27.27.1.1">𝑎</ci><ci id="S4.E34.m1.27.27.27.27.27.27.1.2.cmml" xref="S4.E34.m1.27.27.27.27.27.27.1.2">𝑘</ci></list></apply></apply></apply></apply></apply><apply id="S4.E34.m1.40.40.2.2.3.cmml" xref="S4.E34.m1.42.42.4"><times id="S4.E34.m1.40.40.2.2.3.1.cmml" xref="S4.E34.m1.42.42.4"></times><apply id="S4.E34.m1.31.31.31.2.2.2.cmml" xref="S4.E34.m1.31.31.31.2.2.2"><divide id="S4.E34.m1.31.31.31.2.2.2.1.cmml" xref="S4.E34.m1.31.31.31.2.2.2"></divide><cn id="S4.E34.m1.31.31.31.2.2.2.2.cmml" type="integer" xref="S4.E34.m1.31.31.31.2.2.2.2">1</cn><cn id="S4.E34.m1.31.31.31.2.2.2.3.cmml" type="integer" xref="S4.E34.m1.31.31.31.2.2.2.3">2</cn></apply><apply id="S4.E34.m1.40.40.2.2.3.3.cmml" xref="S4.E34.m1.42.42.4"><apply id="S4.E34.m1.40.40.2.2.3.3.1.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.3.3.1.1.cmml" xref="S4.E34.m1.42.42.4">superscript</csymbol><apply id="S4.E34.m1.40.40.2.2.3.3.1.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.3.3.1.2.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><sum id="S4.E34.m1.32.32.32.3.3.3.cmml" xref="S4.E34.m1.32.32.32.3.3.3"></sum><apply id="S4.E34.m1.33.33.33.4.4.4.1.cmml" xref="S4.E34.m1.33.33.33.4.4.4.1"><eq id="S4.E34.m1.33.33.33.4.4.4.1.1.cmml" xref="S4.E34.m1.33.33.33.4.4.4.1.1"></eq><ci id="S4.E34.m1.33.33.33.4.4.4.1.2.cmml" xref="S4.E34.m1.33.33.33.4.4.4.1.2">𝑘</ci><cn id="S4.E34.m1.33.33.33.4.4.4.1.3.cmml" type="integer" xref="S4.E34.m1.33.33.33.4.4.4.1.3">1</cn></apply></apply><cn id="S4.E34.m1.34.34.34.5.5.5.1.cmml" type="integer" xref="S4.E34.m1.34.34.34.5.5.5.1">4</cn></apply><apply id="S4.E34.m1.40.40.2.2.3.3.2.cmml" xref="S4.E34.m1.42.42.4"><times id="S4.E34.m1.40.40.2.2.3.3.2.1.cmml" xref="S4.E34.m1.42.42.4"></times><apply id="S4.E34.m1.40.40.2.2.3.3.2.2.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.3.3.2.2.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><ci id="S4.E34.m1.35.35.35.6.6.6.cmml" xref="S4.E34.m1.35.35.35.6.6.6">𝕍</ci><list id="S4.E34.m1.36.36.36.7.7.7.1.4.cmml" xref="S4.E34.m1.36.36.36.7.7.7.1.3"><ci id="S4.E34.m1.36.36.36.7.7.7.1.1.cmml" xref="S4.E34.m1.36.36.36.7.7.7.1.1">𝑛</ci><apply id="S4.E34.m1.36.36.36.7.7.7.1.3.1.cmml" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1"><minus id="S4.E34.m1.36.36.36.7.7.7.1.3.1.1.cmml" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.1"></minus><ci id="S4.E34.m1.36.36.36.7.7.7.1.3.1.2.cmml" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.2">𝑛</ci><cn id="S4.E34.m1.36.36.36.7.7.7.1.3.1.3.cmml" type="integer" xref="S4.E34.m1.36.36.36.7.7.7.1.3.1.3">1</cn></apply><ci id="S4.E34.m1.36.36.36.7.7.7.1.2.cmml" xref="S4.E34.m1.36.36.36.7.7.7.1.2">𝑘</ci></list></apply><apply id="S4.E34.m1.40.40.2.2.3.3.2.3.cmml" xref="S4.E34.m1.42.42.4"><csymbol cd="ambiguous" id="S4.E34.m1.40.40.2.2.3.3.2.3.1.cmml" xref="S4.E34.m1.42.42.4">subscript</csymbol><ci id="S4.E34.m1.37.37.37.8.8.8.cmml" xref="S4.E34.m1.37.37.37.8.8.8">𝜆</ci><list id="S4.E34.m1.38.38.38.9.9.9.1.3.cmml" xref="S4.E34.m1.38.38.38.9.9.9.1.4"><ci id="S4.E34.m1.38.38.38.9.9.9.1.1.cmml" xref="S4.E34.m1.38.38.38.9.9.9.1.1">𝑎</ci><ci id="S4.E34.m1.38.38.38.9.9.9.1.2.cmml" xref="S4.E34.m1.38.38.38.9.9.9.1.2">𝑘</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E34.m1.42c">\mathbb{E}[\ln(p(\bm{\phi}_{n}|\bm{\phi}_{n-1},\bm{\Lambda}_{a}))]\propto\frac% {1}{2}\sum_{k=1}^{4}\ln(|{\lambda}_{a,k}|)\\ -\frac{1}{2}\sum_{k=1}^{4}\mathbb{V}_{n,n-1,k}\lambda_{a,k}</annotation><annotation encoding="application/x-llamapun" id="S4.E34.m1.42d">start_ROW start_CELL blackboard_E [ roman_ln ( italic_p ( bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | bold_italic_ϕ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) ] ∝ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_ln ( | italic_λ start_POSTSUBSCRIPT italic_a , italic_k end_POSTSUBSCRIPT | ) end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT blackboard_V start_POSTSUBSCRIPT italic_n , italic_n - 1 , italic_k end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_a , italic_k end_POSTSUBSCRIPT 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">(34)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p5.8">with <math alttext="\bm{\lambda}_{a}=\text{diag}(\bm{\Lambda}_{a})" class="ltx_Math" display="inline" id="S4.p5.8.m1.1"><semantics id="S4.p5.8.m1.1a"><mrow id="S4.p5.8.m1.1.1" xref="S4.p5.8.m1.1.1.cmml"><msub id="S4.p5.8.m1.1.1.3" xref="S4.p5.8.m1.1.1.3.cmml"><mi id="S4.p5.8.m1.1.1.3.2" xref="S4.p5.8.m1.1.1.3.2.cmml">𝝀</mi><mi id="S4.p5.8.m1.1.1.3.3" xref="S4.p5.8.m1.1.1.3.3.cmml">a</mi></msub><mo id="S4.p5.8.m1.1.1.2" xref="S4.p5.8.m1.1.1.2.cmml">=</mo><mrow id="S4.p5.8.m1.1.1.1" xref="S4.p5.8.m1.1.1.1.cmml"><mtext id="S4.p5.8.m1.1.1.1.3" xref="S4.p5.8.m1.1.1.1.3a.cmml">diag</mtext><mo id="S4.p5.8.m1.1.1.1.2" xref="S4.p5.8.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.p5.8.m1.1.1.1.1.1" xref="S4.p5.8.m1.1.1.1.1.1.1.cmml"><mo id="S4.p5.8.m1.1.1.1.1.1.2" stretchy="false" xref="S4.p5.8.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.p5.8.m1.1.1.1.1.1.1" xref="S4.p5.8.m1.1.1.1.1.1.1.cmml"><mi id="S4.p5.8.m1.1.1.1.1.1.1.2" xref="S4.p5.8.m1.1.1.1.1.1.1.2.cmml">𝚲</mi><mi id="S4.p5.8.m1.1.1.1.1.1.1.3" xref="S4.p5.8.m1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.p5.8.m1.1.1.1.1.1.3" stretchy="false" xref="S4.p5.8.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.8.m1.1b"><apply id="S4.p5.8.m1.1.1.cmml" xref="S4.p5.8.m1.1.1"><eq id="S4.p5.8.m1.1.1.2.cmml" xref="S4.p5.8.m1.1.1.2"></eq><apply id="S4.p5.8.m1.1.1.3.cmml" xref="S4.p5.8.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p5.8.m1.1.1.3.1.cmml" xref="S4.p5.8.m1.1.1.3">subscript</csymbol><ci id="S4.p5.8.m1.1.1.3.2.cmml" xref="S4.p5.8.m1.1.1.3.2">𝝀</ci><ci id="S4.p5.8.m1.1.1.3.3.cmml" xref="S4.p5.8.m1.1.1.3.3">𝑎</ci></apply><apply id="S4.p5.8.m1.1.1.1.cmml" xref="S4.p5.8.m1.1.1.1"><times id="S4.p5.8.m1.1.1.1.2.cmml" xref="S4.p5.8.m1.1.1.1.2"></times><ci id="S4.p5.8.m1.1.1.1.3a.cmml" xref="S4.p5.8.m1.1.1.1.3"><mtext id="S4.p5.8.m1.1.1.1.3.cmml" xref="S4.p5.8.m1.1.1.1.3">diag</mtext></ci><apply id="S4.p5.8.m1.1.1.1.1.1.1.cmml" xref="S4.p5.8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p5.8.m1.1.1.1.1.1.1.1.cmml" xref="S4.p5.8.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.p5.8.m1.1.1.1.1.1.1.2.cmml" xref="S4.p5.8.m1.1.1.1.1.1.1.2">𝚲</ci><ci id="S4.p5.8.m1.1.1.1.1.1.1.3.cmml" xref="S4.p5.8.m1.1.1.1.1.1.1.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.8.m1.1c">\bm{\lambda}_{a}=\text{diag}(\bm{\Lambda}_{a})</annotation><annotation encoding="application/x-llamapun" id="S4.p5.8.m1.1d">bold_italic_λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = diag ( bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT )</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S4.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="\mathbb{V}_{n,n-1,i}=\left\|\bm{G}^{-1}\left(\bar{\bm{\phi}}_{n,i}-\bm{T}\bar{% \bm{\phi}}_{n-1,i}\right)\right\|^{2}\\ +\bm{G}^{-1}\left(\bar{\bar{\bm{\phi}}}_{n,i,i}+\bm{T}\bar{\bar{\bm{\phi}}}_{n% -1,i,i}\bm{T}^{\top}\right)\bm{G}^{-\top}," class="ltx_Math" display="block" id="S4.E35.m1.35"><semantics id="S4.E35.m1.35a"><mtable displaystyle="true" id="S4.E35.m1.35.35.3" rowspacing="0pt"><mtr id="S4.E35.m1.35.35.3a"><mtd class="ltx_align_left" columnalign="left" id="S4.E35.m1.35.35.3b"><mrow id="S4.E35.m1.34.34.2.33.17.17"><msub id="S4.E35.m1.34.34.2.33.17.17.18"><mi id="S4.E35.m1.1.1.1.1.1.1" xref="S4.E35.m1.1.1.1.1.1.1.cmml">𝕍</mi><mrow id="S4.E35.m1.2.2.2.2.2.2.1.3" xref="S4.E35.m1.2.2.2.2.2.2.1.4.cmml"><mi id="S4.E35.m1.2.2.2.2.2.2.1.1" xref="S4.E35.m1.2.2.2.2.2.2.1.1.cmml">n</mi><mo id="S4.E35.m1.2.2.2.2.2.2.1.3.2" xref="S4.E35.m1.2.2.2.2.2.2.1.4.cmml">,</mo><mrow id="S4.E35.m1.2.2.2.2.2.2.1.3.1" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.cmml"><mi id="S4.E35.m1.2.2.2.2.2.2.1.3.1.2" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.2.cmml">n</mi><mo id="S4.E35.m1.2.2.2.2.2.2.1.3.1.1" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.1.cmml">−</mo><mn id="S4.E35.m1.2.2.2.2.2.2.1.3.1.3" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.3.cmml">1</mn></mrow><mo id="S4.E35.m1.2.2.2.2.2.2.1.3.3" xref="S4.E35.m1.2.2.2.2.2.2.1.4.cmml">,</mo><mi id="S4.E35.m1.2.2.2.2.2.2.1.2" xref="S4.E35.m1.2.2.2.2.2.2.1.2.cmml">i</mi></mrow></msub><mo id="S4.E35.m1.3.3.3.3.3.3" xref="S4.E35.m1.3.3.3.3.3.3.cmml">=</mo><msup id="S4.E35.m1.34.34.2.33.17.17.17"><mrow id="S4.E35.m1.34.34.2.33.17.17.17.1.1"><mo id="S4.E35.m1.4.4.4.4.4.4" xref="S4.E35.m1.33.33.1.1.1.cmml">‖</mo><mrow id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1"><msup id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.3"><mi id="S4.E35.m1.5.5.5.5.5.5" xref="S4.E35.m1.5.5.5.5.5.5.cmml">𝑮</mi><mrow id="S4.E35.m1.6.6.6.6.6.6.1" xref="S4.E35.m1.6.6.6.6.6.6.1.cmml"><mo id="S4.E35.m1.6.6.6.6.6.6.1a" xref="S4.E35.m1.6.6.6.6.6.6.1.cmml">−</mo><mn id="S4.E35.m1.6.6.6.6.6.6.1.2" xref="S4.E35.m1.6.6.6.6.6.6.1.2.cmml">1</mn></mrow></msup><mo id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.2" xref="S4.E35.m1.33.33.1.1.1.cmml">⁢</mo><mrow id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.1.1"><mo id="S4.E35.m1.7.7.7.7.7.7" xref="S4.E35.m1.33.33.1.1.1.cmml">(</mo><mrow id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.1.1.1"><msub id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.1.1.1.1"><mover accent="true" id="S4.E35.m1.8.8.8.8.8.8" xref="S4.E35.m1.8.8.8.8.8.8.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E35.m1.8.8.8.8.8.8.2" mathvariant="bold-italic" xref="S4.E35.m1.8.8.8.8.8.8.2.cmml">ϕ</mi><mo id="S4.E35.m1.8.8.8.8.8.8.1" xref="S4.E35.m1.8.8.8.8.8.8.1.cmml">¯</mo></mover><mrow id="S4.E35.m1.9.9.9.9.9.9.1.4" xref="S4.E35.m1.9.9.9.9.9.9.1.3.cmml"><mi id="S4.E35.m1.9.9.9.9.9.9.1.1" xref="S4.E35.m1.9.9.9.9.9.9.1.1.cmml">n</mi><mo id="S4.E35.m1.9.9.9.9.9.9.1.4.1" xref="S4.E35.m1.9.9.9.9.9.9.1.3.cmml">,</mo><mi id="S4.E35.m1.9.9.9.9.9.9.1.2" xref="S4.E35.m1.9.9.9.9.9.9.1.2.cmml">i</mi></mrow></msub><mo id="S4.E35.m1.10.10.10.10.10.10" xref="S4.E35.m1.10.10.10.10.10.10.cmml">−</mo><mrow id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.1.1.1.2"><mi id="S4.E35.m1.11.11.11.11.11.11" xref="S4.E35.m1.11.11.11.11.11.11.cmml">𝑻</mi><mo id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.1.1.1.2.1" xref="S4.E35.m1.33.33.1.1.1.cmml">⁢</mo><msub id="S4.E35.m1.34.34.2.33.17.17.17.1.1.1.1.1.1.2.2"><mover accent="true" id="S4.E35.m1.12.12.12.12.12.12" xref="S4.E35.m1.12.12.12.12.12.12.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E35.m1.12.12.12.12.12.12.2" mathvariant="bold-italic" xref="S4.E35.m1.12.12.12.12.12.12.2.cmml">ϕ</mi><mo id="S4.E35.m1.12.12.12.12.12.12.1" xref="S4.E35.m1.12.12.12.12.12.12.1.cmml">¯</mo></mover><mrow id="S4.E35.m1.13.13.13.13.13.13.1.2" xref="S4.E35.m1.13.13.13.13.13.13.1.3.cmml"><mrow id="S4.E35.m1.13.13.13.13.13.13.1.2.1" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.cmml"><mi id="S4.E35.m1.13.13.13.13.13.13.1.2.1.2" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.2.cmml">n</mi><mo id="S4.E35.m1.13.13.13.13.13.13.1.2.1.1" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.1.cmml">−</mo><mn id="S4.E35.m1.13.13.13.13.13.13.1.2.1.3" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.3.cmml">1</mn></mrow><mo id="S4.E35.m1.13.13.13.13.13.13.1.2.2" xref="S4.E35.m1.13.13.13.13.13.13.1.3.cmml">,</mo><mi id="S4.E35.m1.13.13.13.13.13.13.1.1" xref="S4.E35.m1.13.13.13.13.13.13.1.1.cmml">i</mi></mrow></msub></mrow></mrow><mo id="S4.E35.m1.14.14.14.14.14.14" xref="S4.E35.m1.33.33.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E35.m1.15.15.15.15.15.15" xref="S4.E35.m1.33.33.1.1.1.cmml">‖</mo></mrow><mn id="S4.E35.m1.16.16.16.16.16.16.1" xref="S4.E35.m1.16.16.16.16.16.16.1.cmml">2</mn></msup></mrow></mtd></mtr><mtr id="S4.E35.m1.35.35.3c"><mtd class="ltx_align_right" columnalign="right" id="S4.E35.m1.35.35.3d"><mrow id="S4.E35.m1.35.35.3.34.17.17.17"><mrow id="S4.E35.m1.35.35.3.34.17.17.17.1"><mo id="S4.E35.m1.35.35.3.34.17.17.17.1a" xref="S4.E35.m1.33.33.1.1.1.cmml">+</mo><mrow id="S4.E35.m1.35.35.3.34.17.17.17.1.1"><msup id="S4.E35.m1.35.35.3.34.17.17.17.1.1.3"><mi id="S4.E35.m1.18.18.18.2.2.2" xref="S4.E35.m1.18.18.18.2.2.2.cmml">𝑮</mi><mrow id="S4.E35.m1.19.19.19.3.3.3.1" xref="S4.E35.m1.19.19.19.3.3.3.1.cmml"><mo id="S4.E35.m1.19.19.19.3.3.3.1a" xref="S4.E35.m1.19.19.19.3.3.3.1.cmml">−</mo><mn id="S4.E35.m1.19.19.19.3.3.3.1.2" xref="S4.E35.m1.19.19.19.3.3.3.1.2.cmml">1</mn></mrow></msup><mo id="S4.E35.m1.35.35.3.34.17.17.17.1.1.2" xref="S4.E35.m1.33.33.1.1.1.cmml">⁢</mo><mrow id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1"><mo id="S4.E35.m1.20.20.20.4.4.4" xref="S4.E35.m1.33.33.1.1.1.cmml">(</mo><mrow id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1"><msub id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1.1"><mover accent="true" id="S4.E35.m1.21.21.21.5.5.5" xref="S4.E35.m1.21.21.21.5.5.5.cmml"><mover accent="true" id="S4.E35.m1.21.21.21.5.5.5.2" xref="S4.E35.m1.21.21.21.5.5.5.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E35.m1.21.21.21.5.5.5.2.2" mathvariant="bold-italic" xref="S4.E35.m1.21.21.21.5.5.5.2.2.cmml">ϕ</mi><mo id="S4.E35.m1.21.21.21.5.5.5.2.1" xref="S4.E35.m1.21.21.21.5.5.5.2.1.cmml">¯</mo></mover><mo id="S4.E35.m1.21.21.21.5.5.5.1" xref="S4.E35.m1.21.21.21.5.5.5.1.cmml">¯</mo></mover><mrow id="S4.E35.m1.22.22.22.6.6.6.1.5" xref="S4.E35.m1.22.22.22.6.6.6.1.4.cmml"><mi id="S4.E35.m1.22.22.22.6.6.6.1.1" xref="S4.E35.m1.22.22.22.6.6.6.1.1.cmml">n</mi><mo id="S4.E35.m1.22.22.22.6.6.6.1.5.1" xref="S4.E35.m1.22.22.22.6.6.6.1.4.cmml">,</mo><mi id="S4.E35.m1.22.22.22.6.6.6.1.2" xref="S4.E35.m1.22.22.22.6.6.6.1.2.cmml">i</mi><mo id="S4.E35.m1.22.22.22.6.6.6.1.5.2" xref="S4.E35.m1.22.22.22.6.6.6.1.4.cmml">,</mo><mi id="S4.E35.m1.22.22.22.6.6.6.1.3" xref="S4.E35.m1.22.22.22.6.6.6.1.3.cmml">i</mi></mrow></msub><mo id="S4.E35.m1.23.23.23.7.7.7" xref="S4.E35.m1.23.23.23.7.7.7.cmml">+</mo><mrow id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1.2"><mi id="S4.E35.m1.24.24.24.8.8.8" xref="S4.E35.m1.24.24.24.8.8.8.cmml">𝑻</mi><mo id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1.2.1" xref="S4.E35.m1.33.33.1.1.1.cmml">⁢</mo><msub id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1.2.2"><mover accent="true" id="S4.E35.m1.25.25.25.9.9.9" xref="S4.E35.m1.25.25.25.9.9.9.cmml"><mover accent="true" id="S4.E35.m1.25.25.25.9.9.9.2" xref="S4.E35.m1.25.25.25.9.9.9.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S4.E35.m1.25.25.25.9.9.9.2.2" mathvariant="bold-italic" xref="S4.E35.m1.25.25.25.9.9.9.2.2.cmml">ϕ</mi><mo id="S4.E35.m1.25.25.25.9.9.9.2.1" xref="S4.E35.m1.25.25.25.9.9.9.2.1.cmml">¯</mo></mover><mo id="S4.E35.m1.25.25.25.9.9.9.1" xref="S4.E35.m1.25.25.25.9.9.9.1.cmml">¯</mo></mover><mrow id="S4.E35.m1.26.26.26.10.10.10.1.3" xref="S4.E35.m1.26.26.26.10.10.10.1.4.cmml"><mrow id="S4.E35.m1.26.26.26.10.10.10.1.3.1" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.cmml"><mi id="S4.E35.m1.26.26.26.10.10.10.1.3.1.2" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.2.cmml">n</mi><mo id="S4.E35.m1.26.26.26.10.10.10.1.3.1.1" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.1.cmml">−</mo><mn id="S4.E35.m1.26.26.26.10.10.10.1.3.1.3" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.3.cmml">1</mn></mrow><mo id="S4.E35.m1.26.26.26.10.10.10.1.3.2" xref="S4.E35.m1.26.26.26.10.10.10.1.4.cmml">,</mo><mi id="S4.E35.m1.26.26.26.10.10.10.1.1" xref="S4.E35.m1.26.26.26.10.10.10.1.1.cmml">i</mi><mo id="S4.E35.m1.26.26.26.10.10.10.1.3.3" xref="S4.E35.m1.26.26.26.10.10.10.1.4.cmml">,</mo><mi id="S4.E35.m1.26.26.26.10.10.10.1.2" xref="S4.E35.m1.26.26.26.10.10.10.1.2.cmml">i</mi></mrow></msub><mo id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1.2.1a" xref="S4.E35.m1.33.33.1.1.1.cmml">⁢</mo><msup id="S4.E35.m1.35.35.3.34.17.17.17.1.1.1.1.1.2.3"><mi id="S4.E35.m1.27.27.27.11.11.11" xref="S4.E35.m1.27.27.27.11.11.11.cmml">𝑻</mi><mo id="S4.E35.m1.28.28.28.12.12.12.1" xref="S4.E35.m1.28.28.28.12.12.12.1.cmml">⊤</mo></msup></mrow></mrow><mo id="S4.E35.m1.29.29.29.13.13.13" xref="S4.E35.m1.33.33.1.1.1.cmml">)</mo></mrow><mo id="S4.E35.m1.35.35.3.34.17.17.17.1.1.2a" xref="S4.E35.m1.33.33.1.1.1.cmml">⁢</mo><msup id="S4.E35.m1.35.35.3.34.17.17.17.1.1.4"><mi id="S4.E35.m1.30.30.30.14.14.14" xref="S4.E35.m1.30.30.30.14.14.14.cmml">𝑮</mi><mrow id="S4.E35.m1.31.31.31.15.15.15.1" xref="S4.E35.m1.31.31.31.15.15.15.1.cmml"><mi id="S4.E35.m1.31.31.31.15.15.15.1.2" xref="S4.E35.m1.31.31.31.15.15.15.1.2.cmml"></mi><mo id="S4.E35.m1.31.31.31.15.15.15.1.1" rspace="0em" xref="S4.E35.m1.31.31.31.15.15.15.1.1.cmml">−</mo><mo id="S4.E35.m1.31.31.31.15.15.15.1.3" lspace="0em" xref="S4.E35.m1.31.31.31.15.15.15.1.3.cmml">⊤</mo></mrow></msup></mrow></mrow><mo id="S4.E35.m1.32.32.32.16.16.16" xref="S4.E35.m1.33.33.1.1.1.cmml">,</mo></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E35.m1.35b"><apply id="S4.E35.m1.33.33.1.1.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><eq id="S4.E35.m1.3.3.3.3.3.3.cmml" xref="S4.E35.m1.3.3.3.3.3.3"></eq><apply id="S4.E35.m1.33.33.1.1.1.4.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.4.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">subscript</csymbol><ci id="S4.E35.m1.1.1.1.1.1.1.cmml" xref="S4.E35.m1.1.1.1.1.1.1">𝕍</ci><list id="S4.E35.m1.2.2.2.2.2.2.1.4.cmml" xref="S4.E35.m1.2.2.2.2.2.2.1.3"><ci id="S4.E35.m1.2.2.2.2.2.2.1.1.cmml" xref="S4.E35.m1.2.2.2.2.2.2.1.1">𝑛</ci><apply id="S4.E35.m1.2.2.2.2.2.2.1.3.1.cmml" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1"><minus id="S4.E35.m1.2.2.2.2.2.2.1.3.1.1.cmml" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.1"></minus><ci id="S4.E35.m1.2.2.2.2.2.2.1.3.1.2.cmml" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.2">𝑛</ci><cn id="S4.E35.m1.2.2.2.2.2.2.1.3.1.3.cmml" type="integer" xref="S4.E35.m1.2.2.2.2.2.2.1.3.1.3">1</cn></apply><ci id="S4.E35.m1.2.2.2.2.2.2.1.2.cmml" xref="S4.E35.m1.2.2.2.2.2.2.1.2">𝑖</ci></list></apply><apply id="S4.E35.m1.33.33.1.1.1.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><plus id="S4.E35.m1.17.17.17.1.1.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"></plus><apply id="S4.E35.m1.33.33.1.1.1.1.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.1.1.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4">superscript</csymbol><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="latexml" id="S4.E35.m1.33.33.1.1.1.1.1.1.2.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">delimited-∥∥</csymbol><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><times id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"></times><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">superscript</csymbol><ci id="S4.E35.m1.5.5.5.5.5.5.cmml" xref="S4.E35.m1.5.5.5.5.5.5">𝑮</ci><apply id="S4.E35.m1.6.6.6.6.6.6.1.cmml" xref="S4.E35.m1.6.6.6.6.6.6.1"><minus id="S4.E35.m1.6.6.6.6.6.6.1.1.cmml" xref="S4.E35.m1.6.6.6.6.6.6.1"></minus><cn id="S4.E35.m1.6.6.6.6.6.6.1.2.cmml" type="integer" xref="S4.E35.m1.6.6.6.6.6.6.1.2">1</cn></apply></apply><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><minus id="S4.E35.m1.10.10.10.10.10.10.cmml" xref="S4.E35.m1.10.10.10.10.10.10"></minus><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">subscript</csymbol><apply id="S4.E35.m1.8.8.8.8.8.8.cmml" xref="S4.E35.m1.8.8.8.8.8.8"><ci id="S4.E35.m1.8.8.8.8.8.8.1.cmml" xref="S4.E35.m1.8.8.8.8.8.8.1">¯</ci><ci id="S4.E35.m1.8.8.8.8.8.8.2.cmml" xref="S4.E35.m1.8.8.8.8.8.8.2">bold-italic-ϕ</ci></apply><list id="S4.E35.m1.9.9.9.9.9.9.1.3.cmml" xref="S4.E35.m1.9.9.9.9.9.9.1.4"><ci id="S4.E35.m1.9.9.9.9.9.9.1.1.cmml" xref="S4.E35.m1.9.9.9.9.9.9.1.1">𝑛</ci><ci id="S4.E35.m1.9.9.9.9.9.9.1.2.cmml" xref="S4.E35.m1.9.9.9.9.9.9.1.2">𝑖</ci></list></apply><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><times id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"></times><ci id="S4.E35.m1.11.11.11.11.11.11.cmml" xref="S4.E35.m1.11.11.11.11.11.11">𝑻</ci><apply id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">subscript</csymbol><apply id="S4.E35.m1.12.12.12.12.12.12.cmml" xref="S4.E35.m1.12.12.12.12.12.12"><ci id="S4.E35.m1.12.12.12.12.12.12.1.cmml" xref="S4.E35.m1.12.12.12.12.12.12.1">¯</ci><ci id="S4.E35.m1.12.12.12.12.12.12.2.cmml" xref="S4.E35.m1.12.12.12.12.12.12.2">bold-italic-ϕ</ci></apply><list id="S4.E35.m1.13.13.13.13.13.13.1.3.cmml" xref="S4.E35.m1.13.13.13.13.13.13.1.2"><apply id="S4.E35.m1.13.13.13.13.13.13.1.2.1.cmml" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1"><minus id="S4.E35.m1.13.13.13.13.13.13.1.2.1.1.cmml" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.1"></minus><ci id="S4.E35.m1.13.13.13.13.13.13.1.2.1.2.cmml" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.2">𝑛</ci><cn id="S4.E35.m1.13.13.13.13.13.13.1.2.1.3.cmml" type="integer" xref="S4.E35.m1.13.13.13.13.13.13.1.2.1.3">1</cn></apply><ci id="S4.E35.m1.13.13.13.13.13.13.1.1.cmml" xref="S4.E35.m1.13.13.13.13.13.13.1.1">𝑖</ci></list></apply></apply></apply></apply></apply><cn id="S4.E35.m1.16.16.16.16.16.16.1.cmml" type="integer" xref="S4.E35.m1.16.16.16.16.16.16.1">2</cn></apply><apply id="S4.E35.m1.33.33.1.1.1.2.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><times id="S4.E35.m1.33.33.1.1.1.2.2.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"></times><apply id="S4.E35.m1.33.33.1.1.1.2.2.3.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.2.2.3.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">superscript</csymbol><ci id="S4.E35.m1.18.18.18.2.2.2.cmml" xref="S4.E35.m1.18.18.18.2.2.2">𝑮</ci><apply id="S4.E35.m1.19.19.19.3.3.3.1.cmml" xref="S4.E35.m1.19.19.19.3.3.3.1"><minus id="S4.E35.m1.19.19.19.3.3.3.1.1.cmml" xref="S4.E35.m1.19.19.19.3.3.3.1"></minus><cn id="S4.E35.m1.19.19.19.3.3.3.1.2.cmml" type="integer" xref="S4.E35.m1.19.19.19.3.3.3.1.2">1</cn></apply></apply><apply id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><plus id="S4.E35.m1.23.23.23.7.7.7.cmml" xref="S4.E35.m1.23.23.23.7.7.7"></plus><apply id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.2.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.2.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">subscript</csymbol><apply id="S4.E35.m1.21.21.21.5.5.5.cmml" xref="S4.E35.m1.21.21.21.5.5.5"><ci id="S4.E35.m1.21.21.21.5.5.5.1.cmml" xref="S4.E35.m1.21.21.21.5.5.5.1">¯</ci><apply id="S4.E35.m1.21.21.21.5.5.5.2.cmml" xref="S4.E35.m1.21.21.21.5.5.5.2"><ci id="S4.E35.m1.21.21.21.5.5.5.2.1.cmml" xref="S4.E35.m1.21.21.21.5.5.5.2.1">¯</ci><ci id="S4.E35.m1.21.21.21.5.5.5.2.2.cmml" xref="S4.E35.m1.21.21.21.5.5.5.2.2">bold-italic-ϕ</ci></apply></apply><list id="S4.E35.m1.22.22.22.6.6.6.1.4.cmml" xref="S4.E35.m1.22.22.22.6.6.6.1.5"><ci id="S4.E35.m1.22.22.22.6.6.6.1.1.cmml" xref="S4.E35.m1.22.22.22.6.6.6.1.1">𝑛</ci><ci id="S4.E35.m1.22.22.22.6.6.6.1.2.cmml" xref="S4.E35.m1.22.22.22.6.6.6.1.2">𝑖</ci><ci id="S4.E35.m1.22.22.22.6.6.6.1.3.cmml" xref="S4.E35.m1.22.22.22.6.6.6.1.3">𝑖</ci></list></apply><apply id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.3.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><times id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.3.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4"></times><ci id="S4.E35.m1.24.24.24.8.8.8.cmml" xref="S4.E35.m1.24.24.24.8.8.8">𝑻</ci><apply id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.3.3.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.3.3.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">subscript</csymbol><apply id="S4.E35.m1.25.25.25.9.9.9.cmml" xref="S4.E35.m1.25.25.25.9.9.9"><ci id="S4.E35.m1.25.25.25.9.9.9.1.cmml" xref="S4.E35.m1.25.25.25.9.9.9.1">¯</ci><apply id="S4.E35.m1.25.25.25.9.9.9.2.cmml" xref="S4.E35.m1.25.25.25.9.9.9.2"><ci id="S4.E35.m1.25.25.25.9.9.9.2.1.cmml" xref="S4.E35.m1.25.25.25.9.9.9.2.1">¯</ci><ci id="S4.E35.m1.25.25.25.9.9.9.2.2.cmml" xref="S4.E35.m1.25.25.25.9.9.9.2.2">bold-italic-ϕ</ci></apply></apply><list id="S4.E35.m1.26.26.26.10.10.10.1.4.cmml" xref="S4.E35.m1.26.26.26.10.10.10.1.3"><apply id="S4.E35.m1.26.26.26.10.10.10.1.3.1.cmml" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1"><minus id="S4.E35.m1.26.26.26.10.10.10.1.3.1.1.cmml" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.1"></minus><ci id="S4.E35.m1.26.26.26.10.10.10.1.3.1.2.cmml" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.2">𝑛</ci><cn id="S4.E35.m1.26.26.26.10.10.10.1.3.1.3.cmml" type="integer" xref="S4.E35.m1.26.26.26.10.10.10.1.3.1.3">1</cn></apply><ci id="S4.E35.m1.26.26.26.10.10.10.1.1.cmml" xref="S4.E35.m1.26.26.26.10.10.10.1.1">𝑖</ci><ci id="S4.E35.m1.26.26.26.10.10.10.1.2.cmml" xref="S4.E35.m1.26.26.26.10.10.10.1.2">𝑖</ci></list></apply><apply id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.3.4.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.2.2.1.1.1.3.4.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">superscript</csymbol><ci id="S4.E35.m1.27.27.27.11.11.11.cmml" xref="S4.E35.m1.27.27.27.11.11.11">𝑻</ci><csymbol cd="latexml" id="S4.E35.m1.28.28.28.12.12.12.1.cmml" xref="S4.E35.m1.28.28.28.12.12.12.1">top</csymbol></apply></apply></apply><apply id="S4.E35.m1.33.33.1.1.1.2.2.4.cmml" xref="S4.E35.m1.4.4.4.4.4.4"><csymbol cd="ambiguous" id="S4.E35.m1.33.33.1.1.1.2.2.4.1.cmml" xref="S4.E35.m1.4.4.4.4.4.4">superscript</csymbol><ci id="S4.E35.m1.30.30.30.14.14.14.cmml" xref="S4.E35.m1.30.30.30.14.14.14">𝑮</ci><apply id="S4.E35.m1.31.31.31.15.15.15.1.cmml" xref="S4.E35.m1.31.31.31.15.15.15.1"><minus id="S4.E35.m1.31.31.31.15.15.15.1.1.cmml" xref="S4.E35.m1.31.31.31.15.15.15.1.1"></minus><csymbol cd="latexml" id="S4.E35.m1.31.31.31.15.15.15.1.2.cmml" xref="S4.E35.m1.31.31.31.15.15.15.1.2">absent</csymbol><csymbol cd="latexml" id="S4.E35.m1.31.31.31.15.15.15.1.3.cmml" xref="S4.E35.m1.31.31.31.15.15.15.1.3">top</csymbol></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E35.m1.35c">\mathbb{V}_{n,n-1,i}=\left\|\bm{G}^{-1}\left(\bar{\bm{\phi}}_{n,i}-\bm{T}\bar{% \bm{\phi}}_{n-1,i}\right)\right\|^{2}\\ +\bm{G}^{-1}\left(\bar{\bar{\bm{\phi}}}_{n,i,i}+\bm{T}\bar{\bar{\bm{\phi}}}_{n% -1,i,i}\bm{T}^{\top}\right)\bm{G}^{-\top},</annotation><annotation encoding="application/x-llamapun" id="S4.E35.m1.35d">start_ROW start_CELL blackboard_V start_POSTSUBSCRIPT italic_n , italic_n - 1 , italic_i end_POSTSUBSCRIPT = ∥ bold_italic_G start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n , italic_i end_POSTSUBSCRIPT - bold_italic_T over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n - 1 , italic_i end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + bold_italic_G start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over¯ start_ARG over¯ start_ARG bold_italic_ϕ end_ARG end_ARG start_POSTSUBSCRIPT italic_n , italic_i , italic_i end_POSTSUBSCRIPT + bold_italic_T over¯ start_ARG over¯ start_ARG bold_italic_ϕ end_ARG end_ARG start_POSTSUBSCRIPT italic_n - 1 , italic_i , italic_i end_POSTSUBSCRIPT bold_italic_T start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) bold_italic_G start_POSTSUPERSCRIPT - ⊤ 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">(35)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p5.12">which is a gamma distribution in <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S4.p5.9.m1.1"><semantics id="S4.p5.9.m1.1a"><msub id="S4.p5.9.m1.1.1" xref="S4.p5.9.m1.1.1.cmml"><mi id="S4.p5.9.m1.1.1.2" xref="S4.p5.9.m1.1.1.2.cmml">λ</mi><mi id="S4.p5.9.m1.1.1.3" xref="S4.p5.9.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p5.9.m1.1b"><apply id="S4.p5.9.m1.1.1.cmml" xref="S4.p5.9.m1.1.1"><csymbol cd="ambiguous" id="S4.p5.9.m1.1.1.1.cmml" xref="S4.p5.9.m1.1.1">subscript</csymbol><ci id="S4.p5.9.m1.1.1.2.cmml" xref="S4.p5.9.m1.1.1.2">𝜆</ci><ci id="S4.p5.9.m1.1.1.3.cmml" xref="S4.p5.9.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.9.m1.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.9.m1.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Hence, after carrying out the sum in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E8" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">8</span></a>), we obtain a gamma distribution with shape parameter <math alttext="\alpha=(N+\zeta)/2" class="ltx_Math" display="inline" id="S4.p5.10.m2.1"><semantics id="S4.p5.10.m2.1a"><mrow id="S4.p5.10.m2.1.1" xref="S4.p5.10.m2.1.1.cmml"><mi id="S4.p5.10.m2.1.1.3" xref="S4.p5.10.m2.1.1.3.cmml">α</mi><mo id="S4.p5.10.m2.1.1.2" xref="S4.p5.10.m2.1.1.2.cmml">=</mo><mrow id="S4.p5.10.m2.1.1.1" xref="S4.p5.10.m2.1.1.1.cmml"><mrow id="S4.p5.10.m2.1.1.1.1.1" xref="S4.p5.10.m2.1.1.1.1.1.1.cmml"><mo id="S4.p5.10.m2.1.1.1.1.1.2" stretchy="false" xref="S4.p5.10.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.p5.10.m2.1.1.1.1.1.1" xref="S4.p5.10.m2.1.1.1.1.1.1.cmml"><mi id="S4.p5.10.m2.1.1.1.1.1.1.2" xref="S4.p5.10.m2.1.1.1.1.1.1.2.cmml">N</mi><mo id="S4.p5.10.m2.1.1.1.1.1.1.1" xref="S4.p5.10.m2.1.1.1.1.1.1.1.cmml">+</mo><mi id="S4.p5.10.m2.1.1.1.1.1.1.3" xref="S4.p5.10.m2.1.1.1.1.1.1.3.cmml">ζ</mi></mrow><mo id="S4.p5.10.m2.1.1.1.1.1.3" stretchy="false" xref="S4.p5.10.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.p5.10.m2.1.1.1.2" xref="S4.p5.10.m2.1.1.1.2.cmml">/</mo><mn id="S4.p5.10.m2.1.1.1.3" xref="S4.p5.10.m2.1.1.1.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.10.m2.1b"><apply id="S4.p5.10.m2.1.1.cmml" xref="S4.p5.10.m2.1.1"><eq id="S4.p5.10.m2.1.1.2.cmml" xref="S4.p5.10.m2.1.1.2"></eq><ci id="S4.p5.10.m2.1.1.3.cmml" xref="S4.p5.10.m2.1.1.3">𝛼</ci><apply id="S4.p5.10.m2.1.1.1.cmml" xref="S4.p5.10.m2.1.1.1"><divide id="S4.p5.10.m2.1.1.1.2.cmml" xref="S4.p5.10.m2.1.1.1.2"></divide><apply id="S4.p5.10.m2.1.1.1.1.1.1.cmml" xref="S4.p5.10.m2.1.1.1.1.1"><plus id="S4.p5.10.m2.1.1.1.1.1.1.1.cmml" xref="S4.p5.10.m2.1.1.1.1.1.1.1"></plus><ci id="S4.p5.10.m2.1.1.1.1.1.1.2.cmml" xref="S4.p5.10.m2.1.1.1.1.1.1.2">𝑁</ci><ci id="S4.p5.10.m2.1.1.1.1.1.1.3.cmml" xref="S4.p5.10.m2.1.1.1.1.1.1.3">𝜁</ci></apply><cn id="S4.p5.10.m2.1.1.1.3.cmml" type="integer" xref="S4.p5.10.m2.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.10.m2.1c">\alpha=(N+\zeta)/2</annotation><annotation encoding="application/x-llamapun" id="S4.p5.10.m2.1d">italic_α = ( italic_N + italic_ζ ) / 2</annotation></semantics></math> and rate parameter <math alttext="\beta=(\chi+\sum_{n=1}^{N}\mathbb{V}_{n,n-1})/2" class="ltx_Math" display="inline" id="S4.p5.11.m3.3"><semantics id="S4.p5.11.m3.3a"><mrow id="S4.p5.11.m3.3.3" xref="S4.p5.11.m3.3.3.cmml"><mi id="S4.p5.11.m3.3.3.3" xref="S4.p5.11.m3.3.3.3.cmml">β</mi><mo id="S4.p5.11.m3.3.3.2" xref="S4.p5.11.m3.3.3.2.cmml">=</mo><mrow id="S4.p5.11.m3.3.3.1" xref="S4.p5.11.m3.3.3.1.cmml"><mrow id="S4.p5.11.m3.3.3.1.1.1" xref="S4.p5.11.m3.3.3.1.1.1.1.cmml"><mo id="S4.p5.11.m3.3.3.1.1.1.2" stretchy="false" xref="S4.p5.11.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.p5.11.m3.3.3.1.1.1.1" xref="S4.p5.11.m3.3.3.1.1.1.1.cmml"><mi id="S4.p5.11.m3.3.3.1.1.1.1.2" xref="S4.p5.11.m3.3.3.1.1.1.1.2.cmml">χ</mi><mo id="S4.p5.11.m3.3.3.1.1.1.1.1" rspace="0.055em" xref="S4.p5.11.m3.3.3.1.1.1.1.1.cmml">+</mo><mrow id="S4.p5.11.m3.3.3.1.1.1.1.3" xref="S4.p5.11.m3.3.3.1.1.1.1.3.cmml"><msubsup id="S4.p5.11.m3.3.3.1.1.1.1.3.1" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.cmml"><mo id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.2" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.2.cmml">∑</mo><mrow id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.cmml"><mi id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.2" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.2.cmml">n</mi><mo id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.1" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.3" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S4.p5.11.m3.3.3.1.1.1.1.3.1.3" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.3.cmml">N</mi></msubsup><msub id="S4.p5.11.m3.3.3.1.1.1.1.3.2" xref="S4.p5.11.m3.3.3.1.1.1.1.3.2.cmml"><mi id="S4.p5.11.m3.3.3.1.1.1.1.3.2.2" xref="S4.p5.11.m3.3.3.1.1.1.1.3.2.2.cmml">𝕍</mi><mrow id="S4.p5.11.m3.2.2.2.2" xref="S4.p5.11.m3.2.2.2.3.cmml"><mi id="S4.p5.11.m3.1.1.1.1" xref="S4.p5.11.m3.1.1.1.1.cmml">n</mi><mo id="S4.p5.11.m3.2.2.2.2.2" xref="S4.p5.11.m3.2.2.2.3.cmml">,</mo><mrow id="S4.p5.11.m3.2.2.2.2.1" xref="S4.p5.11.m3.2.2.2.2.1.cmml"><mi id="S4.p5.11.m3.2.2.2.2.1.2" xref="S4.p5.11.m3.2.2.2.2.1.2.cmml">n</mi><mo id="S4.p5.11.m3.2.2.2.2.1.1" xref="S4.p5.11.m3.2.2.2.2.1.1.cmml">−</mo><mn id="S4.p5.11.m3.2.2.2.2.1.3" xref="S4.p5.11.m3.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub></mrow></mrow><mo id="S4.p5.11.m3.3.3.1.1.1.3" stretchy="false" xref="S4.p5.11.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.p5.11.m3.3.3.1.2" xref="S4.p5.11.m3.3.3.1.2.cmml">/</mo><mn id="S4.p5.11.m3.3.3.1.3" xref="S4.p5.11.m3.3.3.1.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.11.m3.3b"><apply id="S4.p5.11.m3.3.3.cmml" xref="S4.p5.11.m3.3.3"><eq id="S4.p5.11.m3.3.3.2.cmml" xref="S4.p5.11.m3.3.3.2"></eq><ci id="S4.p5.11.m3.3.3.3.cmml" xref="S4.p5.11.m3.3.3.3">𝛽</ci><apply id="S4.p5.11.m3.3.3.1.cmml" xref="S4.p5.11.m3.3.3.1"><divide id="S4.p5.11.m3.3.3.1.2.cmml" xref="S4.p5.11.m3.3.3.1.2"></divide><apply id="S4.p5.11.m3.3.3.1.1.1.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1"><plus id="S4.p5.11.m3.3.3.1.1.1.1.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.1"></plus><ci id="S4.p5.11.m3.3.3.1.1.1.1.2.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.2">𝜒</ci><apply id="S4.p5.11.m3.3.3.1.1.1.1.3.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3"><apply id="S4.p5.11.m3.3.3.1.1.1.1.3.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.p5.11.m3.3.3.1.1.1.1.3.1.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1">superscript</csymbol><apply id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1">subscript</csymbol><sum id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.2.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.2"></sum><apply id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3"><eq id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.1"></eq><ci id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.2.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.2">𝑛</ci><cn id="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="S4.p5.11.m3.3.3.1.1.1.1.3.1.3.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.1.3">𝑁</ci></apply><apply id="S4.p5.11.m3.3.3.1.1.1.1.3.2.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.p5.11.m3.3.3.1.1.1.1.3.2.1.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.2">subscript</csymbol><ci id="S4.p5.11.m3.3.3.1.1.1.1.3.2.2.cmml" xref="S4.p5.11.m3.3.3.1.1.1.1.3.2.2">𝕍</ci><list id="S4.p5.11.m3.2.2.2.3.cmml" xref="S4.p5.11.m3.2.2.2.2"><ci id="S4.p5.11.m3.1.1.1.1.cmml" xref="S4.p5.11.m3.1.1.1.1">𝑛</ci><apply id="S4.p5.11.m3.2.2.2.2.1.cmml" xref="S4.p5.11.m3.2.2.2.2.1"><minus id="S4.p5.11.m3.2.2.2.2.1.1.cmml" xref="S4.p5.11.m3.2.2.2.2.1.1"></minus><ci id="S4.p5.11.m3.2.2.2.2.1.2.cmml" xref="S4.p5.11.m3.2.2.2.2.1.2">𝑛</ci><cn id="S4.p5.11.m3.2.2.2.2.1.3.cmml" type="integer" xref="S4.p5.11.m3.2.2.2.2.1.3">1</cn></apply></list></apply></apply></apply><cn id="S4.p5.11.m3.3.3.1.3.cmml" type="integer" xref="S4.p5.11.m3.3.3.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.11.m3.3c">\beta=(\chi+\sum_{n=1}^{N}\mathbb{V}_{n,n-1})/2</annotation><annotation encoding="application/x-llamapun" id="S4.p5.11.m3.3d">italic_β = ( italic_χ + ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT blackboard_V start_POSTSUBSCRIPT italic_n , italic_n - 1 end_POSTSUBSCRIPT ) / 2</annotation></semantics></math>. In this article, <math alttext="\zeta=\chi=1" class="ltx_Math" display="inline" id="S4.p5.12.m4.1"><semantics id="S4.p5.12.m4.1a"><mrow id="S4.p5.12.m4.1.1" xref="S4.p5.12.m4.1.1.cmml"><mi id="S4.p5.12.m4.1.1.2" xref="S4.p5.12.m4.1.1.2.cmml">ζ</mi><mo id="S4.p5.12.m4.1.1.3" xref="S4.p5.12.m4.1.1.3.cmml">=</mo><mi id="S4.p5.12.m4.1.1.4" xref="S4.p5.12.m4.1.1.4.cmml">χ</mi><mo id="S4.p5.12.m4.1.1.5" xref="S4.p5.12.m4.1.1.5.cmml">=</mo><mn id="S4.p5.12.m4.1.1.6" xref="S4.p5.12.m4.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.12.m4.1b"><apply id="S4.p5.12.m4.1.1.cmml" xref="S4.p5.12.m4.1.1"><and id="S4.p5.12.m4.1.1a.cmml" xref="S4.p5.12.m4.1.1"></and><apply id="S4.p5.12.m4.1.1b.cmml" xref="S4.p5.12.m4.1.1"><eq id="S4.p5.12.m4.1.1.3.cmml" xref="S4.p5.12.m4.1.1.3"></eq><ci id="S4.p5.12.m4.1.1.2.cmml" xref="S4.p5.12.m4.1.1.2">𝜁</ci><ci id="S4.p5.12.m4.1.1.4.cmml" xref="S4.p5.12.m4.1.1.4">𝜒</ci></apply><apply id="S4.p5.12.m4.1.1c.cmml" xref="S4.p5.12.m4.1.1"><eq id="S4.p5.12.m4.1.1.5.cmml" xref="S4.p5.12.m4.1.1.5"></eq><share href="https://arxiv.org/html/2503.16236v1#S4.p5.12.m4.1.1.4.cmml" id="S4.p5.12.m4.1.1d.cmml" xref="S4.p5.12.m4.1.1"></share><cn id="S4.p5.12.m4.1.1.6.cmml" type="integer" xref="S4.p5.12.m4.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.12.m4.1c">\zeta=\chi=1</annotation><annotation encoding="application/x-llamapun" id="S4.p5.12.m4.1d">italic_ζ = italic_χ = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">With all messages computed, the complete algorithm can now be formulated as Alg. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#alg1" title="Algorithm 1 ‣ IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.2.1.1">Algorithm 1</span> </span> Multiple Radar Bayesian Localization and Tracking</figcaption> <div class="ltx_listing ltx_listing" id="alg1.3"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l1.2">procedure</span> <span class="ltx_text ltx_font_smallcaps" id="alg1.l1.3">MRBLaT</span>(<math alttext="\bm{Z}_{N}^{(1)},\ldots,\bm{Z}_{N}^{(N_{\text{radar}})}" class="ltx_Math" display="inline" id="alg1.l1.m1.5"><semantics id="alg1.l1.m1.5a"><mrow id="alg1.l1.m1.5.5.2" xref="alg1.l1.m1.5.5.3.cmml"><msubsup id="alg1.l1.m1.4.4.1.1" xref="alg1.l1.m1.4.4.1.1.cmml"><mi id="alg1.l1.m1.4.4.1.1.2.2" xref="alg1.l1.m1.4.4.1.1.2.2.cmml">𝒁</mi><mi id="alg1.l1.m1.4.4.1.1.2.3" xref="alg1.l1.m1.4.4.1.1.2.3.cmml">N</mi><mrow id="alg1.l1.m1.1.1.1.3" xref="alg1.l1.m1.4.4.1.1.cmml"><mo id="alg1.l1.m1.1.1.1.3.1" stretchy="false" xref="alg1.l1.m1.4.4.1.1.cmml">(</mo><mn id="alg1.l1.m1.1.1.1.1" xref="alg1.l1.m1.1.1.1.1.cmml">1</mn><mo id="alg1.l1.m1.1.1.1.3.2" stretchy="false" xref="alg1.l1.m1.4.4.1.1.cmml">)</mo></mrow></msubsup><mo id="alg1.l1.m1.5.5.2.3" xref="alg1.l1.m1.5.5.3.cmml">,</mo><mi id="alg1.l1.m1.3.3" mathvariant="normal" xref="alg1.l1.m1.3.3.cmml">…</mi><mo id="alg1.l1.m1.5.5.2.4" xref="alg1.l1.m1.5.5.3.cmml">,</mo><msubsup id="alg1.l1.m1.5.5.2.2" xref="alg1.l1.m1.5.5.2.2.cmml"><mi id="alg1.l1.m1.5.5.2.2.2.2" xref="alg1.l1.m1.5.5.2.2.2.2.cmml">𝒁</mi><mi id="alg1.l1.m1.5.5.2.2.2.3" xref="alg1.l1.m1.5.5.2.2.2.3.cmml">N</mi><mrow id="alg1.l1.m1.2.2.1.1" xref="alg1.l1.m1.2.2.1.1.1.cmml"><mo id="alg1.l1.m1.2.2.1.1.2" stretchy="false" xref="alg1.l1.m1.2.2.1.1.1.cmml">(</mo><msub id="alg1.l1.m1.2.2.1.1.1" xref="alg1.l1.m1.2.2.1.1.1.cmml"><mi id="alg1.l1.m1.2.2.1.1.1.2" xref="alg1.l1.m1.2.2.1.1.1.2.cmml">N</mi><mtext id="alg1.l1.m1.2.2.1.1.1.3" xref="alg1.l1.m1.2.2.1.1.1.3a.cmml">radar</mtext></msub><mo id="alg1.l1.m1.2.2.1.1.3" stretchy="false" xref="alg1.l1.m1.2.2.1.1.1.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.5b"><list id="alg1.l1.m1.5.5.3.cmml" xref="alg1.l1.m1.5.5.2"><apply id="alg1.l1.m1.4.4.1.1.cmml" xref="alg1.l1.m1.4.4.1.1"><csymbol cd="ambiguous" id="alg1.l1.m1.4.4.1.1.1.cmml" xref="alg1.l1.m1.4.4.1.1">superscript</csymbol><apply id="alg1.l1.m1.4.4.1.1.2.cmml" xref="alg1.l1.m1.4.4.1.1"><csymbol cd="ambiguous" id="alg1.l1.m1.4.4.1.1.2.1.cmml" xref="alg1.l1.m1.4.4.1.1">subscript</csymbol><ci id="alg1.l1.m1.4.4.1.1.2.2.cmml" xref="alg1.l1.m1.4.4.1.1.2.2">𝒁</ci><ci id="alg1.l1.m1.4.4.1.1.2.3.cmml" xref="alg1.l1.m1.4.4.1.1.2.3">𝑁</ci></apply><cn id="alg1.l1.m1.1.1.1.1.cmml" type="integer" xref="alg1.l1.m1.1.1.1.1">1</cn></apply><ci id="alg1.l1.m1.3.3.cmml" xref="alg1.l1.m1.3.3">…</ci><apply id="alg1.l1.m1.5.5.2.2.cmml" xref="alg1.l1.m1.5.5.2.2"><csymbol cd="ambiguous" id="alg1.l1.m1.5.5.2.2.1.cmml" xref="alg1.l1.m1.5.5.2.2">superscript</csymbol><apply id="alg1.l1.m1.5.5.2.2.2.cmml" xref="alg1.l1.m1.5.5.2.2"><csymbol cd="ambiguous" id="alg1.l1.m1.5.5.2.2.2.1.cmml" xref="alg1.l1.m1.5.5.2.2">subscript</csymbol><ci id="alg1.l1.m1.5.5.2.2.2.2.cmml" xref="alg1.l1.m1.5.5.2.2.2.2">𝒁</ci><ci id="alg1.l1.m1.5.5.2.2.2.3.cmml" xref="alg1.l1.m1.5.5.2.2.2.3">𝑁</ci></apply><apply id="alg1.l1.m1.2.2.1.1.1.cmml" xref="alg1.l1.m1.2.2.1.1"><csymbol cd="ambiguous" id="alg1.l1.m1.2.2.1.1.1.1.cmml" xref="alg1.l1.m1.2.2.1.1">subscript</csymbol><ci id="alg1.l1.m1.2.2.1.1.1.2.cmml" xref="alg1.l1.m1.2.2.1.1.1.2">𝑁</ci><ci id="alg1.l1.m1.2.2.1.1.1.3a.cmml" xref="alg1.l1.m1.2.2.1.1.1.3"><mtext id="alg1.l1.m1.2.2.1.1.1.3.cmml" mathsize="50%" xref="alg1.l1.m1.2.2.1.1.1.3">radar</mtext></ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.5c">\bm{Z}_{N}^{(1)},\ldots,\bm{Z}_{N}^{(N_{\text{radar}})}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.5d">bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , … , bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span>In parallel at each radar <math alttext="k" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mi id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><ci id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">italic_k</annotation></semantics></math>: </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span> <math alttext="\left(\bar{\bm{\epsilon}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})},\bar{% \bar{\bm{\epsilon}}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})}\right)% \leftarrow\underset{\overline{{\bm{\epsilon}}},\overline{\overline{{\bm{% \epsilon}}}}}{\text{argmin}}\,D_{KL}\!\left(\bm{Z}_{N}^{(k)}\right)" class="ltx_Math" display="inline" id="alg1.l3.m1.10"><semantics id="alg1.l3.m1.10a"><mrow id="alg1.l3.m1.10.10" xref="alg1.l3.m1.10.10.cmml"><mrow id="alg1.l3.m1.9.9.2.2" xref="alg1.l3.m1.9.9.2.3.cmml"><mo id="alg1.l3.m1.9.9.2.2.3" xref="alg1.l3.m1.9.9.2.3.cmml">(</mo><msup id="alg1.l3.m1.8.8.1.1.1" xref="alg1.l3.m1.8.8.1.1.1.cmml"><mover accent="true" id="alg1.l3.m1.8.8.1.1.1.2" xref="alg1.l3.m1.8.8.1.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l3.m1.8.8.1.1.1.2.2" mathvariant="bold-italic" xref="alg1.l3.m1.8.8.1.1.1.2.2.cmml">ϵ</mi><mo id="alg1.l3.m1.8.8.1.1.1.2.1" xref="alg1.l3.m1.8.8.1.1.1.2.1.cmml">¯</mo></mover><mrow id="alg1.l3.m1.2.2.2.2" xref="alg1.l3.m1.2.2.2.2.1.cmml"><mo id="alg1.l3.m1.2.2.2.2.2" stretchy="false" xref="alg1.l3.m1.2.2.2.2.1.cmml">(</mo><mrow id="alg1.l3.m1.2.2.2.2.1" xref="alg1.l3.m1.2.2.2.2.1.cmml"><msubsup id="alg1.l3.m1.2.2.2.2.1.2" xref="alg1.l3.m1.2.2.2.2.1.2.cmml"><mi id="alg1.l3.m1.2.2.2.2.1.2.2.2" xref="alg1.l3.m1.2.2.2.2.1.2.2.2.cmml">𝒁</mi><mi id="alg1.l3.m1.2.2.2.2.1.2.2.3" xref="alg1.l3.m1.2.2.2.2.1.2.2.3.cmml">N</mi><mrow id="alg1.l3.m1.1.1.1.1.1.3" xref="alg1.l3.m1.2.2.2.2.1.2.cmml"><mo id="alg1.l3.m1.1.1.1.1.1.3.1" stretchy="false" xref="alg1.l3.m1.2.2.2.2.1.2.cmml">(</mo><mi id="alg1.l3.m1.1.1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.1.cmml">k</mi><mo id="alg1.l3.m1.1.1.1.1.1.3.2" stretchy="false" xref="alg1.l3.m1.2.2.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="alg1.l3.m1.2.2.2.2.1.1" stretchy="false" xref="alg1.l3.m1.2.2.2.2.1.1.cmml">→</mo><msub id="alg1.l3.m1.2.2.2.2.1.3" xref="alg1.l3.m1.2.2.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l3.m1.2.2.2.2.1.3.2" mathvariant="bold-italic" xref="alg1.l3.m1.2.2.2.2.1.3.2.cmml">ϕ</mi><mi id="alg1.l3.m1.2.2.2.2.1.3.3" xref="alg1.l3.m1.2.2.2.2.1.3.3.cmml">N</mi></msub></mrow><mo id="alg1.l3.m1.2.2.2.2.3" stretchy="false" xref="alg1.l3.m1.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="alg1.l3.m1.9.9.2.2.4" xref="alg1.l3.m1.9.9.2.3.cmml">,</mo><msup id="alg1.l3.m1.9.9.2.2.2" xref="alg1.l3.m1.9.9.2.2.2.cmml"><mover accent="true" id="alg1.l3.m1.9.9.2.2.2.2" xref="alg1.l3.m1.9.9.2.2.2.2.cmml"><mover accent="true" id="alg1.l3.m1.9.9.2.2.2.2.2" xref="alg1.l3.m1.9.9.2.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l3.m1.9.9.2.2.2.2.2.2" mathvariant="bold-italic" xref="alg1.l3.m1.9.9.2.2.2.2.2.2.cmml">ϵ</mi><mo id="alg1.l3.m1.9.9.2.2.2.2.2.1" xref="alg1.l3.m1.9.9.2.2.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l3.m1.9.9.2.2.2.2.1" xref="alg1.l3.m1.9.9.2.2.2.2.1.cmml">¯</mo></mover><mrow id="alg1.l3.m1.4.4.2.2" xref="alg1.l3.m1.4.4.2.2.1.cmml"><mo id="alg1.l3.m1.4.4.2.2.2" stretchy="false" xref="alg1.l3.m1.4.4.2.2.1.cmml">(</mo><mrow id="alg1.l3.m1.4.4.2.2.1" xref="alg1.l3.m1.4.4.2.2.1.cmml"><msubsup id="alg1.l3.m1.4.4.2.2.1.2" xref="alg1.l3.m1.4.4.2.2.1.2.cmml"><mi id="alg1.l3.m1.4.4.2.2.1.2.2.2" xref="alg1.l3.m1.4.4.2.2.1.2.2.2.cmml">𝒁</mi><mi id="alg1.l3.m1.4.4.2.2.1.2.2.3" xref="alg1.l3.m1.4.4.2.2.1.2.2.3.cmml">N</mi><mrow id="alg1.l3.m1.3.3.1.1.1.3" xref="alg1.l3.m1.4.4.2.2.1.2.cmml"><mo id="alg1.l3.m1.3.3.1.1.1.3.1" stretchy="false" xref="alg1.l3.m1.4.4.2.2.1.2.cmml">(</mo><mi id="alg1.l3.m1.3.3.1.1.1.1" xref="alg1.l3.m1.3.3.1.1.1.1.cmml">k</mi><mo id="alg1.l3.m1.3.3.1.1.1.3.2" stretchy="false" xref="alg1.l3.m1.4.4.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="alg1.l3.m1.4.4.2.2.1.1" stretchy="false" xref="alg1.l3.m1.4.4.2.2.1.1.cmml">→</mo><msub id="alg1.l3.m1.4.4.2.2.1.3" xref="alg1.l3.m1.4.4.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l3.m1.4.4.2.2.1.3.2" mathvariant="bold-italic" xref="alg1.l3.m1.4.4.2.2.1.3.2.cmml">ϕ</mi><mi id="alg1.l3.m1.4.4.2.2.1.3.3" xref="alg1.l3.m1.4.4.2.2.1.3.3.cmml">N</mi></msub></mrow><mo id="alg1.l3.m1.4.4.2.2.3" stretchy="false" xref="alg1.l3.m1.4.4.2.2.1.cmml">)</mo></mrow></msup><mo id="alg1.l3.m1.9.9.2.2.5" xref="alg1.l3.m1.9.9.2.3.cmml">)</mo></mrow><mo id="alg1.l3.m1.10.10.4" stretchy="false" xref="alg1.l3.m1.10.10.4.cmml">←</mo><mrow id="alg1.l3.m1.10.10.3" xref="alg1.l3.m1.10.10.3.cmml"><munder accentunder="true" id="alg1.l3.m1.6.6" xref="alg1.l3.m1.6.6.cmml"><mtext id="alg1.l3.m1.6.6.3" xref="alg1.l3.m1.6.6.3a.cmml">argmin</mtext><mrow id="alg1.l3.m1.6.6.2.4" xref="alg1.l3.m1.6.6.2.3.cmml"><mover accent="true" id="alg1.l3.m1.5.5.1.1" xref="alg1.l3.m1.5.5.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l3.m1.5.5.1.1.2" mathvariant="bold-italic" xref="alg1.l3.m1.5.5.1.1.2.cmml">ϵ</mi><mo id="alg1.l3.m1.5.5.1.1.1" xref="alg1.l3.m1.5.5.1.1.1.cmml">¯</mo></mover><mo id="alg1.l3.m1.6.6.2.4.1" xref="alg1.l3.m1.6.6.2.3.cmml">,</mo><mover accent="true" id="alg1.l3.m1.6.6.2.2" xref="alg1.l3.m1.6.6.2.2.cmml"><mover accent="true" id="alg1.l3.m1.6.6.2.2.2" xref="alg1.l3.m1.6.6.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l3.m1.6.6.2.2.2.2" mathvariant="bold-italic" xref="alg1.l3.m1.6.6.2.2.2.2.cmml">ϵ</mi><mo id="alg1.l3.m1.6.6.2.2.2.1" xref="alg1.l3.m1.6.6.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l3.m1.6.6.2.2.1" xref="alg1.l3.m1.6.6.2.2.1.cmml">¯</mo></mover></mrow></munder><mo id="alg1.l3.m1.10.10.3.2" xref="alg1.l3.m1.10.10.3.2.cmml">⁢</mo><msub id="alg1.l3.m1.10.10.3.3" xref="alg1.l3.m1.10.10.3.3.cmml"><mi id="alg1.l3.m1.10.10.3.3.2" xref="alg1.l3.m1.10.10.3.3.2.cmml">D</mi><mrow id="alg1.l3.m1.10.10.3.3.3" xref="alg1.l3.m1.10.10.3.3.3.cmml"><mi id="alg1.l3.m1.10.10.3.3.3.2" xref="alg1.l3.m1.10.10.3.3.3.2.cmml">K</mi><mo id="alg1.l3.m1.10.10.3.3.3.1" xref="alg1.l3.m1.10.10.3.3.3.1.cmml">⁢</mo><mi id="alg1.l3.m1.10.10.3.3.3.3" xref="alg1.l3.m1.10.10.3.3.3.3.cmml">L</mi></mrow></msub><mo id="alg1.l3.m1.10.10.3.2a" xref="alg1.l3.m1.10.10.3.2.cmml">⁢</mo><mrow id="alg1.l3.m1.10.10.3.1.1" xref="alg1.l3.m1.10.10.3.1.1.1.cmml"><mo id="alg1.l3.m1.10.10.3.1.1.2" xref="alg1.l3.m1.10.10.3.1.1.1.cmml">(</mo><msubsup id="alg1.l3.m1.10.10.3.1.1.1" xref="alg1.l3.m1.10.10.3.1.1.1.cmml"><mi id="alg1.l3.m1.10.10.3.1.1.1.2.2" xref="alg1.l3.m1.10.10.3.1.1.1.2.2.cmml">𝒁</mi><mi id="alg1.l3.m1.10.10.3.1.1.1.2.3" xref="alg1.l3.m1.10.10.3.1.1.1.2.3.cmml">N</mi><mrow id="alg1.l3.m1.7.7.1.3" xref="alg1.l3.m1.10.10.3.1.1.1.cmml"><mo id="alg1.l3.m1.7.7.1.3.1" stretchy="false" xref="alg1.l3.m1.10.10.3.1.1.1.cmml">(</mo><mi id="alg1.l3.m1.7.7.1.1" xref="alg1.l3.m1.7.7.1.1.cmml">k</mi><mo id="alg1.l3.m1.7.7.1.3.2" stretchy="false" xref="alg1.l3.m1.10.10.3.1.1.1.cmml">)</mo></mrow></msubsup><mo id="alg1.l3.m1.10.10.3.1.1.3" xref="alg1.l3.m1.10.10.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.10b"><apply id="alg1.l3.m1.10.10.cmml" xref="alg1.l3.m1.10.10"><ci id="alg1.l3.m1.10.10.4.cmml" xref="alg1.l3.m1.10.10.4">←</ci><interval closure="open" id="alg1.l3.m1.9.9.2.3.cmml" xref="alg1.l3.m1.9.9.2.2"><apply id="alg1.l3.m1.8.8.1.1.1.cmml" xref="alg1.l3.m1.8.8.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.8.8.1.1.1.1.cmml" xref="alg1.l3.m1.8.8.1.1.1">superscript</csymbol><apply id="alg1.l3.m1.8.8.1.1.1.2.cmml" xref="alg1.l3.m1.8.8.1.1.1.2"><ci id="alg1.l3.m1.8.8.1.1.1.2.1.cmml" xref="alg1.l3.m1.8.8.1.1.1.2.1">¯</ci><ci id="alg1.l3.m1.8.8.1.1.1.2.2.cmml" xref="alg1.l3.m1.8.8.1.1.1.2.2">bold-italic-ϵ</ci></apply><apply id="alg1.l3.m1.2.2.2.2.1.cmml" xref="alg1.l3.m1.2.2.2.2"><ci id="alg1.l3.m1.2.2.2.2.1.1.cmml" xref="alg1.l3.m1.2.2.2.2.1.1">→</ci><apply id="alg1.l3.m1.2.2.2.2.1.2.cmml" xref="alg1.l3.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.2.2.1.2.1.cmml" xref="alg1.l3.m1.2.2.2.2.1.2">superscript</csymbol><apply id="alg1.l3.m1.2.2.2.2.1.2.2.cmml" xref="alg1.l3.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.2.2.1.2.2.1.cmml" xref="alg1.l3.m1.2.2.2.2.1.2">subscript</csymbol><ci id="alg1.l3.m1.2.2.2.2.1.2.2.2.cmml" xref="alg1.l3.m1.2.2.2.2.1.2.2.2">𝒁</ci><ci id="alg1.l3.m1.2.2.2.2.1.2.2.3.cmml" xref="alg1.l3.m1.2.2.2.2.1.2.2.3">𝑁</ci></apply><ci id="alg1.l3.m1.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1.1.1">𝑘</ci></apply><apply id="alg1.l3.m1.2.2.2.2.1.3.cmml" xref="alg1.l3.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.2.2.1.3.1.cmml" xref="alg1.l3.m1.2.2.2.2.1.3">subscript</csymbol><ci id="alg1.l3.m1.2.2.2.2.1.3.2.cmml" xref="alg1.l3.m1.2.2.2.2.1.3.2">bold-italic-ϕ</ci><ci id="alg1.l3.m1.2.2.2.2.1.3.3.cmml" xref="alg1.l3.m1.2.2.2.2.1.3.3">𝑁</ci></apply></apply></apply><apply id="alg1.l3.m1.9.9.2.2.2.cmml" xref="alg1.l3.m1.9.9.2.2.2"><csymbol cd="ambiguous" id="alg1.l3.m1.9.9.2.2.2.1.cmml" xref="alg1.l3.m1.9.9.2.2.2">superscript</csymbol><apply id="alg1.l3.m1.9.9.2.2.2.2.cmml" xref="alg1.l3.m1.9.9.2.2.2.2"><ci id="alg1.l3.m1.9.9.2.2.2.2.1.cmml" xref="alg1.l3.m1.9.9.2.2.2.2.1">¯</ci><apply id="alg1.l3.m1.9.9.2.2.2.2.2.cmml" xref="alg1.l3.m1.9.9.2.2.2.2.2"><ci id="alg1.l3.m1.9.9.2.2.2.2.2.1.cmml" xref="alg1.l3.m1.9.9.2.2.2.2.2.1">¯</ci><ci id="alg1.l3.m1.9.9.2.2.2.2.2.2.cmml" xref="alg1.l3.m1.9.9.2.2.2.2.2.2">bold-italic-ϵ</ci></apply></apply><apply id="alg1.l3.m1.4.4.2.2.1.cmml" xref="alg1.l3.m1.4.4.2.2"><ci id="alg1.l3.m1.4.4.2.2.1.1.cmml" xref="alg1.l3.m1.4.4.2.2.1.1">→</ci><apply id="alg1.l3.m1.4.4.2.2.1.2.cmml" xref="alg1.l3.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l3.m1.4.4.2.2.1.2.1.cmml" xref="alg1.l3.m1.4.4.2.2.1.2">superscript</csymbol><apply id="alg1.l3.m1.4.4.2.2.1.2.2.cmml" xref="alg1.l3.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l3.m1.4.4.2.2.1.2.2.1.cmml" xref="alg1.l3.m1.4.4.2.2.1.2">subscript</csymbol><ci id="alg1.l3.m1.4.4.2.2.1.2.2.2.cmml" xref="alg1.l3.m1.4.4.2.2.1.2.2.2">𝒁</ci><ci id="alg1.l3.m1.4.4.2.2.1.2.2.3.cmml" xref="alg1.l3.m1.4.4.2.2.1.2.2.3">𝑁</ci></apply><ci id="alg1.l3.m1.3.3.1.1.1.1.cmml" xref="alg1.l3.m1.3.3.1.1.1.1">𝑘</ci></apply><apply id="alg1.l3.m1.4.4.2.2.1.3.cmml" xref="alg1.l3.m1.4.4.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l3.m1.4.4.2.2.1.3.1.cmml" xref="alg1.l3.m1.4.4.2.2.1.3">subscript</csymbol><ci id="alg1.l3.m1.4.4.2.2.1.3.2.cmml" xref="alg1.l3.m1.4.4.2.2.1.3.2">bold-italic-ϕ</ci><ci id="alg1.l3.m1.4.4.2.2.1.3.3.cmml" xref="alg1.l3.m1.4.4.2.2.1.3.3">𝑁</ci></apply></apply></apply></interval><apply id="alg1.l3.m1.10.10.3.cmml" xref="alg1.l3.m1.10.10.3"><times id="alg1.l3.m1.10.10.3.2.cmml" xref="alg1.l3.m1.10.10.3.2"></times><apply id="alg1.l3.m1.6.6.cmml" xref="alg1.l3.m1.6.6"><list id="alg1.l3.m1.6.6.2.3.cmml" xref="alg1.l3.m1.6.6.2.4"><apply id="alg1.l3.m1.5.5.1.1.cmml" xref="alg1.l3.m1.5.5.1.1"><ci id="alg1.l3.m1.5.5.1.1.1.cmml" xref="alg1.l3.m1.5.5.1.1.1">¯</ci><ci id="alg1.l3.m1.5.5.1.1.2.cmml" xref="alg1.l3.m1.5.5.1.1.2">bold-italic-ϵ</ci></apply><apply id="alg1.l3.m1.6.6.2.2.cmml" xref="alg1.l3.m1.6.6.2.2"><ci id="alg1.l3.m1.6.6.2.2.1.cmml" xref="alg1.l3.m1.6.6.2.2.1">¯</ci><apply id="alg1.l3.m1.6.6.2.2.2.cmml" xref="alg1.l3.m1.6.6.2.2.2"><ci id="alg1.l3.m1.6.6.2.2.2.1.cmml" xref="alg1.l3.m1.6.6.2.2.2.1">¯</ci><ci id="alg1.l3.m1.6.6.2.2.2.2.cmml" xref="alg1.l3.m1.6.6.2.2.2.2">bold-italic-ϵ</ci></apply></apply></list><ci id="alg1.l3.m1.6.6.3a.cmml" xref="alg1.l3.m1.6.6.3"><mtext id="alg1.l3.m1.6.6.3.cmml" xref="alg1.l3.m1.6.6.3">argmin</mtext></ci></apply><apply id="alg1.l3.m1.10.10.3.3.cmml" xref="alg1.l3.m1.10.10.3.3"><csymbol cd="ambiguous" id="alg1.l3.m1.10.10.3.3.1.cmml" xref="alg1.l3.m1.10.10.3.3">subscript</csymbol><ci id="alg1.l3.m1.10.10.3.3.2.cmml" xref="alg1.l3.m1.10.10.3.3.2">𝐷</ci><apply id="alg1.l3.m1.10.10.3.3.3.cmml" xref="alg1.l3.m1.10.10.3.3.3"><times id="alg1.l3.m1.10.10.3.3.3.1.cmml" xref="alg1.l3.m1.10.10.3.3.3.1"></times><ci id="alg1.l3.m1.10.10.3.3.3.2.cmml" xref="alg1.l3.m1.10.10.3.3.3.2">𝐾</ci><ci id="alg1.l3.m1.10.10.3.3.3.3.cmml" xref="alg1.l3.m1.10.10.3.3.3.3">𝐿</ci></apply></apply><apply id="alg1.l3.m1.10.10.3.1.1.1.cmml" xref="alg1.l3.m1.10.10.3.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.10.10.3.1.1.1.1.cmml" xref="alg1.l3.m1.10.10.3.1.1">superscript</csymbol><apply id="alg1.l3.m1.10.10.3.1.1.1.2.cmml" xref="alg1.l3.m1.10.10.3.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.10.10.3.1.1.1.2.1.cmml" xref="alg1.l3.m1.10.10.3.1.1">subscript</csymbol><ci id="alg1.l3.m1.10.10.3.1.1.1.2.2.cmml" xref="alg1.l3.m1.10.10.3.1.1.1.2.2">𝒁</ci><ci id="alg1.l3.m1.10.10.3.1.1.1.2.3.cmml" xref="alg1.l3.m1.10.10.3.1.1.1.2.3">𝑁</ci></apply><ci id="alg1.l3.m1.7.7.1.1.cmml" xref="alg1.l3.m1.7.7.1.1">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.10c">\left(\bar{\bm{\epsilon}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})},\bar{% \bar{\bm{\epsilon}}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})}\right)% \leftarrow\underset{\overline{{\bm{\epsilon}}},\overline{\overline{{\bm{% \epsilon}}}}}{\text{argmin}}\,D_{KL}\!\left(\bm{Z}_{N}^{(k)}\right)</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.10d">( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ) ← start_UNDERACCENT over¯ start_ARG bold_italic_ϵ end_ARG , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG end_UNDERACCENT start_ARG argmin end_ARG italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math> using (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E27" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">27</span></a>) </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span> Broadcast to all radars <math alttext="\left(\bar{\bm{\epsilon}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})},\bar{% \bar{\bm{\epsilon}}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})}\right)" class="ltx_Math" display="inline" id="alg1.l4.m1.6"><semantics id="alg1.l4.m1.6a"><mrow id="alg1.l4.m1.6.6.2" xref="alg1.l4.m1.6.6.3.cmml"><mo id="alg1.l4.m1.6.6.2.3" xref="alg1.l4.m1.6.6.3.cmml">(</mo><msup id="alg1.l4.m1.5.5.1.1" xref="alg1.l4.m1.5.5.1.1.cmml"><mover accent="true" id="alg1.l4.m1.5.5.1.1.2" xref="alg1.l4.m1.5.5.1.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l4.m1.5.5.1.1.2.2" mathvariant="bold-italic" xref="alg1.l4.m1.5.5.1.1.2.2.cmml">ϵ</mi><mo id="alg1.l4.m1.5.5.1.1.2.1" xref="alg1.l4.m1.5.5.1.1.2.1.cmml">¯</mo></mover><mrow id="alg1.l4.m1.2.2.2.2" xref="alg1.l4.m1.2.2.2.2.1.cmml"><mo id="alg1.l4.m1.2.2.2.2.2" stretchy="false" xref="alg1.l4.m1.2.2.2.2.1.cmml">(</mo><mrow id="alg1.l4.m1.2.2.2.2.1" xref="alg1.l4.m1.2.2.2.2.1.cmml"><msubsup id="alg1.l4.m1.2.2.2.2.1.2" xref="alg1.l4.m1.2.2.2.2.1.2.cmml"><mi id="alg1.l4.m1.2.2.2.2.1.2.2.2" xref="alg1.l4.m1.2.2.2.2.1.2.2.2.cmml">𝒁</mi><mi id="alg1.l4.m1.2.2.2.2.1.2.2.3" xref="alg1.l4.m1.2.2.2.2.1.2.2.3.cmml">N</mi><mrow id="alg1.l4.m1.1.1.1.1.1.3" xref="alg1.l4.m1.2.2.2.2.1.2.cmml"><mo id="alg1.l4.m1.1.1.1.1.1.3.1" stretchy="false" xref="alg1.l4.m1.2.2.2.2.1.2.cmml">(</mo><mi id="alg1.l4.m1.1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">k</mi><mo id="alg1.l4.m1.1.1.1.1.1.3.2" stretchy="false" xref="alg1.l4.m1.2.2.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="alg1.l4.m1.2.2.2.2.1.1" stretchy="false" xref="alg1.l4.m1.2.2.2.2.1.1.cmml">→</mo><msub id="alg1.l4.m1.2.2.2.2.1.3" xref="alg1.l4.m1.2.2.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l4.m1.2.2.2.2.1.3.2" mathvariant="bold-italic" xref="alg1.l4.m1.2.2.2.2.1.3.2.cmml">ϕ</mi><mi id="alg1.l4.m1.2.2.2.2.1.3.3" xref="alg1.l4.m1.2.2.2.2.1.3.3.cmml">N</mi></msub></mrow><mo id="alg1.l4.m1.2.2.2.2.3" stretchy="false" xref="alg1.l4.m1.2.2.2.2.1.cmml">)</mo></mrow></msup><mo id="alg1.l4.m1.6.6.2.4" xref="alg1.l4.m1.6.6.3.cmml">,</mo><msup id="alg1.l4.m1.6.6.2.2" xref="alg1.l4.m1.6.6.2.2.cmml"><mover accent="true" id="alg1.l4.m1.6.6.2.2.2" xref="alg1.l4.m1.6.6.2.2.2.cmml"><mover accent="true" id="alg1.l4.m1.6.6.2.2.2.2" xref="alg1.l4.m1.6.6.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l4.m1.6.6.2.2.2.2.2" mathvariant="bold-italic" xref="alg1.l4.m1.6.6.2.2.2.2.2.cmml">ϵ</mi><mo id="alg1.l4.m1.6.6.2.2.2.2.1" xref="alg1.l4.m1.6.6.2.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l4.m1.6.6.2.2.2.1" xref="alg1.l4.m1.6.6.2.2.2.1.cmml">¯</mo></mover><mrow id="alg1.l4.m1.4.4.2.2" xref="alg1.l4.m1.4.4.2.2.1.cmml"><mo id="alg1.l4.m1.4.4.2.2.2" stretchy="false" xref="alg1.l4.m1.4.4.2.2.1.cmml">(</mo><mrow id="alg1.l4.m1.4.4.2.2.1" xref="alg1.l4.m1.4.4.2.2.1.cmml"><msubsup id="alg1.l4.m1.4.4.2.2.1.2" xref="alg1.l4.m1.4.4.2.2.1.2.cmml"><mi id="alg1.l4.m1.4.4.2.2.1.2.2.2" xref="alg1.l4.m1.4.4.2.2.1.2.2.2.cmml">𝒁</mi><mi id="alg1.l4.m1.4.4.2.2.1.2.2.3" xref="alg1.l4.m1.4.4.2.2.1.2.2.3.cmml">N</mi><mrow id="alg1.l4.m1.3.3.1.1.1.3" xref="alg1.l4.m1.4.4.2.2.1.2.cmml"><mo id="alg1.l4.m1.3.3.1.1.1.3.1" stretchy="false" xref="alg1.l4.m1.4.4.2.2.1.2.cmml">(</mo><mi id="alg1.l4.m1.3.3.1.1.1.1" xref="alg1.l4.m1.3.3.1.1.1.1.cmml">k</mi><mo id="alg1.l4.m1.3.3.1.1.1.3.2" stretchy="false" xref="alg1.l4.m1.4.4.2.2.1.2.cmml">)</mo></mrow></msubsup><mo id="alg1.l4.m1.4.4.2.2.1.1" stretchy="false" xref="alg1.l4.m1.4.4.2.2.1.1.cmml">→</mo><msub id="alg1.l4.m1.4.4.2.2.1.3" xref="alg1.l4.m1.4.4.2.2.1.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l4.m1.4.4.2.2.1.3.2" mathvariant="bold-italic" xref="alg1.l4.m1.4.4.2.2.1.3.2.cmml">ϕ</mi><mi id="alg1.l4.m1.4.4.2.2.1.3.3" xref="alg1.l4.m1.4.4.2.2.1.3.3.cmml">N</mi></msub></mrow><mo id="alg1.l4.m1.4.4.2.2.3" stretchy="false" xref="alg1.l4.m1.4.4.2.2.1.cmml">)</mo></mrow></msup><mo id="alg1.l4.m1.6.6.2.5" xref="alg1.l4.m1.6.6.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.6b"><interval closure="open" id="alg1.l4.m1.6.6.3.cmml" xref="alg1.l4.m1.6.6.2"><apply id="alg1.l4.m1.5.5.1.1.cmml" xref="alg1.l4.m1.5.5.1.1"><csymbol cd="ambiguous" id="alg1.l4.m1.5.5.1.1.1.cmml" xref="alg1.l4.m1.5.5.1.1">superscript</csymbol><apply id="alg1.l4.m1.5.5.1.1.2.cmml" xref="alg1.l4.m1.5.5.1.1.2"><ci id="alg1.l4.m1.5.5.1.1.2.1.cmml" xref="alg1.l4.m1.5.5.1.1.2.1">¯</ci><ci id="alg1.l4.m1.5.5.1.1.2.2.cmml" xref="alg1.l4.m1.5.5.1.1.2.2">bold-italic-ϵ</ci></apply><apply id="alg1.l4.m1.2.2.2.2.1.cmml" xref="alg1.l4.m1.2.2.2.2"><ci id="alg1.l4.m1.2.2.2.2.1.1.cmml" xref="alg1.l4.m1.2.2.2.2.1.1">→</ci><apply id="alg1.l4.m1.2.2.2.2.1.2.cmml" xref="alg1.l4.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.2.2.2.2.1.2.1.cmml" xref="alg1.l4.m1.2.2.2.2.1.2">superscript</csymbol><apply id="alg1.l4.m1.2.2.2.2.1.2.2.cmml" xref="alg1.l4.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.2.2.2.2.1.2.2.1.cmml" xref="alg1.l4.m1.2.2.2.2.1.2">subscript</csymbol><ci id="alg1.l4.m1.2.2.2.2.1.2.2.2.cmml" xref="alg1.l4.m1.2.2.2.2.1.2.2.2">𝒁</ci><ci id="alg1.l4.m1.2.2.2.2.1.2.2.3.cmml" xref="alg1.l4.m1.2.2.2.2.1.2.2.3">𝑁</ci></apply><ci id="alg1.l4.m1.1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1.1.1">𝑘</ci></apply><apply id="alg1.l4.m1.2.2.2.2.1.3.cmml" xref="alg1.l4.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l4.m1.2.2.2.2.1.3.1.cmml" xref="alg1.l4.m1.2.2.2.2.1.3">subscript</csymbol><ci id="alg1.l4.m1.2.2.2.2.1.3.2.cmml" xref="alg1.l4.m1.2.2.2.2.1.3.2">bold-italic-ϕ</ci><ci id="alg1.l4.m1.2.2.2.2.1.3.3.cmml" xref="alg1.l4.m1.2.2.2.2.1.3.3">𝑁</ci></apply></apply></apply><apply id="alg1.l4.m1.6.6.2.2.cmml" xref="alg1.l4.m1.6.6.2.2"><csymbol cd="ambiguous" id="alg1.l4.m1.6.6.2.2.1.cmml" xref="alg1.l4.m1.6.6.2.2">superscript</csymbol><apply id="alg1.l4.m1.6.6.2.2.2.cmml" xref="alg1.l4.m1.6.6.2.2.2"><ci id="alg1.l4.m1.6.6.2.2.2.1.cmml" xref="alg1.l4.m1.6.6.2.2.2.1">¯</ci><apply id="alg1.l4.m1.6.6.2.2.2.2.cmml" xref="alg1.l4.m1.6.6.2.2.2.2"><ci id="alg1.l4.m1.6.6.2.2.2.2.1.cmml" xref="alg1.l4.m1.6.6.2.2.2.2.1">¯</ci><ci id="alg1.l4.m1.6.6.2.2.2.2.2.cmml" xref="alg1.l4.m1.6.6.2.2.2.2.2">bold-italic-ϵ</ci></apply></apply><apply id="alg1.l4.m1.4.4.2.2.1.cmml" xref="alg1.l4.m1.4.4.2.2"><ci id="alg1.l4.m1.4.4.2.2.1.1.cmml" xref="alg1.l4.m1.4.4.2.2.1.1">→</ci><apply id="alg1.l4.m1.4.4.2.2.1.2.cmml" xref="alg1.l4.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.4.4.2.2.1.2.1.cmml" xref="alg1.l4.m1.4.4.2.2.1.2">superscript</csymbol><apply id="alg1.l4.m1.4.4.2.2.1.2.2.cmml" xref="alg1.l4.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.4.4.2.2.1.2.2.1.cmml" xref="alg1.l4.m1.4.4.2.2.1.2">subscript</csymbol><ci id="alg1.l4.m1.4.4.2.2.1.2.2.2.cmml" xref="alg1.l4.m1.4.4.2.2.1.2.2.2">𝒁</ci><ci id="alg1.l4.m1.4.4.2.2.1.2.2.3.cmml" xref="alg1.l4.m1.4.4.2.2.1.2.2.3">𝑁</ci></apply><ci id="alg1.l4.m1.3.3.1.1.1.1.cmml" xref="alg1.l4.m1.3.3.1.1.1.1">𝑘</ci></apply><apply id="alg1.l4.m1.4.4.2.2.1.3.cmml" xref="alg1.l4.m1.4.4.2.2.1.3"><csymbol cd="ambiguous" id="alg1.l4.m1.4.4.2.2.1.3.1.cmml" xref="alg1.l4.m1.4.4.2.2.1.3">subscript</csymbol><ci id="alg1.l4.m1.4.4.2.2.1.3.2.cmml" xref="alg1.l4.m1.4.4.2.2.1.3.2">bold-italic-ϕ</ci><ci id="alg1.l4.m1.4.4.2.2.1.3.3.cmml" xref="alg1.l4.m1.4.4.2.2.1.3.3">𝑁</ci></apply></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.6c">\left(\bar{\bm{\epsilon}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})},\bar{% \bar{\bm{\epsilon}}}^{(\bm{Z}_{N}^{(k)}\rightarrow\bm{\phi}_{N})}\right)</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.6d">( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT ( bold_italic_Z start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → bold_italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT )</annotation></semantics></math> and save to memory </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.1.1.1" style="font-size:80%;">5:</span></span>Local message passing at each radar: </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l6.2">for</span> ite <math alttext="\leftarrow 1" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><mrow id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><mi id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml"></mi><mo id="alg1.l6.m1.1.1.1" stretchy="false" xref="alg1.l6.m1.1.1.1.cmml">←</mo><mn id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><ci id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1">←</ci><csymbol cd="latexml" id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2">absent</csymbol><cn id="alg1.l6.m1.1.1.3.cmml" type="integer" xref="alg1.l6.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">\leftarrow 1</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">← 1</annotation></semantics></math> to <math alttext="N_{\text{ite}}" class="ltx_Math" display="inline" id="alg1.l6.m2.1"><semantics id="alg1.l6.m2.1a"><msub id="alg1.l6.m2.1.1" xref="alg1.l6.m2.1.1.cmml"><mi id="alg1.l6.m2.1.1.2" xref="alg1.l6.m2.1.1.2.cmml">N</mi><mtext id="alg1.l6.m2.1.1.3" xref="alg1.l6.m2.1.1.3a.cmml">ite</mtext></msub><annotation-xml encoding="MathML-Content" id="alg1.l6.m2.1b"><apply id="alg1.l6.m2.1.1.cmml" xref="alg1.l6.m2.1.1"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.1.cmml" xref="alg1.l6.m2.1.1">subscript</csymbol><ci id="alg1.l6.m2.1.1.2.cmml" xref="alg1.l6.m2.1.1.2">𝑁</ci><ci id="alg1.l6.m2.1.1.3a.cmml" xref="alg1.l6.m2.1.1.3"><mtext id="alg1.l6.m2.1.1.3.cmml" mathsize="70%" xref="alg1.l6.m2.1.1.3">ite</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m2.1c">N_{\text{ite}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m2.1d">italic_N start_POSTSUBSCRIPT ite end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l6.3">do</span> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l7.1.1.1" style="font-size:80%;">7:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l7.2">for</span> <math alttext="n\leftarrow 0\,\,\text{to}\,\,N" class="ltx_Math" display="inline" id="alg1.l7.m1.1"><semantics id="alg1.l7.m1.1a"><mrow id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml"><mi id="alg1.l7.m1.1.1.2" xref="alg1.l7.m1.1.1.2.cmml">n</mi><mo id="alg1.l7.m1.1.1.1" stretchy="false" xref="alg1.l7.m1.1.1.1.cmml">←</mo><mrow id="alg1.l7.m1.1.1.3" xref="alg1.l7.m1.1.1.3.cmml"><mn id="alg1.l7.m1.1.1.3.2" xref="alg1.l7.m1.1.1.3.2.cmml">0</mn><mo id="alg1.l7.m1.1.1.3.1" lspace="0.330em" xref="alg1.l7.m1.1.1.3.1.cmml">⁢</mo><mtext id="alg1.l7.m1.1.1.3.3" xref="alg1.l7.m1.1.1.3.3a.cmml">to</mtext><mo id="alg1.l7.m1.1.1.3.1a" lspace="0.330em" xref="alg1.l7.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l7.m1.1.1.3.4" xref="alg1.l7.m1.1.1.3.4.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.1b"><apply id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1"><ci id="alg1.l7.m1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1">←</ci><ci id="alg1.l7.m1.1.1.2.cmml" xref="alg1.l7.m1.1.1.2">𝑛</ci><apply id="alg1.l7.m1.1.1.3.cmml" xref="alg1.l7.m1.1.1.3"><times id="alg1.l7.m1.1.1.3.1.cmml" xref="alg1.l7.m1.1.1.3.1"></times><cn id="alg1.l7.m1.1.1.3.2.cmml" type="integer" xref="alg1.l7.m1.1.1.3.2">0</cn><ci id="alg1.l7.m1.1.1.3.3a.cmml" xref="alg1.l7.m1.1.1.3.3"><mtext id="alg1.l7.m1.1.1.3.3.cmml" xref="alg1.l7.m1.1.1.3.3">to</mtext></ci><ci id="alg1.l7.m1.1.1.3.4.cmml" xref="alg1.l7.m1.1.1.3.4">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.1c">n\leftarrow 0\,\,\text{to}\,\,N</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.1d">italic_n ← 0 to italic_N</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l7.3">do</span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l8.1.1.1" style="font-size:80%;">8:</span></span> <math alttext="\bar{\bar{\bm{\phi}}}_{n}^{-1}\leftarrow\sum_{\epsilon\in\mathcal{N}_{\bm{\phi% }_{n}}}\bar{\bar{\epsilon}}_{n}^{-1}" class="ltx_Math" display="inline" id="alg1.l8.m1.1"><semantics id="alg1.l8.m1.1a"><mrow id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml"><msubsup id="alg1.l8.m1.1.1.2" xref="alg1.l8.m1.1.1.2.cmml"><mover accent="true" id="alg1.l8.m1.1.1.2.2.2" xref="alg1.l8.m1.1.1.2.2.2.cmml"><mover accent="true" id="alg1.l8.m1.1.1.2.2.2.2" xref="alg1.l8.m1.1.1.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l8.m1.1.1.2.2.2.2.2" mathvariant="bold-italic" xref="alg1.l8.m1.1.1.2.2.2.2.2.cmml">ϕ</mi><mo id="alg1.l8.m1.1.1.2.2.2.2.1" xref="alg1.l8.m1.1.1.2.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l8.m1.1.1.2.2.2.1" xref="alg1.l8.m1.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="alg1.l8.m1.1.1.2.2.3" xref="alg1.l8.m1.1.1.2.2.3.cmml">n</mi><mrow id="alg1.l8.m1.1.1.2.3" xref="alg1.l8.m1.1.1.2.3.cmml"><mo id="alg1.l8.m1.1.1.2.3a" xref="alg1.l8.m1.1.1.2.3.cmml">−</mo><mn id="alg1.l8.m1.1.1.2.3.2" xref="alg1.l8.m1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="alg1.l8.m1.1.1.1" rspace="0.111em" stretchy="false" xref="alg1.l8.m1.1.1.1.cmml">←</mo><mrow id="alg1.l8.m1.1.1.3" xref="alg1.l8.m1.1.1.3.cmml"><msub id="alg1.l8.m1.1.1.3.1" xref="alg1.l8.m1.1.1.3.1.cmml"><mo id="alg1.l8.m1.1.1.3.1.2" xref="alg1.l8.m1.1.1.3.1.2.cmml">∑</mo><mrow id="alg1.l8.m1.1.1.3.1.3" xref="alg1.l8.m1.1.1.3.1.3.cmml"><mi id="alg1.l8.m1.1.1.3.1.3.2" xref="alg1.l8.m1.1.1.3.1.3.2.cmml">ϵ</mi><mo id="alg1.l8.m1.1.1.3.1.3.1" xref="alg1.l8.m1.1.1.3.1.3.1.cmml">∈</mo><msub id="alg1.l8.m1.1.1.3.1.3.3" xref="alg1.l8.m1.1.1.3.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l8.m1.1.1.3.1.3.3.2" xref="alg1.l8.m1.1.1.3.1.3.3.2.cmml">𝒩</mi><msub id="alg1.l8.m1.1.1.3.1.3.3.3" xref="alg1.l8.m1.1.1.3.1.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l8.m1.1.1.3.1.3.3.3.2" mathvariant="bold-italic" xref="alg1.l8.m1.1.1.3.1.3.3.3.2.cmml">ϕ</mi><mi id="alg1.l8.m1.1.1.3.1.3.3.3.3" xref="alg1.l8.m1.1.1.3.1.3.3.3.3.cmml">n</mi></msub></msub></mrow></msub><msubsup id="alg1.l8.m1.1.1.3.2" xref="alg1.l8.m1.1.1.3.2.cmml"><mover accent="true" id="alg1.l8.m1.1.1.3.2.2.2" xref="alg1.l8.m1.1.1.3.2.2.2.cmml"><mover accent="true" id="alg1.l8.m1.1.1.3.2.2.2.2" xref="alg1.l8.m1.1.1.3.2.2.2.2.cmml"><mi id="alg1.l8.m1.1.1.3.2.2.2.2.2" xref="alg1.l8.m1.1.1.3.2.2.2.2.2.cmml">ϵ</mi><mo id="alg1.l8.m1.1.1.3.2.2.2.2.1" xref="alg1.l8.m1.1.1.3.2.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l8.m1.1.1.3.2.2.2.1" xref="alg1.l8.m1.1.1.3.2.2.2.1.cmml">¯</mo></mover><mi id="alg1.l8.m1.1.1.3.2.2.3" xref="alg1.l8.m1.1.1.3.2.2.3.cmml">n</mi><mrow id="alg1.l8.m1.1.1.3.2.3" xref="alg1.l8.m1.1.1.3.2.3.cmml"><mo id="alg1.l8.m1.1.1.3.2.3a" xref="alg1.l8.m1.1.1.3.2.3.cmml">−</mo><mn id="alg1.l8.m1.1.1.3.2.3.2" xref="alg1.l8.m1.1.1.3.2.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.1b"><apply id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1"><ci id="alg1.l8.m1.1.1.1.cmml" xref="alg1.l8.m1.1.1.1">←</ci><apply id="alg1.l8.m1.1.1.2.cmml" xref="alg1.l8.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.2.1.cmml" xref="alg1.l8.m1.1.1.2">superscript</csymbol><apply id="alg1.l8.m1.1.1.2.2.cmml" xref="alg1.l8.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.2.2.1.cmml" xref="alg1.l8.m1.1.1.2">subscript</csymbol><apply id="alg1.l8.m1.1.1.2.2.2.cmml" xref="alg1.l8.m1.1.1.2.2.2"><ci id="alg1.l8.m1.1.1.2.2.2.1.cmml" xref="alg1.l8.m1.1.1.2.2.2.1">¯</ci><apply id="alg1.l8.m1.1.1.2.2.2.2.cmml" xref="alg1.l8.m1.1.1.2.2.2.2"><ci id="alg1.l8.m1.1.1.2.2.2.2.1.cmml" xref="alg1.l8.m1.1.1.2.2.2.2.1">¯</ci><ci id="alg1.l8.m1.1.1.2.2.2.2.2.cmml" xref="alg1.l8.m1.1.1.2.2.2.2.2">bold-italic-ϕ</ci></apply></apply><ci id="alg1.l8.m1.1.1.2.2.3.cmml" xref="alg1.l8.m1.1.1.2.2.3">𝑛</ci></apply><apply id="alg1.l8.m1.1.1.2.3.cmml" xref="alg1.l8.m1.1.1.2.3"><minus id="alg1.l8.m1.1.1.2.3.1.cmml" xref="alg1.l8.m1.1.1.2.3"></minus><cn id="alg1.l8.m1.1.1.2.3.2.cmml" type="integer" xref="alg1.l8.m1.1.1.2.3.2">1</cn></apply></apply><apply id="alg1.l8.m1.1.1.3.cmml" xref="alg1.l8.m1.1.1.3"><apply id="alg1.l8.m1.1.1.3.1.cmml" xref="alg1.l8.m1.1.1.3.1"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.1.1.cmml" xref="alg1.l8.m1.1.1.3.1">subscript</csymbol><sum id="alg1.l8.m1.1.1.3.1.2.cmml" xref="alg1.l8.m1.1.1.3.1.2"></sum><apply id="alg1.l8.m1.1.1.3.1.3.cmml" xref="alg1.l8.m1.1.1.3.1.3"><in id="alg1.l8.m1.1.1.3.1.3.1.cmml" xref="alg1.l8.m1.1.1.3.1.3.1"></in><ci id="alg1.l8.m1.1.1.3.1.3.2.cmml" xref="alg1.l8.m1.1.1.3.1.3.2">italic-ϵ</ci><apply id="alg1.l8.m1.1.1.3.1.3.3.cmml" xref="alg1.l8.m1.1.1.3.1.3.3"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.1.3.3.1.cmml" xref="alg1.l8.m1.1.1.3.1.3.3">subscript</csymbol><ci id="alg1.l8.m1.1.1.3.1.3.3.2.cmml" xref="alg1.l8.m1.1.1.3.1.3.3.2">𝒩</ci><apply id="alg1.l8.m1.1.1.3.1.3.3.3.cmml" xref="alg1.l8.m1.1.1.3.1.3.3.3"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.1.3.3.3.1.cmml" xref="alg1.l8.m1.1.1.3.1.3.3.3">subscript</csymbol><ci id="alg1.l8.m1.1.1.3.1.3.3.3.2.cmml" xref="alg1.l8.m1.1.1.3.1.3.3.3.2">bold-italic-ϕ</ci><ci id="alg1.l8.m1.1.1.3.1.3.3.3.3.cmml" xref="alg1.l8.m1.1.1.3.1.3.3.3.3">𝑛</ci></apply></apply></apply></apply><apply id="alg1.l8.m1.1.1.3.2.cmml" xref="alg1.l8.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.2.1.cmml" xref="alg1.l8.m1.1.1.3.2">superscript</csymbol><apply id="alg1.l8.m1.1.1.3.2.2.cmml" xref="alg1.l8.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.2.2.1.cmml" xref="alg1.l8.m1.1.1.3.2">subscript</csymbol><apply id="alg1.l8.m1.1.1.3.2.2.2.cmml" xref="alg1.l8.m1.1.1.3.2.2.2"><ci id="alg1.l8.m1.1.1.3.2.2.2.1.cmml" xref="alg1.l8.m1.1.1.3.2.2.2.1">¯</ci><apply id="alg1.l8.m1.1.1.3.2.2.2.2.cmml" xref="alg1.l8.m1.1.1.3.2.2.2.2"><ci id="alg1.l8.m1.1.1.3.2.2.2.2.1.cmml" xref="alg1.l8.m1.1.1.3.2.2.2.2.1">¯</ci><ci id="alg1.l8.m1.1.1.3.2.2.2.2.2.cmml" xref="alg1.l8.m1.1.1.3.2.2.2.2.2">italic-ϵ</ci></apply></apply><ci id="alg1.l8.m1.1.1.3.2.2.3.cmml" xref="alg1.l8.m1.1.1.3.2.2.3">𝑛</ci></apply><apply id="alg1.l8.m1.1.1.3.2.3.cmml" xref="alg1.l8.m1.1.1.3.2.3"><minus id="alg1.l8.m1.1.1.3.2.3.1.cmml" xref="alg1.l8.m1.1.1.3.2.3"></minus><cn id="alg1.l8.m1.1.1.3.2.3.2.cmml" type="integer" xref="alg1.l8.m1.1.1.3.2.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.1c">\bar{\bar{\bm{\phi}}}_{n}^{-1}\leftarrow\sum_{\epsilon\in\mathcal{N}_{\bm{\phi% }_{n}}}\bar{\bar{\epsilon}}_{n}^{-1}</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.1d">over¯ start_ARG over¯ start_ARG bold_italic_ϕ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ← ∑ start_POSTSUBSCRIPT italic_ϵ ∈ caligraphic_N start_POSTSUBSCRIPT bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT over¯ start_ARG over¯ start_ARG italic_ϵ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l9.1.1.1" style="font-size:80%;">9:</span></span> <math alttext="\bar{\bm{\phi}}_{n}\leftarrow\bar{\bar{\bm{\phi}}}_{n}\sum_{\epsilon\in% \mathcal{N}_{\bm{\phi}_{n}}}\bar{\bar{\epsilon}}^{-1}\bar{\epsilon}" class="ltx_Math" display="inline" id="alg1.l9.m1.1"><semantics id="alg1.l9.m1.1a"><mrow id="alg1.l9.m1.1.1" xref="alg1.l9.m1.1.1.cmml"><msub id="alg1.l9.m1.1.1.2" xref="alg1.l9.m1.1.1.2.cmml"><mover accent="true" id="alg1.l9.m1.1.1.2.2" xref="alg1.l9.m1.1.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l9.m1.1.1.2.2.2" mathvariant="bold-italic" xref="alg1.l9.m1.1.1.2.2.2.cmml">ϕ</mi><mo id="alg1.l9.m1.1.1.2.2.1" xref="alg1.l9.m1.1.1.2.2.1.cmml">¯</mo></mover><mi id="alg1.l9.m1.1.1.2.3" xref="alg1.l9.m1.1.1.2.3.cmml">n</mi></msub><mo id="alg1.l9.m1.1.1.1" stretchy="false" xref="alg1.l9.m1.1.1.1.cmml">←</mo><mrow id="alg1.l9.m1.1.1.3" xref="alg1.l9.m1.1.1.3.cmml"><msub id="alg1.l9.m1.1.1.3.2" xref="alg1.l9.m1.1.1.3.2.cmml"><mover accent="true" id="alg1.l9.m1.1.1.3.2.2" xref="alg1.l9.m1.1.1.3.2.2.cmml"><mover accent="true" id="alg1.l9.m1.1.1.3.2.2.2" xref="alg1.l9.m1.1.1.3.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l9.m1.1.1.3.2.2.2.2" mathvariant="bold-italic" xref="alg1.l9.m1.1.1.3.2.2.2.2.cmml">ϕ</mi><mo id="alg1.l9.m1.1.1.3.2.2.2.1" xref="alg1.l9.m1.1.1.3.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l9.m1.1.1.3.2.2.1" xref="alg1.l9.m1.1.1.3.2.2.1.cmml">¯</mo></mover><mi id="alg1.l9.m1.1.1.3.2.3" xref="alg1.l9.m1.1.1.3.2.3.cmml">n</mi></msub><mo id="alg1.l9.m1.1.1.3.1" xref="alg1.l9.m1.1.1.3.1.cmml">⁢</mo><mrow id="alg1.l9.m1.1.1.3.3" xref="alg1.l9.m1.1.1.3.3.cmml"><msub id="alg1.l9.m1.1.1.3.3.1" xref="alg1.l9.m1.1.1.3.3.1.cmml"><mo id="alg1.l9.m1.1.1.3.3.1.2" xref="alg1.l9.m1.1.1.3.3.1.2.cmml">∑</mo><mrow id="alg1.l9.m1.1.1.3.3.1.3" xref="alg1.l9.m1.1.1.3.3.1.3.cmml"><mi id="alg1.l9.m1.1.1.3.3.1.3.2" xref="alg1.l9.m1.1.1.3.3.1.3.2.cmml">ϵ</mi><mo id="alg1.l9.m1.1.1.3.3.1.3.1" xref="alg1.l9.m1.1.1.3.3.1.3.1.cmml">∈</mo><msub id="alg1.l9.m1.1.1.3.3.1.3.3" xref="alg1.l9.m1.1.1.3.3.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l9.m1.1.1.3.3.1.3.3.2" xref="alg1.l9.m1.1.1.3.3.1.3.3.2.cmml">𝒩</mi><msub id="alg1.l9.m1.1.1.3.3.1.3.3.3" xref="alg1.l9.m1.1.1.3.3.1.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l9.m1.1.1.3.3.1.3.3.3.2" mathvariant="bold-italic" xref="alg1.l9.m1.1.1.3.3.1.3.3.3.2.cmml">ϕ</mi><mi id="alg1.l9.m1.1.1.3.3.1.3.3.3.3" xref="alg1.l9.m1.1.1.3.3.1.3.3.3.3.cmml">n</mi></msub></msub></mrow></msub><mrow id="alg1.l9.m1.1.1.3.3.2" xref="alg1.l9.m1.1.1.3.3.2.cmml"><msup id="alg1.l9.m1.1.1.3.3.2.2" xref="alg1.l9.m1.1.1.3.3.2.2.cmml"><mover accent="true" id="alg1.l9.m1.1.1.3.3.2.2.2" xref="alg1.l9.m1.1.1.3.3.2.2.2.cmml"><mover accent="true" id="alg1.l9.m1.1.1.3.3.2.2.2.2" xref="alg1.l9.m1.1.1.3.3.2.2.2.2.cmml"><mi id="alg1.l9.m1.1.1.3.3.2.2.2.2.2" xref="alg1.l9.m1.1.1.3.3.2.2.2.2.2.cmml">ϵ</mi><mo id="alg1.l9.m1.1.1.3.3.2.2.2.2.1" xref="alg1.l9.m1.1.1.3.3.2.2.2.2.1.cmml">¯</mo></mover><mo id="alg1.l9.m1.1.1.3.3.2.2.2.1" xref="alg1.l9.m1.1.1.3.3.2.2.2.1.cmml">¯</mo></mover><mrow id="alg1.l9.m1.1.1.3.3.2.2.3" xref="alg1.l9.m1.1.1.3.3.2.2.3.cmml"><mo id="alg1.l9.m1.1.1.3.3.2.2.3a" xref="alg1.l9.m1.1.1.3.3.2.2.3.cmml">−</mo><mn id="alg1.l9.m1.1.1.3.3.2.2.3.2" xref="alg1.l9.m1.1.1.3.3.2.2.3.2.cmml">1</mn></mrow></msup><mo id="alg1.l9.m1.1.1.3.3.2.1" xref="alg1.l9.m1.1.1.3.3.2.1.cmml">⁢</mo><mover accent="true" id="alg1.l9.m1.1.1.3.3.2.3" xref="alg1.l9.m1.1.1.3.3.2.3.cmml"><mi id="alg1.l9.m1.1.1.3.3.2.3.2" xref="alg1.l9.m1.1.1.3.3.2.3.2.cmml">ϵ</mi><mo id="alg1.l9.m1.1.1.3.3.2.3.1" xref="alg1.l9.m1.1.1.3.3.2.3.1.cmml">¯</mo></mover></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m1.1b"><apply id="alg1.l9.m1.1.1.cmml" xref="alg1.l9.m1.1.1"><ci id="alg1.l9.m1.1.1.1.cmml" xref="alg1.l9.m1.1.1.1">←</ci><apply id="alg1.l9.m1.1.1.2.cmml" xref="alg1.l9.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.2.1.cmml" xref="alg1.l9.m1.1.1.2">subscript</csymbol><apply id="alg1.l9.m1.1.1.2.2.cmml" xref="alg1.l9.m1.1.1.2.2"><ci id="alg1.l9.m1.1.1.2.2.1.cmml" xref="alg1.l9.m1.1.1.2.2.1">¯</ci><ci id="alg1.l9.m1.1.1.2.2.2.cmml" xref="alg1.l9.m1.1.1.2.2.2">bold-italic-ϕ</ci></apply><ci id="alg1.l9.m1.1.1.2.3.cmml" xref="alg1.l9.m1.1.1.2.3">𝑛</ci></apply><apply id="alg1.l9.m1.1.1.3.cmml" xref="alg1.l9.m1.1.1.3"><times id="alg1.l9.m1.1.1.3.1.cmml" xref="alg1.l9.m1.1.1.3.1"></times><apply id="alg1.l9.m1.1.1.3.2.cmml" xref="alg1.l9.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.3.2.1.cmml" xref="alg1.l9.m1.1.1.3.2">subscript</csymbol><apply id="alg1.l9.m1.1.1.3.2.2.cmml" xref="alg1.l9.m1.1.1.3.2.2"><ci id="alg1.l9.m1.1.1.3.2.2.1.cmml" xref="alg1.l9.m1.1.1.3.2.2.1">¯</ci><apply id="alg1.l9.m1.1.1.3.2.2.2.cmml" xref="alg1.l9.m1.1.1.3.2.2.2"><ci id="alg1.l9.m1.1.1.3.2.2.2.1.cmml" xref="alg1.l9.m1.1.1.3.2.2.2.1">¯</ci><ci id="alg1.l9.m1.1.1.3.2.2.2.2.cmml" xref="alg1.l9.m1.1.1.3.2.2.2.2">bold-italic-ϕ</ci></apply></apply><ci id="alg1.l9.m1.1.1.3.2.3.cmml" xref="alg1.l9.m1.1.1.3.2.3">𝑛</ci></apply><apply id="alg1.l9.m1.1.1.3.3.cmml" xref="alg1.l9.m1.1.1.3.3"><apply id="alg1.l9.m1.1.1.3.3.1.cmml" xref="alg1.l9.m1.1.1.3.3.1"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.3.3.1.1.cmml" xref="alg1.l9.m1.1.1.3.3.1">subscript</csymbol><sum id="alg1.l9.m1.1.1.3.3.1.2.cmml" xref="alg1.l9.m1.1.1.3.3.1.2"></sum><apply id="alg1.l9.m1.1.1.3.3.1.3.cmml" xref="alg1.l9.m1.1.1.3.3.1.3"><in id="alg1.l9.m1.1.1.3.3.1.3.1.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.1"></in><ci id="alg1.l9.m1.1.1.3.3.1.3.2.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.2">italic-ϵ</ci><apply id="alg1.l9.m1.1.1.3.3.1.3.3.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.3.3.1.3.3.1.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3">subscript</csymbol><ci id="alg1.l9.m1.1.1.3.3.1.3.3.2.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3.2">𝒩</ci><apply id="alg1.l9.m1.1.1.3.3.1.3.3.3.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3.3"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.3.3.1.3.3.3.1.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3.3">subscript</csymbol><ci id="alg1.l9.m1.1.1.3.3.1.3.3.3.2.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3.3.2">bold-italic-ϕ</ci><ci id="alg1.l9.m1.1.1.3.3.1.3.3.3.3.cmml" xref="alg1.l9.m1.1.1.3.3.1.3.3.3.3">𝑛</ci></apply></apply></apply></apply><apply id="alg1.l9.m1.1.1.3.3.2.cmml" xref="alg1.l9.m1.1.1.3.3.2"><times id="alg1.l9.m1.1.1.3.3.2.1.cmml" xref="alg1.l9.m1.1.1.3.3.2.1"></times><apply id="alg1.l9.m1.1.1.3.3.2.2.cmml" xref="alg1.l9.m1.1.1.3.3.2.2"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.3.3.2.2.1.cmml" xref="alg1.l9.m1.1.1.3.3.2.2">superscript</csymbol><apply id="alg1.l9.m1.1.1.3.3.2.2.2.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.2"><ci id="alg1.l9.m1.1.1.3.3.2.2.2.1.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.2.1">¯</ci><apply id="alg1.l9.m1.1.1.3.3.2.2.2.2.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.2.2"><ci id="alg1.l9.m1.1.1.3.3.2.2.2.2.1.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.2.2.1">¯</ci><ci id="alg1.l9.m1.1.1.3.3.2.2.2.2.2.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.2.2.2">italic-ϵ</ci></apply></apply><apply id="alg1.l9.m1.1.1.3.3.2.2.3.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.3"><minus id="alg1.l9.m1.1.1.3.3.2.2.3.1.cmml" xref="alg1.l9.m1.1.1.3.3.2.2.3"></minus><cn id="alg1.l9.m1.1.1.3.3.2.2.3.2.cmml" type="integer" xref="alg1.l9.m1.1.1.3.3.2.2.3.2">1</cn></apply></apply><apply id="alg1.l9.m1.1.1.3.3.2.3.cmml" xref="alg1.l9.m1.1.1.3.3.2.3"><ci id="alg1.l9.m1.1.1.3.3.2.3.1.cmml" xref="alg1.l9.m1.1.1.3.3.2.3.1">¯</ci><ci id="alg1.l9.m1.1.1.3.3.2.3.2.cmml" xref="alg1.l9.m1.1.1.3.3.2.3.2">italic-ϵ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.1c">\bar{\bm{\phi}}_{n}\leftarrow\bar{\bar{\bm{\phi}}}_{n}\sum_{\epsilon\in% \mathcal{N}_{\bm{\phi}_{n}}}\bar{\bar{\epsilon}}^{-1}\bar{\epsilon}</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.1d">over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ← over¯ start_ARG over¯ start_ARG bold_italic_ϕ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_ϵ ∈ caligraphic_N start_POSTSUBSCRIPT bold_italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT over¯ start_ARG over¯ start_ARG italic_ϵ end_ARG end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over¯ start_ARG italic_ϵ end_ARG</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l10.1.1.1" style="font-size:80%;">10:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l10.2">if</span> <math alttext="N>1" class="ltx_Math" display="inline" id="alg1.l10.m1.1"><semantics id="alg1.l10.m1.1a"><mrow id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml"><mi id="alg1.l10.m1.1.1.2" xref="alg1.l10.m1.1.1.2.cmml">N</mi><mo id="alg1.l10.m1.1.1.1" xref="alg1.l10.m1.1.1.1.cmml">></mo><mn id="alg1.l10.m1.1.1.3" xref="alg1.l10.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.1b"><apply id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1"><gt id="alg1.l10.m1.1.1.1.cmml" xref="alg1.l10.m1.1.1.1"></gt><ci id="alg1.l10.m1.1.1.2.cmml" xref="alg1.l10.m1.1.1.2">𝑁</ci><cn id="alg1.l10.m1.1.1.3.cmml" type="integer" xref="alg1.l10.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.1c">N>1</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.1d">italic_N > 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l10.3">then</span> </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l11.1.1.1" style="font-size:80%;">11:</span></span> <math alttext="\alpha=N+1" class="ltx_Math" display="inline" id="alg1.l11.m1.1"><semantics id="alg1.l11.m1.1a"><mrow id="alg1.l11.m1.1.1" xref="alg1.l11.m1.1.1.cmml"><mi id="alg1.l11.m1.1.1.2" xref="alg1.l11.m1.1.1.2.cmml">α</mi><mo id="alg1.l11.m1.1.1.1" xref="alg1.l11.m1.1.1.1.cmml">=</mo><mrow id="alg1.l11.m1.1.1.3" xref="alg1.l11.m1.1.1.3.cmml"><mi id="alg1.l11.m1.1.1.3.2" xref="alg1.l11.m1.1.1.3.2.cmml">N</mi><mo id="alg1.l11.m1.1.1.3.1" xref="alg1.l11.m1.1.1.3.1.cmml">+</mo><mn id="alg1.l11.m1.1.1.3.3" xref="alg1.l11.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.1b"><apply id="alg1.l11.m1.1.1.cmml" xref="alg1.l11.m1.1.1"><eq id="alg1.l11.m1.1.1.1.cmml" xref="alg1.l11.m1.1.1.1"></eq><ci id="alg1.l11.m1.1.1.2.cmml" xref="alg1.l11.m1.1.1.2">𝛼</ci><apply id="alg1.l11.m1.1.1.3.cmml" xref="alg1.l11.m1.1.1.3"><plus id="alg1.l11.m1.1.1.3.1.cmml" xref="alg1.l11.m1.1.1.3.1"></plus><ci id="alg1.l11.m1.1.1.3.2.cmml" xref="alg1.l11.m1.1.1.3.2">𝑁</ci><cn id="alg1.l11.m1.1.1.3.3.cmml" type="integer" xref="alg1.l11.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.1c">\alpha=N+1</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m1.1d">italic_α = italic_N + 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l12.1.1.1" style="font-size:80%;">12:</span></span> <math alttext="\bm{\beta}=\sum_{n=1}^{N}\bm{\mathbb{V}}_{n,n-1}" class="ltx_Math" display="inline" id="alg1.l12.m1.2"><semantics id="alg1.l12.m1.2a"><mrow id="alg1.l12.m1.2.3" xref="alg1.l12.m1.2.3.cmml"><mi id="alg1.l12.m1.2.3.2" xref="alg1.l12.m1.2.3.2.cmml">𝜷</mi><mo id="alg1.l12.m1.2.3.1" rspace="0.111em" xref="alg1.l12.m1.2.3.1.cmml">=</mo><mrow id="alg1.l12.m1.2.3.3" xref="alg1.l12.m1.2.3.3.cmml"><msubsup id="alg1.l12.m1.2.3.3.1" xref="alg1.l12.m1.2.3.3.1.cmml"><mo id="alg1.l12.m1.2.3.3.1.2.2" xref="alg1.l12.m1.2.3.3.1.2.2.cmml">∑</mo><mrow id="alg1.l12.m1.2.3.3.1.2.3" xref="alg1.l12.m1.2.3.3.1.2.3.cmml"><mi id="alg1.l12.m1.2.3.3.1.2.3.2" xref="alg1.l12.m1.2.3.3.1.2.3.2.cmml">n</mi><mo id="alg1.l12.m1.2.3.3.1.2.3.1" xref="alg1.l12.m1.2.3.3.1.2.3.1.cmml">=</mo><mn id="alg1.l12.m1.2.3.3.1.2.3.3" xref="alg1.l12.m1.2.3.3.1.2.3.3.cmml">1</mn></mrow><mi id="alg1.l12.m1.2.3.3.1.3" xref="alg1.l12.m1.2.3.3.1.3.cmml">N</mi></msubsup><msub id="alg1.l12.m1.2.3.3.2" xref="alg1.l12.m1.2.3.3.2.cmml"><mi id="alg1.l12.m1.2.3.3.2.2" xref="alg1.l12.m1.2.3.3.2.2.cmml">𝕍</mi><mrow id="alg1.l12.m1.2.2.2.2" xref="alg1.l12.m1.2.2.2.3.cmml"><mi id="alg1.l12.m1.1.1.1.1" xref="alg1.l12.m1.1.1.1.1.cmml">n</mi><mo id="alg1.l12.m1.2.2.2.2.2" xref="alg1.l12.m1.2.2.2.3.cmml">,</mo><mrow id="alg1.l12.m1.2.2.2.2.1" xref="alg1.l12.m1.2.2.2.2.1.cmml"><mi id="alg1.l12.m1.2.2.2.2.1.2" xref="alg1.l12.m1.2.2.2.2.1.2.cmml">n</mi><mo id="alg1.l12.m1.2.2.2.2.1.1" xref="alg1.l12.m1.2.2.2.2.1.1.cmml">−</mo><mn id="alg1.l12.m1.2.2.2.2.1.3" xref="alg1.l12.m1.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l12.m1.2b"><apply id="alg1.l12.m1.2.3.cmml" xref="alg1.l12.m1.2.3"><eq id="alg1.l12.m1.2.3.1.cmml" xref="alg1.l12.m1.2.3.1"></eq><ci id="alg1.l12.m1.2.3.2.cmml" xref="alg1.l12.m1.2.3.2">𝜷</ci><apply id="alg1.l12.m1.2.3.3.cmml" xref="alg1.l12.m1.2.3.3"><apply id="alg1.l12.m1.2.3.3.1.cmml" xref="alg1.l12.m1.2.3.3.1"><csymbol cd="ambiguous" id="alg1.l12.m1.2.3.3.1.1.cmml" xref="alg1.l12.m1.2.3.3.1">superscript</csymbol><apply id="alg1.l12.m1.2.3.3.1.2.cmml" xref="alg1.l12.m1.2.3.3.1"><csymbol cd="ambiguous" id="alg1.l12.m1.2.3.3.1.2.1.cmml" xref="alg1.l12.m1.2.3.3.1">subscript</csymbol><sum id="alg1.l12.m1.2.3.3.1.2.2.cmml" xref="alg1.l12.m1.2.3.3.1.2.2"></sum><apply id="alg1.l12.m1.2.3.3.1.2.3.cmml" xref="alg1.l12.m1.2.3.3.1.2.3"><eq id="alg1.l12.m1.2.3.3.1.2.3.1.cmml" xref="alg1.l12.m1.2.3.3.1.2.3.1"></eq><ci id="alg1.l12.m1.2.3.3.1.2.3.2.cmml" xref="alg1.l12.m1.2.3.3.1.2.3.2">𝑛</ci><cn id="alg1.l12.m1.2.3.3.1.2.3.3.cmml" type="integer" xref="alg1.l12.m1.2.3.3.1.2.3.3">1</cn></apply></apply><ci id="alg1.l12.m1.2.3.3.1.3.cmml" xref="alg1.l12.m1.2.3.3.1.3">𝑁</ci></apply><apply id="alg1.l12.m1.2.3.3.2.cmml" xref="alg1.l12.m1.2.3.3.2"><csymbol cd="ambiguous" id="alg1.l12.m1.2.3.3.2.1.cmml" xref="alg1.l12.m1.2.3.3.2">subscript</csymbol><ci id="alg1.l12.m1.2.3.3.2.2.cmml" xref="alg1.l12.m1.2.3.3.2.2">𝕍</ci><list id="alg1.l12.m1.2.2.2.3.cmml" xref="alg1.l12.m1.2.2.2.2"><ci id="alg1.l12.m1.1.1.1.1.cmml" xref="alg1.l12.m1.1.1.1.1">𝑛</ci><apply id="alg1.l12.m1.2.2.2.2.1.cmml" xref="alg1.l12.m1.2.2.2.2.1"><minus id="alg1.l12.m1.2.2.2.2.1.1.cmml" xref="alg1.l12.m1.2.2.2.2.1.1"></minus><ci id="alg1.l12.m1.2.2.2.2.1.2.cmml" xref="alg1.l12.m1.2.2.2.2.1.2">𝑛</ci><cn id="alg1.l12.m1.2.2.2.2.1.3.cmml" type="integer" xref="alg1.l12.m1.2.2.2.2.1.3">1</cn></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l12.m1.2c">\bm{\beta}=\sum_{n=1}^{N}\bm{\mathbb{V}}_{n,n-1}</annotation><annotation encoding="application/x-llamapun" id="alg1.l12.m1.2d">bold_italic_β = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT blackboard_bold_V start_POSTSUBSCRIPT italic_n , italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> using (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E35" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">35</span></a>) </div> <div class="ltx_listingline" id="alg1.l13"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l13.1.1.1" style="font-size:80%;">13:</span></span> <math alttext="\bar{\Lambda}_{a}^{-1}=\bm{\beta}/\alpha" class="ltx_Math" display="inline" id="alg1.l13.m1.1"><semantics id="alg1.l13.m1.1a"><mrow id="alg1.l13.m1.1.1" xref="alg1.l13.m1.1.1.cmml"><msubsup id="alg1.l13.m1.1.1.2" xref="alg1.l13.m1.1.1.2.cmml"><mover accent="true" id="alg1.l13.m1.1.1.2.2.2" xref="alg1.l13.m1.1.1.2.2.2.cmml"><mi id="alg1.l13.m1.1.1.2.2.2.2" mathvariant="normal" xref="alg1.l13.m1.1.1.2.2.2.2.cmml">Λ</mi><mo id="alg1.l13.m1.1.1.2.2.2.1" xref="alg1.l13.m1.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="alg1.l13.m1.1.1.2.2.3" xref="alg1.l13.m1.1.1.2.2.3.cmml">a</mi><mrow id="alg1.l13.m1.1.1.2.3" xref="alg1.l13.m1.1.1.2.3.cmml"><mo id="alg1.l13.m1.1.1.2.3a" xref="alg1.l13.m1.1.1.2.3.cmml">−</mo><mn id="alg1.l13.m1.1.1.2.3.2" xref="alg1.l13.m1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="alg1.l13.m1.1.1.1" xref="alg1.l13.m1.1.1.1.cmml">=</mo><mrow id="alg1.l13.m1.1.1.3" xref="alg1.l13.m1.1.1.3.cmml"><mi id="alg1.l13.m1.1.1.3.2" xref="alg1.l13.m1.1.1.3.2.cmml">𝜷</mi><mo id="alg1.l13.m1.1.1.3.1" xref="alg1.l13.m1.1.1.3.1.cmml">/</mo><mi id="alg1.l13.m1.1.1.3.3" xref="alg1.l13.m1.1.1.3.3.cmml">α</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l13.m1.1b"><apply id="alg1.l13.m1.1.1.cmml" xref="alg1.l13.m1.1.1"><eq id="alg1.l13.m1.1.1.1.cmml" xref="alg1.l13.m1.1.1.1"></eq><apply id="alg1.l13.m1.1.1.2.cmml" xref="alg1.l13.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l13.m1.1.1.2.1.cmml" xref="alg1.l13.m1.1.1.2">superscript</csymbol><apply id="alg1.l13.m1.1.1.2.2.cmml" xref="alg1.l13.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l13.m1.1.1.2.2.1.cmml" xref="alg1.l13.m1.1.1.2">subscript</csymbol><apply id="alg1.l13.m1.1.1.2.2.2.cmml" xref="alg1.l13.m1.1.1.2.2.2"><ci id="alg1.l13.m1.1.1.2.2.2.1.cmml" xref="alg1.l13.m1.1.1.2.2.2.1">¯</ci><ci id="alg1.l13.m1.1.1.2.2.2.2.cmml" xref="alg1.l13.m1.1.1.2.2.2.2">Λ</ci></apply><ci id="alg1.l13.m1.1.1.2.2.3.cmml" xref="alg1.l13.m1.1.1.2.2.3">𝑎</ci></apply><apply id="alg1.l13.m1.1.1.2.3.cmml" xref="alg1.l13.m1.1.1.2.3"><minus id="alg1.l13.m1.1.1.2.3.1.cmml" xref="alg1.l13.m1.1.1.2.3"></minus><cn id="alg1.l13.m1.1.1.2.3.2.cmml" type="integer" xref="alg1.l13.m1.1.1.2.3.2">1</cn></apply></apply><apply id="alg1.l13.m1.1.1.3.cmml" xref="alg1.l13.m1.1.1.3"><divide id="alg1.l13.m1.1.1.3.1.cmml" xref="alg1.l13.m1.1.1.3.1"></divide><ci id="alg1.l13.m1.1.1.3.2.cmml" xref="alg1.l13.m1.1.1.3.2">𝜷</ci><ci id="alg1.l13.m1.1.1.3.3.cmml" xref="alg1.l13.m1.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l13.m1.1c">\bar{\Lambda}_{a}^{-1}=\bm{\beta}/\alpha</annotation><annotation encoding="application/x-llamapun" id="alg1.l13.m1.1d">over¯ start_ARG roman_Λ end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = bold_italic_β / italic_α</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l14"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l14.1.1.1" style="font-size:80%;">14:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l14.2">else</span> </div> <div class="ltx_listingline" id="alg1.l15"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l15.1.1.1" style="font-size:80%;">15:</span></span> <math alttext="\bm{\Lambda}_{a}\leftarrow\bm{\Lambda}_{a}^{(\text{init})}" class="ltx_Math" display="inline" id="alg1.l15.m1.1"><semantics id="alg1.l15.m1.1a"><mrow id="alg1.l15.m1.1.2" xref="alg1.l15.m1.1.2.cmml"><msub id="alg1.l15.m1.1.2.2" xref="alg1.l15.m1.1.2.2.cmml"><mi id="alg1.l15.m1.1.2.2.2" xref="alg1.l15.m1.1.2.2.2.cmml">𝚲</mi><mi id="alg1.l15.m1.1.2.2.3" xref="alg1.l15.m1.1.2.2.3.cmml">a</mi></msub><mo id="alg1.l15.m1.1.2.1" stretchy="false" xref="alg1.l15.m1.1.2.1.cmml">←</mo><msubsup id="alg1.l15.m1.1.2.3" xref="alg1.l15.m1.1.2.3.cmml"><mi id="alg1.l15.m1.1.2.3.2.2" xref="alg1.l15.m1.1.2.3.2.2.cmml">𝚲</mi><mi id="alg1.l15.m1.1.2.3.2.3" xref="alg1.l15.m1.1.2.3.2.3.cmml">a</mi><mrow id="alg1.l15.m1.1.1.1.3" xref="alg1.l15.m1.1.1.1.1a.cmml"><mo id="alg1.l15.m1.1.1.1.3.1" stretchy="false" xref="alg1.l15.m1.1.1.1.1a.cmml">(</mo><mtext id="alg1.l15.m1.1.1.1.1" xref="alg1.l15.m1.1.1.1.1.cmml">init</mtext><mo id="alg1.l15.m1.1.1.1.3.2" stretchy="false" xref="alg1.l15.m1.1.1.1.1a.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l15.m1.1b"><apply id="alg1.l15.m1.1.2.cmml" xref="alg1.l15.m1.1.2"><ci id="alg1.l15.m1.1.2.1.cmml" xref="alg1.l15.m1.1.2.1">←</ci><apply id="alg1.l15.m1.1.2.2.cmml" xref="alg1.l15.m1.1.2.2"><csymbol cd="ambiguous" id="alg1.l15.m1.1.2.2.1.cmml" xref="alg1.l15.m1.1.2.2">subscript</csymbol><ci id="alg1.l15.m1.1.2.2.2.cmml" xref="alg1.l15.m1.1.2.2.2">𝚲</ci><ci id="alg1.l15.m1.1.2.2.3.cmml" xref="alg1.l15.m1.1.2.2.3">𝑎</ci></apply><apply id="alg1.l15.m1.1.2.3.cmml" xref="alg1.l15.m1.1.2.3"><csymbol cd="ambiguous" id="alg1.l15.m1.1.2.3.1.cmml" xref="alg1.l15.m1.1.2.3">superscript</csymbol><apply id="alg1.l15.m1.1.2.3.2.cmml" xref="alg1.l15.m1.1.2.3"><csymbol cd="ambiguous" id="alg1.l15.m1.1.2.3.2.1.cmml" xref="alg1.l15.m1.1.2.3">subscript</csymbol><ci id="alg1.l15.m1.1.2.3.2.2.cmml" xref="alg1.l15.m1.1.2.3.2.2">𝚲</ci><ci id="alg1.l15.m1.1.2.3.2.3.cmml" xref="alg1.l15.m1.1.2.3.2.3">𝑎</ci></apply><ci id="alg1.l15.m1.1.1.1.1a.cmml" xref="alg1.l15.m1.1.1.1.3"><mtext id="alg1.l15.m1.1.1.1.1.cmml" mathsize="70%" xref="alg1.l15.m1.1.1.1.1">init</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l15.m1.1c">\bm{\Lambda}_{a}\leftarrow\bm{\Lambda}_{a}^{(\text{init})}</annotation><annotation encoding="application/x-llamapun" id="alg1.l15.m1.1d">bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ← bold_Λ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( init ) end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l16"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l16.1.1.1" style="font-size:80%;">16:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l16.2">return</span> <math alttext="\left(\bar{\bm{\phi}}_{0}\ldots,\bar{\bm{\phi}}_{N},\bar{\bar{\bm{\phi}}}_{0},% \ldots,\bar{\bar{\bm{\phi}}}_{N}\right)" class="ltx_Math" display="inline" id="alg1.l16.m1.5"><semantics id="alg1.l16.m1.5a"><mrow id="alg1.l16.m1.5.5.4" xref="alg1.l16.m1.5.5.5.cmml"><mo id="alg1.l16.m1.5.5.4.5" xref="alg1.l16.m1.5.5.5.cmml">(</mo><mrow id="alg1.l16.m1.2.2.1.1" xref="alg1.l16.m1.2.2.1.1.cmml"><msub id="alg1.l16.m1.2.2.1.1.2" xref="alg1.l16.m1.2.2.1.1.2.cmml"><mover accent="true" id="alg1.l16.m1.2.2.1.1.2.2" xref="alg1.l16.m1.2.2.1.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l16.m1.2.2.1.1.2.2.2" mathvariant="bold-italic" xref="alg1.l16.m1.2.2.1.1.2.2.2.cmml">ϕ</mi><mo id="alg1.l16.m1.2.2.1.1.2.2.1" xref="alg1.l16.m1.2.2.1.1.2.2.1.cmml">¯</mo></mover><mn id="alg1.l16.m1.2.2.1.1.2.3" xref="alg1.l16.m1.2.2.1.1.2.3.cmml">0</mn></msub><mo id="alg1.l16.m1.2.2.1.1.1" xref="alg1.l16.m1.2.2.1.1.1.cmml">⁢</mo><mi id="alg1.l16.m1.2.2.1.1.3" mathvariant="normal" xref="alg1.l16.m1.2.2.1.1.3.cmml">…</mi></mrow><mo id="alg1.l16.m1.5.5.4.6" xref="alg1.l16.m1.5.5.5.cmml">,</mo><msub id="alg1.l16.m1.3.3.2.2" xref="alg1.l16.m1.3.3.2.2.cmml"><mover accent="true" id="alg1.l16.m1.3.3.2.2.2" xref="alg1.l16.m1.3.3.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l16.m1.3.3.2.2.2.2" mathvariant="bold-italic" xref="alg1.l16.m1.3.3.2.2.2.2.cmml">ϕ</mi><mo id="alg1.l16.m1.3.3.2.2.2.1" xref="alg1.l16.m1.3.3.2.2.2.1.cmml">¯</mo></mover><mi id="alg1.l16.m1.3.3.2.2.3" xref="alg1.l16.m1.3.3.2.2.3.cmml">N</mi></msub><mo id="alg1.l16.m1.5.5.4.7" xref="alg1.l16.m1.5.5.5.cmml">,</mo><msub id="alg1.l16.m1.4.4.3.3" xref="alg1.l16.m1.4.4.3.3.cmml"><mover accent="true" id="alg1.l16.m1.4.4.3.3.2" xref="alg1.l16.m1.4.4.3.3.2.cmml"><mover accent="true" id="alg1.l16.m1.4.4.3.3.2.2" xref="alg1.l16.m1.4.4.3.3.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l16.m1.4.4.3.3.2.2.2" mathvariant="bold-italic" xref="alg1.l16.m1.4.4.3.3.2.2.2.cmml">ϕ</mi><mo id="alg1.l16.m1.4.4.3.3.2.2.1" xref="alg1.l16.m1.4.4.3.3.2.2.1.cmml">¯</mo></mover><mo id="alg1.l16.m1.4.4.3.3.2.1" xref="alg1.l16.m1.4.4.3.3.2.1.cmml">¯</mo></mover><mn id="alg1.l16.m1.4.4.3.3.3" xref="alg1.l16.m1.4.4.3.3.3.cmml">0</mn></msub><mo id="alg1.l16.m1.5.5.4.8" xref="alg1.l16.m1.5.5.5.cmml">,</mo><mi id="alg1.l16.m1.1.1" mathvariant="normal" xref="alg1.l16.m1.1.1.cmml">…</mi><mo id="alg1.l16.m1.5.5.4.9" xref="alg1.l16.m1.5.5.5.cmml">,</mo><msub id="alg1.l16.m1.5.5.4.4" xref="alg1.l16.m1.5.5.4.4.cmml"><mover accent="true" id="alg1.l16.m1.5.5.4.4.2" xref="alg1.l16.m1.5.5.4.4.2.cmml"><mover accent="true" id="alg1.l16.m1.5.5.4.4.2.2" xref="alg1.l16.m1.5.5.4.4.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="alg1.l16.m1.5.5.4.4.2.2.2" mathvariant="bold-italic" xref="alg1.l16.m1.5.5.4.4.2.2.2.cmml">ϕ</mi><mo id="alg1.l16.m1.5.5.4.4.2.2.1" xref="alg1.l16.m1.5.5.4.4.2.2.1.cmml">¯</mo></mover><mo id="alg1.l16.m1.5.5.4.4.2.1" xref="alg1.l16.m1.5.5.4.4.2.1.cmml">¯</mo></mover><mi id="alg1.l16.m1.5.5.4.4.3" xref="alg1.l16.m1.5.5.4.4.3.cmml">N</mi></msub><mo id="alg1.l16.m1.5.5.4.10" xref="alg1.l16.m1.5.5.5.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l16.m1.5b"><vector id="alg1.l16.m1.5.5.5.cmml" xref="alg1.l16.m1.5.5.4"><apply id="alg1.l16.m1.2.2.1.1.cmml" xref="alg1.l16.m1.2.2.1.1"><times id="alg1.l16.m1.2.2.1.1.1.cmml" xref="alg1.l16.m1.2.2.1.1.1"></times><apply id="alg1.l16.m1.2.2.1.1.2.cmml" xref="alg1.l16.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="alg1.l16.m1.2.2.1.1.2.1.cmml" xref="alg1.l16.m1.2.2.1.1.2">subscript</csymbol><apply id="alg1.l16.m1.2.2.1.1.2.2.cmml" xref="alg1.l16.m1.2.2.1.1.2.2"><ci id="alg1.l16.m1.2.2.1.1.2.2.1.cmml" xref="alg1.l16.m1.2.2.1.1.2.2.1">¯</ci><ci id="alg1.l16.m1.2.2.1.1.2.2.2.cmml" xref="alg1.l16.m1.2.2.1.1.2.2.2">bold-italic-ϕ</ci></apply><cn id="alg1.l16.m1.2.2.1.1.2.3.cmml" type="integer" xref="alg1.l16.m1.2.2.1.1.2.3">0</cn></apply><ci id="alg1.l16.m1.2.2.1.1.3.cmml" xref="alg1.l16.m1.2.2.1.1.3">…</ci></apply><apply id="alg1.l16.m1.3.3.2.2.cmml" xref="alg1.l16.m1.3.3.2.2"><csymbol cd="ambiguous" id="alg1.l16.m1.3.3.2.2.1.cmml" xref="alg1.l16.m1.3.3.2.2">subscript</csymbol><apply id="alg1.l16.m1.3.3.2.2.2.cmml" xref="alg1.l16.m1.3.3.2.2.2"><ci id="alg1.l16.m1.3.3.2.2.2.1.cmml" xref="alg1.l16.m1.3.3.2.2.2.1">¯</ci><ci id="alg1.l16.m1.3.3.2.2.2.2.cmml" xref="alg1.l16.m1.3.3.2.2.2.2">bold-italic-ϕ</ci></apply><ci id="alg1.l16.m1.3.3.2.2.3.cmml" xref="alg1.l16.m1.3.3.2.2.3">𝑁</ci></apply><apply id="alg1.l16.m1.4.4.3.3.cmml" xref="alg1.l16.m1.4.4.3.3"><csymbol cd="ambiguous" id="alg1.l16.m1.4.4.3.3.1.cmml" xref="alg1.l16.m1.4.4.3.3">subscript</csymbol><apply id="alg1.l16.m1.4.4.3.3.2.cmml" xref="alg1.l16.m1.4.4.3.3.2"><ci id="alg1.l16.m1.4.4.3.3.2.1.cmml" xref="alg1.l16.m1.4.4.3.3.2.1">¯</ci><apply id="alg1.l16.m1.4.4.3.3.2.2.cmml" xref="alg1.l16.m1.4.4.3.3.2.2"><ci id="alg1.l16.m1.4.4.3.3.2.2.1.cmml" xref="alg1.l16.m1.4.4.3.3.2.2.1">¯</ci><ci id="alg1.l16.m1.4.4.3.3.2.2.2.cmml" xref="alg1.l16.m1.4.4.3.3.2.2.2">bold-italic-ϕ</ci></apply></apply><cn id="alg1.l16.m1.4.4.3.3.3.cmml" type="integer" xref="alg1.l16.m1.4.4.3.3.3">0</cn></apply><ci id="alg1.l16.m1.1.1.cmml" xref="alg1.l16.m1.1.1">…</ci><apply id="alg1.l16.m1.5.5.4.4.cmml" xref="alg1.l16.m1.5.5.4.4"><csymbol cd="ambiguous" id="alg1.l16.m1.5.5.4.4.1.cmml" xref="alg1.l16.m1.5.5.4.4">subscript</csymbol><apply id="alg1.l16.m1.5.5.4.4.2.cmml" xref="alg1.l16.m1.5.5.4.4.2"><ci id="alg1.l16.m1.5.5.4.4.2.1.cmml" xref="alg1.l16.m1.5.5.4.4.2.1">¯</ci><apply id="alg1.l16.m1.5.5.4.4.2.2.cmml" xref="alg1.l16.m1.5.5.4.4.2.2"><ci id="alg1.l16.m1.5.5.4.4.2.2.1.cmml" xref="alg1.l16.m1.5.5.4.4.2.2.1">¯</ci><ci id="alg1.l16.m1.5.5.4.4.2.2.2.cmml" xref="alg1.l16.m1.5.5.4.4.2.2.2">bold-italic-ϕ</ci></apply></apply><ci id="alg1.l16.m1.5.5.4.4.3.cmml" xref="alg1.l16.m1.5.5.4.4.3">𝑁</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="alg1.l16.m1.5c">\left(\bar{\bm{\phi}}_{0}\ldots,\bar{\bm{\phi}}_{N},\bar{\bar{\bm{\phi}}}_{0},% \ldots,\bar{\bar{\bm{\phi}}}_{N}\right)</annotation><annotation encoding="application/x-llamapun" id="alg1.l16.m1.5d">( over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT … , over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , over¯ start_ARG over¯ start_ARG bold_italic_ϕ end_ARG end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , over¯ start_ARG over¯ start_ARG bold_italic_ϕ end_ARG end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> </div> </div> </figure> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Simulations</span> </h2> <div class="ltx_para ltx_noindent" id="S5.p1"> <p class="ltx_p" id="S5.p1.3">We consider three <math alttext="3\times 3" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mrow id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mn id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">3</mn><mo id="S5.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.1.m1.1.1.1.cmml">×</mo><mn id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><times id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1"></times><cn id="S5.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.p1.1.m1.1.1.2">3</cn><cn id="S5.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">3\times 3</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">3 × 3</annotation></semantics></math> MIMO radars operating in TDM mode, transmitting complex baseband signals of duration <math alttext="T_{T_{x}}" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><msub id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><mi id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml">T</mi><msub id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml"><mi id="S5.p1.2.m2.1.1.3.2" xref="S5.p1.2.m2.1.1.3.2.cmml">T</mi><mi id="S5.p1.2.m2.1.1.3.3" xref="S5.p1.2.m2.1.1.3.3.cmml">x</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.1.cmml" xref="S5.p1.2.m2.1.1">subscript</csymbol><ci id="S5.p1.2.m2.1.1.2.cmml" xref="S5.p1.2.m2.1.1.2">𝑇</ci><apply id="S5.p1.2.m2.1.1.3.cmml" xref="S5.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.3.1.cmml" xref="S5.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.p1.2.m2.1.1.3.2.cmml" xref="S5.p1.2.m2.1.1.3.2">𝑇</ci><ci id="S5.p1.2.m2.1.1.3.3.cmml" xref="S5.p1.2.m2.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">T_{T_{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with a linear chirp. The three radars are positioned <span class="ltx_ERROR undefined" id="S5.p1.3.1">\qty</span>50m apart along the <math alttext="x" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><mi id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><ci id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">italic_x</annotation></semantics></math>-axis, with the first radar in Origo<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Optimal radar placement is outside the scope of the paper.</span></span></span>. Parameter settings for the radars can be seen in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.T1" title="TABLE I ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">I</span></a>. The data is generated using the model in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S2.E2" title="In II Signal Model ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">2</span></a>).</p> </div> <figure class="ltx_table" id="S5.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">TABLE I: </span>Parameter settings</figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S5.T1.10"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T1.7.7"> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.1.1.1" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="N_{T,R}" class="ltx_Math" display="inline" id="S5.T1.1.1.1.m1.2"><semantics id="S5.T1.1.1.1.m1.2a"><msub id="S5.T1.1.1.1.m1.2.3" xref="S5.T1.1.1.1.m1.2.3.cmml"><mi id="S5.T1.1.1.1.m1.2.3.2" xref="S5.T1.1.1.1.m1.2.3.2.cmml">N</mi><mrow id="S5.T1.1.1.1.m1.2.2.2.4" xref="S5.T1.1.1.1.m1.2.2.2.3.cmml"><mi id="S5.T1.1.1.1.m1.1.1.1.1" xref="S5.T1.1.1.1.m1.1.1.1.1.cmml">T</mi><mo id="S5.T1.1.1.1.m1.2.2.2.4.1" xref="S5.T1.1.1.1.m1.2.2.2.3.cmml">,</mo><mi id="S5.T1.1.1.1.m1.2.2.2.2" xref="S5.T1.1.1.1.m1.2.2.2.2.cmml">R</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.1.1.1.m1.2b"><apply id="S5.T1.1.1.1.m1.2.3.cmml" xref="S5.T1.1.1.1.m1.2.3"><csymbol cd="ambiguous" id="S5.T1.1.1.1.m1.2.3.1.cmml" xref="S5.T1.1.1.1.m1.2.3">subscript</csymbol><ci id="S5.T1.1.1.1.m1.2.3.2.cmml" xref="S5.T1.1.1.1.m1.2.3.2">𝑁</ci><list id="S5.T1.1.1.1.m1.2.2.2.3.cmml" xref="S5.T1.1.1.1.m1.2.2.2.4"><ci id="S5.T1.1.1.1.m1.1.1.1.1.cmml" xref="S5.T1.1.1.1.m1.1.1.1.1">𝑇</ci><ci id="S5.T1.1.1.1.m1.2.2.2.2.cmml" xref="S5.T1.1.1.1.m1.2.2.2.2">𝑅</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.1.1.1.m1.2c">N_{T,R}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.1.1.1.m1.2d">italic_N start_POSTSUBSCRIPT italic_T , italic_R end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.7.7.8" style="padding-left:11.0pt;padding-right:11.0pt;">PRF</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.2.2.2" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="\rho" class="ltx_Math" display="inline" id="S5.T1.2.2.2.m1.1"><semantics id="S5.T1.2.2.2.m1.1a"><mi id="S5.T1.2.2.2.m1.1.1" xref="S5.T1.2.2.2.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.T1.2.2.2.m1.1b"><ci id="S5.T1.2.2.2.m1.1.1.cmml" xref="S5.T1.2.2.2.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.2.2.2.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.T1.2.2.2.m1.1d">italic_ρ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.7.7.9" style="padding-left:11.0pt;padding-right:11.0pt;">G</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.7.7.10" style="padding-left:11.0pt;padding-right:11.0pt;">P</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.3.3.3" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="R_{\text{max}}" class="ltx_Math" display="inline" id="S5.T1.3.3.3.m1.1"><semantics id="S5.T1.3.3.3.m1.1a"><msub id="S5.T1.3.3.3.m1.1.1" xref="S5.T1.3.3.3.m1.1.1.cmml"><mi id="S5.T1.3.3.3.m1.1.1.2" xref="S5.T1.3.3.3.m1.1.1.2.cmml">R</mi><mtext id="S5.T1.3.3.3.m1.1.1.3" xref="S5.T1.3.3.3.m1.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T1.3.3.3.m1.1b"><apply id="S5.T1.3.3.3.m1.1.1.cmml" xref="S5.T1.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.3.3.3.m1.1.1.1.cmml" xref="S5.T1.3.3.3.m1.1.1">subscript</csymbol><ci id="S5.T1.3.3.3.m1.1.1.2.cmml" xref="S5.T1.3.3.3.m1.1.1.2">𝑅</ci><ci id="S5.T1.3.3.3.m1.1.1.3a.cmml" xref="S5.T1.3.3.3.m1.1.1.3"><mtext id="S5.T1.3.3.3.m1.1.1.3.cmml" mathsize="70%" xref="S5.T1.3.3.3.m1.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.3.3.3.m1.1c">R_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.3.3.3.m1.1d">italic_R start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.4.4.4" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="f_{c}" class="ltx_Math" display="inline" id="S5.T1.4.4.4.m1.1"><semantics id="S5.T1.4.4.4.m1.1a"><msub id="S5.T1.4.4.4.m1.1.1" xref="S5.T1.4.4.4.m1.1.1.cmml"><mi id="S5.T1.4.4.4.m1.1.1.2" xref="S5.T1.4.4.4.m1.1.1.2.cmml">f</mi><mi id="S5.T1.4.4.4.m1.1.1.3" xref="S5.T1.4.4.4.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S5.T1.4.4.4.m1.1b"><apply id="S5.T1.4.4.4.m1.1.1.cmml" xref="S5.T1.4.4.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.4.4.4.m1.1.1.1.cmml" xref="S5.T1.4.4.4.m1.1.1">subscript</csymbol><ci id="S5.T1.4.4.4.m1.1.1.2.cmml" xref="S5.T1.4.4.4.m1.1.1.2">𝑓</ci><ci id="S5.T1.4.4.4.m1.1.1.3.cmml" xref="S5.T1.4.4.4.m1.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.4.4.4.m1.1c">f_{c}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.4.4.4.m1.1d">italic_f start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.7.7.11" style="padding-left:11.0pt;padding-right:11.0pt;">BW</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.5.5.5" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="T_{T_{x}}" class="ltx_Math" display="inline" id="S5.T1.5.5.5.m1.1"><semantics id="S5.T1.5.5.5.m1.1a"><msub id="S5.T1.5.5.5.m1.1.1" xref="S5.T1.5.5.5.m1.1.1.cmml"><mi id="S5.T1.5.5.5.m1.1.1.2" xref="S5.T1.5.5.5.m1.1.1.2.cmml">T</mi><msub id="S5.T1.5.5.5.m1.1.1.3" xref="S5.T1.5.5.5.m1.1.1.3.cmml"><mi id="S5.T1.5.5.5.m1.1.1.3.2" xref="S5.T1.5.5.5.m1.1.1.3.2.cmml">T</mi><mi id="S5.T1.5.5.5.m1.1.1.3.3" xref="S5.T1.5.5.5.m1.1.1.3.3.cmml">x</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S5.T1.5.5.5.m1.1b"><apply id="S5.T1.5.5.5.m1.1.1.cmml" xref="S5.T1.5.5.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.5.5.5.m1.1.1.1.cmml" xref="S5.T1.5.5.5.m1.1.1">subscript</csymbol><ci id="S5.T1.5.5.5.m1.1.1.2.cmml" xref="S5.T1.5.5.5.m1.1.1.2">𝑇</ci><apply id="S5.T1.5.5.5.m1.1.1.3.cmml" xref="S5.T1.5.5.5.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T1.5.5.5.m1.1.1.3.1.cmml" xref="S5.T1.5.5.5.m1.1.1.3">subscript</csymbol><ci id="S5.T1.5.5.5.m1.1.1.3.2.cmml" xref="S5.T1.5.5.5.m1.1.1.3.2">𝑇</ci><ci id="S5.T1.5.5.5.m1.1.1.3.3.cmml" xref="S5.T1.5.5.5.m1.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.5.5.5.m1.1c">T_{T_{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.5.5.5.m1.1d">italic_T start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.6.6.6" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="f_{s}" class="ltx_Math" display="inline" id="S5.T1.6.6.6.m1.1"><semantics id="S5.T1.6.6.6.m1.1a"><msub id="S5.T1.6.6.6.m1.1.1" xref="S5.T1.6.6.6.m1.1.1.cmml"><mi id="S5.T1.6.6.6.m1.1.1.2" xref="S5.T1.6.6.6.m1.1.1.2.cmml">f</mi><mi id="S5.T1.6.6.6.m1.1.1.3" xref="S5.T1.6.6.6.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S5.T1.6.6.6.m1.1b"><apply id="S5.T1.6.6.6.m1.1.1.cmml" xref="S5.T1.6.6.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.6.6.6.m1.1.1.1.cmml" xref="S5.T1.6.6.6.m1.1.1">subscript</csymbol><ci id="S5.T1.6.6.6.m1.1.1.2.cmml" xref="S5.T1.6.6.6.m1.1.1.2">𝑓</ci><ci id="S5.T1.6.6.6.m1.1.1.3.cmml" xref="S5.T1.6.6.6.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.6.6.6.m1.1c">f_{s}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.6.6.6.m1.1d">italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.7.7.7" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="\sigma_{w}^{2}" class="ltx_Math" display="inline" id="S5.T1.7.7.7.m1.1"><semantics id="S5.T1.7.7.7.m1.1a"><msubsup id="S5.T1.7.7.7.m1.1.1" xref="S5.T1.7.7.7.m1.1.1.cmml"><mi id="S5.T1.7.7.7.m1.1.1.2.2" xref="S5.T1.7.7.7.m1.1.1.2.2.cmml">σ</mi><mi id="S5.T1.7.7.7.m1.1.1.2.3" xref="S5.T1.7.7.7.m1.1.1.2.3.cmml">w</mi><mn id="S5.T1.7.7.7.m1.1.1.3" xref="S5.T1.7.7.7.m1.1.1.3.cmml">2</mn></msubsup><annotation-xml encoding="MathML-Content" id="S5.T1.7.7.7.m1.1b"><apply id="S5.T1.7.7.7.m1.1.1.cmml" xref="S5.T1.7.7.7.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.7.7.7.m1.1.1.1.cmml" xref="S5.T1.7.7.7.m1.1.1">superscript</csymbol><apply id="S5.T1.7.7.7.m1.1.1.2.cmml" xref="S5.T1.7.7.7.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.7.7.7.m1.1.1.2.1.cmml" xref="S5.T1.7.7.7.m1.1.1">subscript</csymbol><ci id="S5.T1.7.7.7.m1.1.1.2.2.cmml" xref="S5.T1.7.7.7.m1.1.1.2.2">𝜎</ci><ci id="S5.T1.7.7.7.m1.1.1.2.3.cmml" xref="S5.T1.7.7.7.m1.1.1.2.3">𝑤</ci></apply><cn id="S5.T1.7.7.7.m1.1.1.3.cmml" type="integer" xref="S5.T1.7.7.7.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.7.7.7.m1.1c">\sigma_{w}^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.7.7.7.m1.1d">italic_σ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.9.9"> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.3" style="padding-left:11.0pt;padding-right:11.0pt;">3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.4" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.4.1">\qty</span>10Hz</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.5" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.5.1">\qty</span>0.05m^2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.6" style="padding-left:11.0pt;padding-right:11.0pt;">1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.7" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.7.1">\qty</span>6.99W</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.8" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.8.1">\qty</span>300m</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.9" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.9.1">\qty</span>10GHz</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.10" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.10.1">\qty</span>20MHz</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.8.8.1" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.8.8.1.1">\qty</span>16<math alttext="\mu" class="ltx_Math" display="inline" id="S5.T1.8.8.1.m1.1"><semantics id="S5.T1.8.8.1.m1.1a"><mi id="S5.T1.8.8.1.m1.1.1" xref="S5.T1.8.8.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S5.T1.8.8.1.m1.1b"><ci id="S5.T1.8.8.1.m1.1.1.cmml" xref="S5.T1.8.8.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.8.8.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S5.T1.8.8.1.m1.1d">italic_μ</annotation></semantics></math> s</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.11" style="padding-left:11.0pt;padding-right:11.0pt;"> <span class="ltx_ERROR undefined" id="S5.T1.9.9.11.1">\qty</span>256MHz</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.9.9.2" style="padding-left:11.0pt;padding-right:11.0pt;"><math alttext="\text{BW}\cdot k_{b}\cdot 290" class="ltx_Math" display="inline" id="S5.T1.9.9.2.m1.1"><semantics id="S5.T1.9.9.2.m1.1a"><mrow id="S5.T1.9.9.2.m1.1.1" xref="S5.T1.9.9.2.m1.1.1.cmml"><mtext id="S5.T1.9.9.2.m1.1.1.2" xref="S5.T1.9.9.2.m1.1.1.2a.cmml">BW</mtext><mo id="S5.T1.9.9.2.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.9.9.2.m1.1.1.1.cmml">⋅</mo><msub id="S5.T1.9.9.2.m1.1.1.3" xref="S5.T1.9.9.2.m1.1.1.3.cmml"><mi id="S5.T1.9.9.2.m1.1.1.3.2" xref="S5.T1.9.9.2.m1.1.1.3.2.cmml">k</mi><mi id="S5.T1.9.9.2.m1.1.1.3.3" xref="S5.T1.9.9.2.m1.1.1.3.3.cmml">b</mi></msub><mo id="S5.T1.9.9.2.m1.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S5.T1.9.9.2.m1.1.1.1.cmml">⋅</mo><mn id="S5.T1.9.9.2.m1.1.1.4" xref="S5.T1.9.9.2.m1.1.1.4.cmml">290</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.9.9.2.m1.1b"><apply id="S5.T1.9.9.2.m1.1.1.cmml" xref="S5.T1.9.9.2.m1.1.1"><ci id="S5.T1.9.9.2.m1.1.1.1.cmml" xref="S5.T1.9.9.2.m1.1.1.1">⋅</ci><ci id="S5.T1.9.9.2.m1.1.1.2a.cmml" xref="S5.T1.9.9.2.m1.1.1.2"><mtext id="S5.T1.9.9.2.m1.1.1.2.cmml" xref="S5.T1.9.9.2.m1.1.1.2">BW</mtext></ci><apply id="S5.T1.9.9.2.m1.1.1.3.cmml" xref="S5.T1.9.9.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T1.9.9.2.m1.1.1.3.1.cmml" xref="S5.T1.9.9.2.m1.1.1.3">subscript</csymbol><ci id="S5.T1.9.9.2.m1.1.1.3.2.cmml" xref="S5.T1.9.9.2.m1.1.1.3.2">𝑘</ci><ci id="S5.T1.9.9.2.m1.1.1.3.3.cmml" xref="S5.T1.9.9.2.m1.1.1.3.3">𝑏</ci></apply><cn id="S5.T1.9.9.2.m1.1.1.4.cmml" type="integer" xref="S5.T1.9.9.2.m1.1.1.4">290</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.9.9.2.m1.1c">\text{BW}\cdot k_{b}\cdot 290</annotation><annotation encoding="application/x-llamapun" id="S5.T1.9.9.2.m1.1d">BW ⋅ italic_k start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⋅ 290</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.10.10"> <td class="ltx_td ltx_align_left ltx_border_tt" colspan="11" id="S5.T1.10.10.1" style="padding-left:11.0pt;padding-right:11.0pt;"> <math alttext="k_{b}" class="ltx_Math" display="inline" id="S5.T1.10.10.1.m1.1"><semantics id="S5.T1.10.10.1.m1.1a"><msub id="S5.T1.10.10.1.m1.1.1" xref="S5.T1.10.10.1.m1.1.1.cmml"><mi id="S5.T1.10.10.1.m1.1.1.2" xref="S5.T1.10.10.1.m1.1.1.2.cmml">k</mi><mi id="S5.T1.10.10.1.m1.1.1.3" xref="S5.T1.10.10.1.m1.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S5.T1.10.10.1.m1.1b"><apply id="S5.T1.10.10.1.m1.1.1.cmml" xref="S5.T1.10.10.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.10.10.1.m1.1.1.1.cmml" xref="S5.T1.10.10.1.m1.1.1">subscript</csymbol><ci id="S5.T1.10.10.1.m1.1.1.2.cmml" xref="S5.T1.10.10.1.m1.1.1.2">𝑘</ci><ci id="S5.T1.10.10.1.m1.1.1.3.cmml" xref="S5.T1.10.10.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.10.10.1.m1.1c">k_{b}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.10.10.1.m1.1d">italic_k start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> is the Boltzmann constant.</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.12">As a baseline, a KF using backwards smoothing was chosen, where the observation matrix was extended as follows,</p> <table class="ltx_equation ltx_eqn_table" id="S5.E36"> <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{H}=\begin{bmatrix}\bm{H}_{\text{radar}}^{(1)}\\ \vdots\\ \bm{H}_{\text{radar}}^{(N_{\text{radar}})}\end{bmatrix},\phantom{mm}\bm{H}_{% \text{radar}}^{(k)}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\end{bmatrix}\!." class="ltx_Math" display="block" id="S5.E36.m1.4"><semantics id="S5.E36.m1.4a"><mrow id="S5.E36.m1.4.4.1"><mrow id="S5.E36.m1.4.4.1.1.2" xref="S5.E36.m1.4.4.1.1.3.cmml"><mrow id="S5.E36.m1.4.4.1.1.1.1" xref="S5.E36.m1.4.4.1.1.1.1.cmml"><mi id="S5.E36.m1.4.4.1.1.1.1.2" xref="S5.E36.m1.4.4.1.1.1.1.2.cmml">𝑯</mi><mo id="S5.E36.m1.4.4.1.1.1.1.1" xref="S5.E36.m1.4.4.1.1.1.1.1.cmml">=</mo><mrow id="S5.E36.m1.1.1.3" xref="S5.E36.m1.1.1.2.cmml"><mo id="S5.E36.m1.1.1.3.1" xref="S5.E36.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S5.E36.m1.1.1.1.1" rowspacing="0pt" xref="S5.E36.m1.1.1.1.1.cmml"><mtr id="S5.E36.m1.1.1.1.1a" xref="S5.E36.m1.1.1.1.1.cmml"><mtd id="S5.E36.m1.1.1.1.1b" xref="S5.E36.m1.1.1.1.1.cmml"><msubsup id="S5.E36.m1.1.1.1.1.1.1.1.1" xref="S5.E36.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.E36.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.E36.m1.1.1.1.1.1.1.1.1.3.2.cmml">𝑯</mi><mtext id="S5.E36.m1.1.1.1.1.1.1.1.1.3.3" xref="S5.E36.m1.1.1.1.1.1.1.1.1.3.3a.cmml">radar</mtext><mrow id="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.E36.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.3.1" stretchy="false" xref="S5.E36.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mn id="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">1</mn><mo id="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.3.2" stretchy="false" xref="S5.E36.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr><mtr id="S5.E36.m1.1.1.1.1c" xref="S5.E36.m1.1.1.1.1.cmml"><mtd id="S5.E36.m1.1.1.1.1d" xref="S5.E36.m1.1.1.1.1.cmml"><mi id="S5.E36.m1.1.1.1.1.3.1.1" mathvariant="normal" xref="S5.E36.m1.1.1.1.1.3.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S5.E36.m1.1.1.1.1e" xref="S5.E36.m1.1.1.1.1.cmml"><mtd id="S5.E36.m1.1.1.1.1f" xref="S5.E36.m1.1.1.1.1.cmml"><msubsup id="S5.E36.m1.1.1.1.1.2.2.1.1" xref="S5.E36.m1.1.1.1.1.2.2.1.1.cmml"><mi id="S5.E36.m1.1.1.1.1.2.2.1.1.3.2" xref="S5.E36.m1.1.1.1.1.2.2.1.1.3.2.cmml">𝑯</mi><mtext id="S5.E36.m1.1.1.1.1.2.2.1.1.3.3" xref="S5.E36.m1.1.1.1.1.2.2.1.1.3.3a.cmml">radar</mtext><mrow id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.cmml"><mo id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.cmml"><mi id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.2" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.cmml">N</mi><mtext id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.3" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.3a.cmml">radar</mtext></msub><mo id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.3" stretchy="false" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr></mtable><mo id="S5.E36.m1.1.1.3.2" xref="S5.E36.m1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S5.E36.m1.4.4.1.1.2.3" rspace="1.927em" xref="S5.E36.m1.4.4.1.1.3a.cmml">,</mo><mrow id="S5.E36.m1.4.4.1.1.2.2" xref="S5.E36.m1.4.4.1.1.2.2.cmml"><msubsup id="S5.E36.m1.4.4.1.1.2.2.2" xref="S5.E36.m1.4.4.1.1.2.2.2.cmml"><mi id="S5.E36.m1.4.4.1.1.2.2.2.2.2" xref="S5.E36.m1.4.4.1.1.2.2.2.2.2.cmml">𝑯</mi><mtext id="S5.E36.m1.4.4.1.1.2.2.2.2.3" xref="S5.E36.m1.4.4.1.1.2.2.2.2.3a.cmml">radar</mtext><mrow id="S5.E36.m1.3.3.1.3" xref="S5.E36.m1.4.4.1.1.2.2.2.cmml"><mo id="S5.E36.m1.3.3.1.3.1" stretchy="false" xref="S5.E36.m1.4.4.1.1.2.2.2.cmml">(</mo><mi id="S5.E36.m1.3.3.1.1" xref="S5.E36.m1.3.3.1.1.cmml">k</mi><mo id="S5.E36.m1.3.3.1.3.2" stretchy="false" xref="S5.E36.m1.4.4.1.1.2.2.2.cmml">)</mo></mrow></msubsup><mo id="S5.E36.m1.4.4.1.1.2.2.1" xref="S5.E36.m1.4.4.1.1.2.2.1.cmml">=</mo><mrow id="S5.E36.m1.2.2.3" xref="S5.E36.m1.2.2.2.cmml"><mo id="S5.E36.m1.2.2.3.1" xref="S5.E36.m1.2.2.2.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" id="S5.E36.m1.2.2.1.1" rowspacing="0pt" xref="S5.E36.m1.2.2.1.1.cmml"><mtr id="S5.E36.m1.2.2.1.1a" xref="S5.E36.m1.2.2.1.1.cmml"><mtd id="S5.E36.m1.2.2.1.1b" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.1.1.1" xref="S5.E36.m1.2.2.1.1.1.1.1.cmml">1</mn></mtd><mtd id="S5.E36.m1.2.2.1.1c" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.1.2.1" xref="S5.E36.m1.2.2.1.1.1.2.1.cmml">0</mn></mtd><mtd id="S5.E36.m1.2.2.1.1d" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.1.3.1" xref="S5.E36.m1.2.2.1.1.1.3.1.cmml">0</mn></mtd><mtd id="S5.E36.m1.2.2.1.1e" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.1.4.1" xref="S5.E36.m1.2.2.1.1.1.4.1.cmml">0</mn></mtd></mtr><mtr id="S5.E36.m1.2.2.1.1f" xref="S5.E36.m1.2.2.1.1.cmml"><mtd id="S5.E36.m1.2.2.1.1g" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.2.1.1" xref="S5.E36.m1.2.2.1.1.2.1.1.cmml">0</mn></mtd><mtd id="S5.E36.m1.2.2.1.1h" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.2.2.1" xref="S5.E36.m1.2.2.1.1.2.2.1.cmml">1</mn></mtd><mtd id="S5.E36.m1.2.2.1.1i" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.2.3.1" xref="S5.E36.m1.2.2.1.1.2.3.1.cmml">0</mn></mtd><mtd id="S5.E36.m1.2.2.1.1j" xref="S5.E36.m1.2.2.1.1.cmml"><mn id="S5.E36.m1.2.2.1.1.2.4.1" xref="S5.E36.m1.2.2.1.1.2.4.1.cmml">0</mn></mtd></mtr></mtable><mo id="S5.E36.m1.2.2.3.2" xref="S5.E36.m1.2.2.2.1.cmml">]</mo></mrow></mrow></mrow><mo id="S5.E36.m1.4.4.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E36.m1.4b"><apply id="S5.E36.m1.4.4.1.1.3.cmml" xref="S5.E36.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S5.E36.m1.4.4.1.1.3a.cmml" xref="S5.E36.m1.4.4.1.1.2.3">formulae-sequence</csymbol><apply id="S5.E36.m1.4.4.1.1.1.1.cmml" xref="S5.E36.m1.4.4.1.1.1.1"><eq id="S5.E36.m1.4.4.1.1.1.1.1.cmml" xref="S5.E36.m1.4.4.1.1.1.1.1"></eq><ci id="S5.E36.m1.4.4.1.1.1.1.2.cmml" xref="S5.E36.m1.4.4.1.1.1.1.2">𝑯</ci><apply id="S5.E36.m1.1.1.2.cmml" xref="S5.E36.m1.1.1.3"><csymbol cd="latexml" id="S5.E36.m1.1.1.2.1.cmml" xref="S5.E36.m1.1.1.3.1">matrix</csymbol><matrix id="S5.E36.m1.1.1.1.1.cmml" xref="S5.E36.m1.1.1.1.1"><matrixrow id="S5.E36.m1.1.1.1.1a.cmml" xref="S5.E36.m1.1.1.1.1"><apply id="S5.E36.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.E36.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E36.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E36.m1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.E36.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.E36.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E36.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.E36.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.E36.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.E36.m1.1.1.1.1.1.1.1.1.3.2">𝑯</ci><ci id="S5.E36.m1.1.1.1.1.1.1.1.1.3.3a.cmml" xref="S5.E36.m1.1.1.1.1.1.1.1.1.3.3"><mtext id="S5.E36.m1.1.1.1.1.1.1.1.1.3.3.cmml" mathsize="70%" xref="S5.E36.m1.1.1.1.1.1.1.1.1.3.3">radar</mtext></ci></apply><cn id="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" type="integer" xref="S5.E36.m1.1.1.1.1.1.1.1.1.1.1.1">1</cn></apply></matrixrow><matrixrow id="S5.E36.m1.1.1.1.1b.cmml" xref="S5.E36.m1.1.1.1.1"><ci id="S5.E36.m1.1.1.1.1.3.1.1.cmml" xref="S5.E36.m1.1.1.1.1.3.1.1">⋮</ci></matrixrow><matrixrow id="S5.E36.m1.1.1.1.1c.cmml" xref="S5.E36.m1.1.1.1.1"><apply id="S5.E36.m1.1.1.1.1.2.2.1.1.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S5.E36.m1.1.1.1.1.2.2.1.1.2.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1">superscript</csymbol><apply id="S5.E36.m1.1.1.1.1.2.2.1.1.3.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S5.E36.m1.1.1.1.1.2.2.1.1.3.1.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1">subscript</csymbol><ci id="S5.E36.m1.1.1.1.1.2.2.1.1.3.2.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1.3.2">𝑯</ci><ci id="S5.E36.m1.1.1.1.1.2.2.1.1.3.3a.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1.3.3"><mtext id="S5.E36.m1.1.1.1.1.2.2.1.1.3.3.cmml" mathsize="70%" xref="S5.E36.m1.1.1.1.1.2.2.1.1.3.3">radar</mtext></ci></apply><apply id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.2.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.3a.cmml" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.3"><mtext id="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.3.cmml" mathsize="50%" xref="S5.E36.m1.1.1.1.1.2.2.1.1.1.1.1.1.3">radar</mtext></ci></apply></apply></matrixrow></matrix></apply></apply><apply id="S5.E36.m1.4.4.1.1.2.2.cmml" xref="S5.E36.m1.4.4.1.1.2.2"><eq id="S5.E36.m1.4.4.1.1.2.2.1.cmml" xref="S5.E36.m1.4.4.1.1.2.2.1"></eq><apply id="S5.E36.m1.4.4.1.1.2.2.2.cmml" xref="S5.E36.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.E36.m1.4.4.1.1.2.2.2.1.cmml" xref="S5.E36.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="S5.E36.m1.4.4.1.1.2.2.2.2.cmml" xref="S5.E36.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.E36.m1.4.4.1.1.2.2.2.2.1.cmml" xref="S5.E36.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="S5.E36.m1.4.4.1.1.2.2.2.2.2.cmml" xref="S5.E36.m1.4.4.1.1.2.2.2.2.2">𝑯</ci><ci id="S5.E36.m1.4.4.1.1.2.2.2.2.3a.cmml" xref="S5.E36.m1.4.4.1.1.2.2.2.2.3"><mtext id="S5.E36.m1.4.4.1.1.2.2.2.2.3.cmml" mathsize="70%" xref="S5.E36.m1.4.4.1.1.2.2.2.2.3">radar</mtext></ci></apply><ci id="S5.E36.m1.3.3.1.1.cmml" xref="S5.E36.m1.3.3.1.1">𝑘</ci></apply><apply id="S5.E36.m1.2.2.2.cmml" xref="S5.E36.m1.2.2.3"><csymbol cd="latexml" id="S5.E36.m1.2.2.2.1.cmml" xref="S5.E36.m1.2.2.3.1">matrix</csymbol><matrix id="S5.E36.m1.2.2.1.1.cmml" xref="S5.E36.m1.2.2.1.1"><matrixrow id="S5.E36.m1.2.2.1.1a.cmml" xref="S5.E36.m1.2.2.1.1"><cn id="S5.E36.m1.2.2.1.1.1.1.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.1.1.1">1</cn><cn id="S5.E36.m1.2.2.1.1.1.2.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.1.2.1">0</cn><cn id="S5.E36.m1.2.2.1.1.1.3.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.1.3.1">0</cn><cn id="S5.E36.m1.2.2.1.1.1.4.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.1.4.1">0</cn></matrixrow><matrixrow id="S5.E36.m1.2.2.1.1b.cmml" xref="S5.E36.m1.2.2.1.1"><cn id="S5.E36.m1.2.2.1.1.2.1.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.2.1.1">0</cn><cn id="S5.E36.m1.2.2.1.1.2.2.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.2.2.1">1</cn><cn id="S5.E36.m1.2.2.1.1.2.3.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.2.3.1">0</cn><cn id="S5.E36.m1.2.2.1.1.2.4.1.cmml" type="integer" xref="S5.E36.m1.2.2.1.1.2.4.1">0</cn></matrixrow></matrix></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E36.m1.4c">\bm{H}=\begin{bmatrix}\bm{H}_{\text{radar}}^{(1)}\\ \vdots\\ \bm{H}_{\text{radar}}^{(N_{\text{radar}})}\end{bmatrix},\phantom{mm}\bm{H}_{% \text{radar}}^{(k)}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\end{bmatrix}\!.</annotation><annotation encoding="application/x-llamapun" id="S5.E36.m1.4d">bold_italic_H = [ start_ARG start_ROW start_CELL bold_italic_H start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_italic_H start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] , bold_italic_H start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = [ start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 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">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.13">To have a fair comparison between the algorithms, the KF also assumes linear motion for the target and the observations for the KF presupposes that a target exists. The range to the target can be found as follows,</p> <table class="ltx_equation ltx_eqn_table" id="S5.E37"> <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="r_{n}^{(k)}=\text{median}\!\left(\underset{r}{\text{argmax}}\left|\mathcal{F}^% {-1}\left\{\bm{Z}_{n}^{(k)}(r,\theta)\right\}\right|\right)\!," class="ltx_Math" display="block" id="S5.E37.m1.5"><semantics id="S5.E37.m1.5a"><mrow id="S5.E37.m1.5.5.1" xref="S5.E37.m1.5.5.1.1.cmml"><mrow id="S5.E37.m1.5.5.1.1" xref="S5.E37.m1.5.5.1.1.cmml"><msubsup id="S5.E37.m1.5.5.1.1.3" xref="S5.E37.m1.5.5.1.1.3.cmml"><mi id="S5.E37.m1.5.5.1.1.3.2.2" xref="S5.E37.m1.5.5.1.1.3.2.2.cmml">r</mi><mi id="S5.E37.m1.5.5.1.1.3.2.3" xref="S5.E37.m1.5.5.1.1.3.2.3.cmml">n</mi><mrow id="S5.E37.m1.1.1.1.3" xref="S5.E37.m1.5.5.1.1.3.cmml"><mo id="S5.E37.m1.1.1.1.3.1" stretchy="false" xref="S5.E37.m1.5.5.1.1.3.cmml">(</mo><mi id="S5.E37.m1.1.1.1.1" xref="S5.E37.m1.1.1.1.1.cmml">k</mi><mo id="S5.E37.m1.1.1.1.3.2" stretchy="false" xref="S5.E37.m1.5.5.1.1.3.cmml">)</mo></mrow></msubsup><mo id="S5.E37.m1.5.5.1.1.2" xref="S5.E37.m1.5.5.1.1.2.cmml">=</mo><mrow id="S5.E37.m1.5.5.1.1.1" xref="S5.E37.m1.5.5.1.1.1.cmml"><mpadded width="3.167em"><mtext id="S5.E37.m1.5.5.1.1.1.3" xref="S5.E37.m1.5.5.1.1.1.3a.cmml">median</mtext></mpadded><mo id="S5.E37.m1.5.5.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.cmml"><mo id="S5.E37.m1.5.5.1.1.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.cmml"><munder accentunder="true" id="S5.E37.m1.5.5.1.1.1.1.1.1.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3.cmml"><mtext id="S5.E37.m1.5.5.1.1.1.1.1.1.3.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3.2a.cmml">argmax</mtext><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.3.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3.1.cmml">𝑟</mo></munder><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.2.cmml"><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.2.1.cmml">|</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.cmml"><msup id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.2.cmml">ℱ</mi><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.cmml"><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3a" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.cmml">−</mo><mn id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml">{</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><msubsup id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">𝒁</mi><mi id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.3.cmml">n</mi><mrow id="S5.E37.m1.2.2.1.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S5.E37.m1.2.2.1.3.1" stretchy="false" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">(</mo><mi id="S5.E37.m1.2.2.1.1" xref="S5.E37.m1.2.2.1.1.cmml">k</mi><mo id="S5.E37.m1.2.2.1.3.2" stretchy="false" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></msubsup><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">(</mo><mi id="S5.E37.m1.3.3" xref="S5.E37.m1.3.3.cmml">r</mi><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.2.2" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">,</mo><mi id="S5.E37.m1.4.4" xref="S5.E37.m1.4.4.cmml">θ</mi><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.2.3" stretchy="false" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mpadded width="0.288em"><mo id="S5.E37.m1.5.5.1.1.1.1.1.3" xref="S5.E37.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mpadded></mrow></mrow></mrow><mo id="S5.E37.m1.5.5.1.2" xref="S5.E37.m1.5.5.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E37.m1.5b"><apply id="S5.E37.m1.5.5.1.1.cmml" xref="S5.E37.m1.5.5.1"><eq id="S5.E37.m1.5.5.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.2"></eq><apply id="S5.E37.m1.5.5.1.1.3.cmml" xref="S5.E37.m1.5.5.1.1.3"><csymbol cd="ambiguous" id="S5.E37.m1.5.5.1.1.3.1.cmml" xref="S5.E37.m1.5.5.1.1.3">superscript</csymbol><apply id="S5.E37.m1.5.5.1.1.3.2.cmml" xref="S5.E37.m1.5.5.1.1.3"><csymbol cd="ambiguous" id="S5.E37.m1.5.5.1.1.3.2.1.cmml" xref="S5.E37.m1.5.5.1.1.3">subscript</csymbol><ci id="S5.E37.m1.5.5.1.1.3.2.2.cmml" xref="S5.E37.m1.5.5.1.1.3.2.2">𝑟</ci><ci id="S5.E37.m1.5.5.1.1.3.2.3.cmml" xref="S5.E37.m1.5.5.1.1.3.2.3">𝑛</ci></apply><ci id="S5.E37.m1.1.1.1.1.cmml" xref="S5.E37.m1.1.1.1.1">𝑘</ci></apply><apply id="S5.E37.m1.5.5.1.1.1.cmml" xref="S5.E37.m1.5.5.1.1.1"><times id="S5.E37.m1.5.5.1.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.1.2"></times><ci id="S5.E37.m1.5.5.1.1.1.3a.cmml" xref="S5.E37.m1.5.5.1.1.1.3"><mtext id="S5.E37.m1.5.5.1.1.1.3.cmml" xref="S5.E37.m1.5.5.1.1.1.3">median</mtext></ci><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1"><times id="S5.E37.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.2"></times><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3"><ci id="S5.E37.m1.5.5.1.1.1.1.1.1.3.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3.1">𝑟</ci><ci id="S5.E37.m1.5.5.1.1.1.1.1.1.3.2a.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3.2"><mtext id="S5.E37.m1.5.5.1.1.1.1.1.1.3.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.3.2">argmax</mtext></ci></apply><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1"><abs id="S5.E37.m1.5.5.1.1.1.1.1.1.1.2.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.2"></abs><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1"><times id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.2"></times><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.2">ℱ</ci><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3"><minus id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3"></minus><cn id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.3.3.2">1</cn></apply></apply><set id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1"><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1"><times id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1"></times><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><apply id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2">𝒁</ci><ci id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.2.2.3">𝑛</ci></apply><ci id="S5.E37.m1.2.2.1.1.cmml" xref="S5.E37.m1.2.2.1.1">𝑘</ci></apply><interval closure="open" id="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.E37.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.3.2"><ci id="S5.E37.m1.3.3.cmml" xref="S5.E37.m1.3.3">𝑟</ci><ci id="S5.E37.m1.4.4.cmml" xref="S5.E37.m1.4.4">𝜃</ci></interval></apply></set></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E37.m1.5c">r_{n}^{(k)}=\text{median}\!\left(\underset{r}{\text{argmax}}\left|\mathcal{F}^% {-1}\left\{\bm{Z}_{n}^{(k)}(r,\theta)\right\}\right|\right)\!,</annotation><annotation encoding="application/x-llamapun" id="S5.E37.m1.5d">italic_r start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = median ( underitalic_r start_ARG argmax end_ARG | caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT { bold_italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_r , italic_θ ) } | ) ,</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="S5.p2.3">with <math alttext="\mathcal{F}^{-1}\left\{\cdot\right\}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mrow id="S5.p2.1.m1.1.2" xref="S5.p2.1.m1.1.2.cmml"><msup id="S5.p2.1.m1.1.2.2" xref="S5.p2.1.m1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p2.1.m1.1.2.2.2" xref="S5.p2.1.m1.1.2.2.2.cmml">ℱ</mi><mrow id="S5.p2.1.m1.1.2.2.3" xref="S5.p2.1.m1.1.2.2.3.cmml"><mo id="S5.p2.1.m1.1.2.2.3a" xref="S5.p2.1.m1.1.2.2.3.cmml">−</mo><mn id="S5.p2.1.m1.1.2.2.3.2" xref="S5.p2.1.m1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.p2.1.m1.1.2.1" xref="S5.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.p2.1.m1.1.2.3.2" xref="S5.p2.1.m1.1.2.3.1.cmml"><mo id="S5.p2.1.m1.1.2.3.2.1" xref="S5.p2.1.m1.1.2.3.1.cmml">{</mo><mo id="S5.p2.1.m1.1.1" lspace="0em" rspace="0em" xref="S5.p2.1.m1.1.1.cmml">⋅</mo><mo id="S5.p2.1.m1.1.2.3.2.2" xref="S5.p2.1.m1.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.2.cmml" xref="S5.p2.1.m1.1.2"><times id="S5.p2.1.m1.1.2.1.cmml" xref="S5.p2.1.m1.1.2.1"></times><apply id="S5.p2.1.m1.1.2.2.cmml" xref="S5.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.2.2.1.cmml" xref="S5.p2.1.m1.1.2.2">superscript</csymbol><ci id="S5.p2.1.m1.1.2.2.2.cmml" xref="S5.p2.1.m1.1.2.2.2">ℱ</ci><apply id="S5.p2.1.m1.1.2.2.3.cmml" xref="S5.p2.1.m1.1.2.2.3"><minus id="S5.p2.1.m1.1.2.2.3.1.cmml" xref="S5.p2.1.m1.1.2.2.3"></minus><cn id="S5.p2.1.m1.1.2.2.3.2.cmml" type="integer" xref="S5.p2.1.m1.1.2.2.3.2">1</cn></apply></apply><set id="S5.p2.1.m1.1.2.3.1.cmml" xref="S5.p2.1.m1.1.2.3.2"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">⋅</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\mathcal{F}^{-1}\left\{\cdot\right\}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT { ⋅ }</annotation></semantics></math> being the inverse Fourier transform, and the median and argmax being taken over all <math alttext="N_{T}N_{R}" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mrow id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml"><msub id="S5.p2.2.m2.1.1.2" xref="S5.p2.2.m2.1.1.2.cmml"><mi id="S5.p2.2.m2.1.1.2.2" xref="S5.p2.2.m2.1.1.2.2.cmml">N</mi><mi id="S5.p2.2.m2.1.1.2.3" xref="S5.p2.2.m2.1.1.2.3.cmml">T</mi></msub><mo id="S5.p2.2.m2.1.1.1" xref="S5.p2.2.m2.1.1.1.cmml">⁢</mo><msub id="S5.p2.2.m2.1.1.3" xref="S5.p2.2.m2.1.1.3.cmml"><mi id="S5.p2.2.m2.1.1.3.2" xref="S5.p2.2.m2.1.1.3.2.cmml">N</mi><mi id="S5.p2.2.m2.1.1.3.3" xref="S5.p2.2.m2.1.1.3.3.cmml">R</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1"><times id="S5.p2.2.m2.1.1.1.cmml" xref="S5.p2.2.m2.1.1.1"></times><apply id="S5.p2.2.m2.1.1.2.cmml" xref="S5.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.2.1.cmml" xref="S5.p2.2.m2.1.1.2">subscript</csymbol><ci id="S5.p2.2.m2.1.1.2.2.cmml" xref="S5.p2.2.m2.1.1.2.2">𝑁</ci><ci id="S5.p2.2.m2.1.1.2.3.cmml" xref="S5.p2.2.m2.1.1.2.3">𝑇</ci></apply><apply id="S5.p2.2.m2.1.1.3.cmml" xref="S5.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.3.1.cmml" xref="S5.p2.2.m2.1.1.3">subscript</csymbol><ci id="S5.p2.2.m2.1.1.3.2.cmml" xref="S5.p2.2.m2.1.1.3.2">𝑁</ci><ci id="S5.p2.2.m2.1.1.3.3.cmml" xref="S5.p2.2.m2.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">N_{T}N_{R}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> channels. The direction of arrival <math alttext="\theta^{(k)}_{n}" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><msubsup id="S5.p2.3.m3.1.2" xref="S5.p2.3.m3.1.2.cmml"><mi id="S5.p2.3.m3.1.2.2.2" xref="S5.p2.3.m3.1.2.2.2.cmml">θ</mi><mi id="S5.p2.3.m3.1.2.3" xref="S5.p2.3.m3.1.2.3.cmml">n</mi><mrow id="S5.p2.3.m3.1.1.1.3" xref="S5.p2.3.m3.1.2.cmml"><mo id="S5.p2.3.m3.1.1.1.3.1" stretchy="false" xref="S5.p2.3.m3.1.2.cmml">(</mo><mi id="S5.p2.3.m3.1.1.1.1" xref="S5.p2.3.m3.1.1.1.1.cmml">k</mi><mo id="S5.p2.3.m3.1.1.1.3.2" stretchy="false" xref="S5.p2.3.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><apply id="S5.p2.3.m3.1.2.cmml" xref="S5.p2.3.m3.1.2"><csymbol cd="ambiguous" id="S5.p2.3.m3.1.2.1.cmml" xref="S5.p2.3.m3.1.2">subscript</csymbol><apply id="S5.p2.3.m3.1.2.2.cmml" xref="S5.p2.3.m3.1.2"><csymbol cd="ambiguous" id="S5.p2.3.m3.1.2.2.1.cmml" xref="S5.p2.3.m3.1.2">superscript</csymbol><ci id="S5.p2.3.m3.1.2.2.2.cmml" xref="S5.p2.3.m3.1.2.2.2">𝜃</ci><ci id="S5.p2.3.m3.1.1.1.1.cmml" xref="S5.p2.3.m3.1.1.1.1">𝑘</ci></apply><ci id="S5.p2.3.m3.1.2.3.cmml" xref="S5.p2.3.m3.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">\theta^{(k)}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">italic_θ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, is estimated using a Capon beamformer. The Cartesian coordinates for the radar under consideration are obtained as</p> <table class="ltx_equation ltx_eqn_table" id="S5.E38"> <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="u_{n}^{(k)}=r_{n}^{(k)}\text{sin}\big{(}\theta_{n}^{(k)}\big{)},\phantom{mm}v_% {n}^{(k)}=r_{n}^{(k)}\text{cos}\big{(}\theta_{n}^{(k)}\big{)}." class="ltx_Math" display="block" id="S5.E38.m1.7"><semantics id="S5.E38.m1.7a"><mrow id="S5.E38.m1.7.7.1"><mrow id="S5.E38.m1.7.7.1.1.2" xref="S5.E38.m1.7.7.1.1.3.cmml"><mrow id="S5.E38.m1.7.7.1.1.1.1" xref="S5.E38.m1.7.7.1.1.1.1.cmml"><msubsup id="S5.E38.m1.7.7.1.1.1.1.3" xref="S5.E38.m1.7.7.1.1.1.1.3.cmml"><mi id="S5.E38.m1.7.7.1.1.1.1.3.2.2" xref="S5.E38.m1.7.7.1.1.1.1.3.2.2.cmml">u</mi><mi id="S5.E38.m1.7.7.1.1.1.1.3.2.3" xref="S5.E38.m1.7.7.1.1.1.1.3.2.3.cmml">n</mi><mrow id="S5.E38.m1.1.1.1.3" xref="S5.E38.m1.7.7.1.1.1.1.3.cmml"><mo id="S5.E38.m1.1.1.1.3.1" stretchy="false" xref="S5.E38.m1.7.7.1.1.1.1.3.cmml">(</mo><mi id="S5.E38.m1.1.1.1.1" xref="S5.E38.m1.1.1.1.1.cmml">k</mi><mo id="S5.E38.m1.1.1.1.3.2" stretchy="false" xref="S5.E38.m1.7.7.1.1.1.1.3.cmml">)</mo></mrow></msubsup><mo id="S5.E38.m1.7.7.1.1.1.1.2" xref="S5.E38.m1.7.7.1.1.1.1.2.cmml">=</mo><mrow id="S5.E38.m1.7.7.1.1.1.1.1" xref="S5.E38.m1.7.7.1.1.1.1.1.cmml"><msubsup id="S5.E38.m1.7.7.1.1.1.1.1.3" xref="S5.E38.m1.7.7.1.1.1.1.1.3.cmml"><mi id="S5.E38.m1.7.7.1.1.1.1.1.3.2.2" xref="S5.E38.m1.7.7.1.1.1.1.1.3.2.2.cmml">r</mi><mi id="S5.E38.m1.7.7.1.1.1.1.1.3.2.3" xref="S5.E38.m1.7.7.1.1.1.1.1.3.2.3.cmml">n</mi><mrow id="S5.E38.m1.2.2.1.3" xref="S5.E38.m1.7.7.1.1.1.1.1.3.cmml"><mo id="S5.E38.m1.2.2.1.3.1" stretchy="false" xref="S5.E38.m1.7.7.1.1.1.1.1.3.cmml">(</mo><mi id="S5.E38.m1.2.2.1.1" xref="S5.E38.m1.2.2.1.1.cmml">k</mi><mo id="S5.E38.m1.2.2.1.3.2" stretchy="false" xref="S5.E38.m1.7.7.1.1.1.1.1.3.cmml">)</mo></mrow></msubsup><mo id="S5.E38.m1.7.7.1.1.1.1.1.2" xref="S5.E38.m1.7.7.1.1.1.1.1.2.cmml">⁢</mo><mtext id="S5.E38.m1.7.7.1.1.1.1.1.4" xref="S5.E38.m1.7.7.1.1.1.1.1.4a.cmml">sin</mtext><mo id="S5.E38.m1.7.7.1.1.1.1.1.2a" xref="S5.E38.m1.7.7.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E38.m1.7.7.1.1.1.1.1.1.1" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E38.m1.7.7.1.1.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml"><mi id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.2" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.2.cmml">θ</mi><mi id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.3" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.3.cmml">n</mi><mrow id="S5.E38.m1.3.3.1.3" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E38.m1.3.3.1.3.1" stretchy="false" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S5.E38.m1.3.3.1.1" xref="S5.E38.m1.3.3.1.1.cmml">k</mi><mo id="S5.E38.m1.3.3.1.3.2" stretchy="false" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.E38.m1.7.7.1.1.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E38.m1.7.7.1.1.2.3" rspace="1.927em" xref="S5.E38.m1.7.7.1.1.3a.cmml">,</mo><mrow id="S5.E38.m1.7.7.1.1.2.2" xref="S5.E38.m1.7.7.1.1.2.2.cmml"><msubsup id="S5.E38.m1.7.7.1.1.2.2.3" xref="S5.E38.m1.7.7.1.1.2.2.3.cmml"><mi id="S5.E38.m1.7.7.1.1.2.2.3.2.2" xref="S5.E38.m1.7.7.1.1.2.2.3.2.2.cmml">v</mi><mi id="S5.E38.m1.7.7.1.1.2.2.3.2.3" xref="S5.E38.m1.7.7.1.1.2.2.3.2.3.cmml">n</mi><mrow id="S5.E38.m1.4.4.1.3" xref="S5.E38.m1.7.7.1.1.2.2.3.cmml"><mo id="S5.E38.m1.4.4.1.3.1" stretchy="false" xref="S5.E38.m1.7.7.1.1.2.2.3.cmml">(</mo><mi id="S5.E38.m1.4.4.1.1" xref="S5.E38.m1.4.4.1.1.cmml">k</mi><mo id="S5.E38.m1.4.4.1.3.2" stretchy="false" xref="S5.E38.m1.7.7.1.1.2.2.3.cmml">)</mo></mrow></msubsup><mo id="S5.E38.m1.7.7.1.1.2.2.2" xref="S5.E38.m1.7.7.1.1.2.2.2.cmml">=</mo><mrow id="S5.E38.m1.7.7.1.1.2.2.1" xref="S5.E38.m1.7.7.1.1.2.2.1.cmml"><msubsup id="S5.E38.m1.7.7.1.1.2.2.1.3" xref="S5.E38.m1.7.7.1.1.2.2.1.3.cmml"><mi id="S5.E38.m1.7.7.1.1.2.2.1.3.2.2" xref="S5.E38.m1.7.7.1.1.2.2.1.3.2.2.cmml">r</mi><mi id="S5.E38.m1.7.7.1.1.2.2.1.3.2.3" xref="S5.E38.m1.7.7.1.1.2.2.1.3.2.3.cmml">n</mi><mrow id="S5.E38.m1.5.5.1.3" xref="S5.E38.m1.7.7.1.1.2.2.1.3.cmml"><mo id="S5.E38.m1.5.5.1.3.1" stretchy="false" xref="S5.E38.m1.7.7.1.1.2.2.1.3.cmml">(</mo><mi id="S5.E38.m1.5.5.1.1" xref="S5.E38.m1.5.5.1.1.cmml">k</mi><mo id="S5.E38.m1.5.5.1.3.2" stretchy="false" xref="S5.E38.m1.7.7.1.1.2.2.1.3.cmml">)</mo></mrow></msubsup><mo id="S5.E38.m1.7.7.1.1.2.2.1.2" xref="S5.E38.m1.7.7.1.1.2.2.1.2.cmml">⁢</mo><mtext id="S5.E38.m1.7.7.1.1.2.2.1.4" xref="S5.E38.m1.7.7.1.1.2.2.1.4a.cmml">cos</mtext><mo id="S5.E38.m1.7.7.1.1.2.2.1.2a" xref="S5.E38.m1.7.7.1.1.2.2.1.2.cmml">⁢</mo><mrow id="S5.E38.m1.7.7.1.1.2.2.1.1.1" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml"><mo id="S5.E38.m1.7.7.1.1.2.2.1.1.1.2" maxsize="120%" minsize="120%" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml">(</mo><msubsup id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml"><mi id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.2" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.2.cmml">θ</mi><mi id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.3" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.3.cmml">n</mi><mrow id="S5.E38.m1.6.6.1.3" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml"><mo id="S5.E38.m1.6.6.1.3.1" stretchy="false" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml">(</mo><mi id="S5.E38.m1.6.6.1.1" xref="S5.E38.m1.6.6.1.1.cmml">k</mi><mo id="S5.E38.m1.6.6.1.3.2" stretchy="false" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.E38.m1.7.7.1.1.2.2.1.1.1.3" maxsize="120%" minsize="120%" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S5.E38.m1.7.7.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E38.m1.7b"><apply id="S5.E38.m1.7.7.1.1.3.cmml" xref="S5.E38.m1.7.7.1.1.2"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.3a.cmml" xref="S5.E38.m1.7.7.1.1.2.3">formulae-sequence</csymbol><apply id="S5.E38.m1.7.7.1.1.1.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1"><eq id="S5.E38.m1.7.7.1.1.1.1.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.2"></eq><apply id="S5.E38.m1.7.7.1.1.1.1.3.cmml" xref="S5.E38.m1.7.7.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.1.1.3.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.3">superscript</csymbol><apply id="S5.E38.m1.7.7.1.1.1.1.3.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.1.1.3.2.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.3">subscript</csymbol><ci id="S5.E38.m1.7.7.1.1.1.1.3.2.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.3.2.2">𝑢</ci><ci id="S5.E38.m1.7.7.1.1.1.1.3.2.3.cmml" xref="S5.E38.m1.7.7.1.1.1.1.3.2.3">𝑛</ci></apply><ci id="S5.E38.m1.1.1.1.1.cmml" xref="S5.E38.m1.1.1.1.1">𝑘</ci></apply><apply id="S5.E38.m1.7.7.1.1.1.1.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1"><times id="S5.E38.m1.7.7.1.1.1.1.1.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.2"></times><apply id="S5.E38.m1.7.7.1.1.1.1.1.3.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.1.1.1.3.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.3">superscript</csymbol><apply id="S5.E38.m1.7.7.1.1.1.1.1.3.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.1.1.1.3.2.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.3">subscript</csymbol><ci id="S5.E38.m1.7.7.1.1.1.1.1.3.2.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.3.2.2">𝑟</ci><ci id="S5.E38.m1.7.7.1.1.1.1.1.3.2.3.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.3.2.3">𝑛</ci></apply><ci id="S5.E38.m1.2.2.1.1.cmml" xref="S5.E38.m1.2.2.1.1">𝑘</ci></apply><ci id="S5.E38.m1.7.7.1.1.1.1.1.4a.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.4"><mtext id="S5.E38.m1.7.7.1.1.1.1.1.4.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.4">sin</mtext></ci><apply id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.2">𝜃</ci><ci id="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.E38.m1.7.7.1.1.1.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S5.E38.m1.3.3.1.1.cmml" xref="S5.E38.m1.3.3.1.1">𝑘</ci></apply></apply></apply><apply id="S5.E38.m1.7.7.1.1.2.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2"><eq id="S5.E38.m1.7.7.1.1.2.2.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.2"></eq><apply id="S5.E38.m1.7.7.1.1.2.2.3.cmml" xref="S5.E38.m1.7.7.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.2.2.3.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.3">superscript</csymbol><apply id="S5.E38.m1.7.7.1.1.2.2.3.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.2.2.3.2.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.3">subscript</csymbol><ci id="S5.E38.m1.7.7.1.1.2.2.3.2.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.3.2.2">𝑣</ci><ci id="S5.E38.m1.7.7.1.1.2.2.3.2.3.cmml" xref="S5.E38.m1.7.7.1.1.2.2.3.2.3">𝑛</ci></apply><ci id="S5.E38.m1.4.4.1.1.cmml" xref="S5.E38.m1.4.4.1.1">𝑘</ci></apply><apply id="S5.E38.m1.7.7.1.1.2.2.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1"><times id="S5.E38.m1.7.7.1.1.2.2.1.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.2"></times><apply id="S5.E38.m1.7.7.1.1.2.2.1.3.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.2.2.1.3.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.3">superscript</csymbol><apply id="S5.E38.m1.7.7.1.1.2.2.1.3.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.3"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.2.2.1.3.2.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.3">subscript</csymbol><ci id="S5.E38.m1.7.7.1.1.2.2.1.3.2.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.3.2.2">𝑟</ci><ci id="S5.E38.m1.7.7.1.1.2.2.1.3.2.3.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.3.2.3">𝑛</ci></apply><ci id="S5.E38.m1.5.5.1.1.cmml" xref="S5.E38.m1.5.5.1.1">𝑘</ci></apply><ci id="S5.E38.m1.7.7.1.1.2.2.1.4a.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.4"><mtext id="S5.E38.m1.7.7.1.1.2.2.1.4.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.4">cos</mtext></ci><apply id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1">superscript</csymbol><apply id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.1.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1">subscript</csymbol><ci id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.2.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.2">𝜃</ci><ci id="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.3.cmml" xref="S5.E38.m1.7.7.1.1.2.2.1.1.1.1.2.3">𝑛</ci></apply><ci id="S5.E38.m1.6.6.1.1.cmml" xref="S5.E38.m1.6.6.1.1">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E38.m1.7c">u_{n}^{(k)}=r_{n}^{(k)}\text{sin}\big{(}\theta_{n}^{(k)}\big{)},\phantom{mm}v_% {n}^{(k)}=r_{n}^{(k)}\text{cos}\big{(}\theta_{n}^{(k)}\big{)}.</annotation><annotation encoding="application/x-llamapun" id="S5.E38.m1.7d">italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = italic_r start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT sin ( italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) , italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = italic_r start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT cos ( italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT 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">(38)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.8">Each coordinate set <math alttext="\big{(}u_{n}^{(k)},v_{n}^{(k)}\big{)}" class="ltx_Math" display="inline" id="S5.p2.4.m1.4"><semantics id="S5.p2.4.m1.4a"><mrow id="S5.p2.4.m1.4.4.2" xref="S5.p2.4.m1.4.4.3.cmml"><mo id="S5.p2.4.m1.4.4.2.3" maxsize="120%" minsize="120%" xref="S5.p2.4.m1.4.4.3.cmml">(</mo><msubsup id="S5.p2.4.m1.3.3.1.1" xref="S5.p2.4.m1.3.3.1.1.cmml"><mi id="S5.p2.4.m1.3.3.1.1.2.2" xref="S5.p2.4.m1.3.3.1.1.2.2.cmml">u</mi><mi id="S5.p2.4.m1.3.3.1.1.2.3" xref="S5.p2.4.m1.3.3.1.1.2.3.cmml">n</mi><mrow id="S5.p2.4.m1.1.1.1.3" xref="S5.p2.4.m1.3.3.1.1.cmml"><mo id="S5.p2.4.m1.1.1.1.3.1" stretchy="false" xref="S5.p2.4.m1.3.3.1.1.cmml">(</mo><mi id="S5.p2.4.m1.1.1.1.1" xref="S5.p2.4.m1.1.1.1.1.cmml">k</mi><mo id="S5.p2.4.m1.1.1.1.3.2" stretchy="false" xref="S5.p2.4.m1.3.3.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.p2.4.m1.4.4.2.4" xref="S5.p2.4.m1.4.4.3.cmml">,</mo><msubsup id="S5.p2.4.m1.4.4.2.2" xref="S5.p2.4.m1.4.4.2.2.cmml"><mi id="S5.p2.4.m1.4.4.2.2.2.2" xref="S5.p2.4.m1.4.4.2.2.2.2.cmml">v</mi><mi id="S5.p2.4.m1.4.4.2.2.2.3" xref="S5.p2.4.m1.4.4.2.2.2.3.cmml">n</mi><mrow id="S5.p2.4.m1.2.2.1.3" xref="S5.p2.4.m1.4.4.2.2.cmml"><mo id="S5.p2.4.m1.2.2.1.3.1" stretchy="false" xref="S5.p2.4.m1.4.4.2.2.cmml">(</mo><mi id="S5.p2.4.m1.2.2.1.1" xref="S5.p2.4.m1.2.2.1.1.cmml">k</mi><mo id="S5.p2.4.m1.2.2.1.3.2" stretchy="false" xref="S5.p2.4.m1.4.4.2.2.cmml">)</mo></mrow></msubsup><mo id="S5.p2.4.m1.4.4.2.5" maxsize="120%" minsize="120%" xref="S5.p2.4.m1.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.4.m1.4b"><interval closure="open" id="S5.p2.4.m1.4.4.3.cmml" xref="S5.p2.4.m1.4.4.2"><apply id="S5.p2.4.m1.3.3.1.1.cmml" xref="S5.p2.4.m1.3.3.1.1"><csymbol cd="ambiguous" id="S5.p2.4.m1.3.3.1.1.1.cmml" xref="S5.p2.4.m1.3.3.1.1">superscript</csymbol><apply id="S5.p2.4.m1.3.3.1.1.2.cmml" xref="S5.p2.4.m1.3.3.1.1"><csymbol cd="ambiguous" id="S5.p2.4.m1.3.3.1.1.2.1.cmml" xref="S5.p2.4.m1.3.3.1.1">subscript</csymbol><ci id="S5.p2.4.m1.3.3.1.1.2.2.cmml" xref="S5.p2.4.m1.3.3.1.1.2.2">𝑢</ci><ci id="S5.p2.4.m1.3.3.1.1.2.3.cmml" xref="S5.p2.4.m1.3.3.1.1.2.3">𝑛</ci></apply><ci id="S5.p2.4.m1.1.1.1.1.cmml" xref="S5.p2.4.m1.1.1.1.1">𝑘</ci></apply><apply id="S5.p2.4.m1.4.4.2.2.cmml" xref="S5.p2.4.m1.4.4.2.2"><csymbol cd="ambiguous" id="S5.p2.4.m1.4.4.2.2.1.cmml" xref="S5.p2.4.m1.4.4.2.2">superscript</csymbol><apply id="S5.p2.4.m1.4.4.2.2.2.cmml" xref="S5.p2.4.m1.4.4.2.2"><csymbol cd="ambiguous" id="S5.p2.4.m1.4.4.2.2.2.1.cmml" xref="S5.p2.4.m1.4.4.2.2">subscript</csymbol><ci id="S5.p2.4.m1.4.4.2.2.2.2.cmml" xref="S5.p2.4.m1.4.4.2.2.2.2">𝑣</ci><ci id="S5.p2.4.m1.4.4.2.2.2.3.cmml" xref="S5.p2.4.m1.4.4.2.2.2.3">𝑛</ci></apply><ci id="S5.p2.4.m1.2.2.1.1.cmml" xref="S5.p2.4.m1.2.2.1.1">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m1.4c">\big{(}u_{n}^{(k)},v_{n}^{(k)}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m1.4d">( italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math> are now expressed in relation to the position and orientation of radar <math alttext="k" class="ltx_Math" display="inline" id="S5.p2.5.m2.1"><semantics id="S5.p2.5.m2.1a"><mi id="S5.p2.5.m2.1.1" xref="S5.p2.5.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.p2.5.m2.1b"><ci id="S5.p2.5.m2.1.1.cmml" xref="S5.p2.5.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m2.1d">italic_k</annotation></semantics></math>. When combining the results, we translate these into a global coordinate system <math alttext="\big{(}x_{n}^{(k)},y_{n}^{(k)}\big{)}" class="ltx_Math" display="inline" id="S5.p2.6.m3.4"><semantics id="S5.p2.6.m3.4a"><mrow id="S5.p2.6.m3.4.4.2" xref="S5.p2.6.m3.4.4.3.cmml"><mo id="S5.p2.6.m3.4.4.2.3" maxsize="120%" minsize="120%" xref="S5.p2.6.m3.4.4.3.cmml">(</mo><msubsup id="S5.p2.6.m3.3.3.1.1" xref="S5.p2.6.m3.3.3.1.1.cmml"><mi id="S5.p2.6.m3.3.3.1.1.2.2" xref="S5.p2.6.m3.3.3.1.1.2.2.cmml">x</mi><mi id="S5.p2.6.m3.3.3.1.1.2.3" xref="S5.p2.6.m3.3.3.1.1.2.3.cmml">n</mi><mrow id="S5.p2.6.m3.1.1.1.3" xref="S5.p2.6.m3.3.3.1.1.cmml"><mo id="S5.p2.6.m3.1.1.1.3.1" stretchy="false" xref="S5.p2.6.m3.3.3.1.1.cmml">(</mo><mi id="S5.p2.6.m3.1.1.1.1" xref="S5.p2.6.m3.1.1.1.1.cmml">k</mi><mo id="S5.p2.6.m3.1.1.1.3.2" stretchy="false" xref="S5.p2.6.m3.3.3.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.p2.6.m3.4.4.2.4" xref="S5.p2.6.m3.4.4.3.cmml">,</mo><msubsup id="S5.p2.6.m3.4.4.2.2" xref="S5.p2.6.m3.4.4.2.2.cmml"><mi id="S5.p2.6.m3.4.4.2.2.2.2" xref="S5.p2.6.m3.4.4.2.2.2.2.cmml">y</mi><mi id="S5.p2.6.m3.4.4.2.2.2.3" xref="S5.p2.6.m3.4.4.2.2.2.3.cmml">n</mi><mrow id="S5.p2.6.m3.2.2.1.3" xref="S5.p2.6.m3.4.4.2.2.cmml"><mo id="S5.p2.6.m3.2.2.1.3.1" stretchy="false" xref="S5.p2.6.m3.4.4.2.2.cmml">(</mo><mi id="S5.p2.6.m3.2.2.1.1" xref="S5.p2.6.m3.2.2.1.1.cmml">k</mi><mo id="S5.p2.6.m3.2.2.1.3.2" stretchy="false" xref="S5.p2.6.m3.4.4.2.2.cmml">)</mo></mrow></msubsup><mo id="S5.p2.6.m3.4.4.2.5" maxsize="120%" minsize="120%" xref="S5.p2.6.m3.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.6.m3.4b"><interval closure="open" id="S5.p2.6.m3.4.4.3.cmml" xref="S5.p2.6.m3.4.4.2"><apply id="S5.p2.6.m3.3.3.1.1.cmml" xref="S5.p2.6.m3.3.3.1.1"><csymbol cd="ambiguous" id="S5.p2.6.m3.3.3.1.1.1.cmml" xref="S5.p2.6.m3.3.3.1.1">superscript</csymbol><apply id="S5.p2.6.m3.3.3.1.1.2.cmml" xref="S5.p2.6.m3.3.3.1.1"><csymbol cd="ambiguous" id="S5.p2.6.m3.3.3.1.1.2.1.cmml" xref="S5.p2.6.m3.3.3.1.1">subscript</csymbol><ci id="S5.p2.6.m3.3.3.1.1.2.2.cmml" xref="S5.p2.6.m3.3.3.1.1.2.2">𝑥</ci><ci id="S5.p2.6.m3.3.3.1.1.2.3.cmml" xref="S5.p2.6.m3.3.3.1.1.2.3">𝑛</ci></apply><ci id="S5.p2.6.m3.1.1.1.1.cmml" xref="S5.p2.6.m3.1.1.1.1">𝑘</ci></apply><apply id="S5.p2.6.m3.4.4.2.2.cmml" xref="S5.p2.6.m3.4.4.2.2"><csymbol cd="ambiguous" id="S5.p2.6.m3.4.4.2.2.1.cmml" xref="S5.p2.6.m3.4.4.2.2">superscript</csymbol><apply id="S5.p2.6.m3.4.4.2.2.2.cmml" xref="S5.p2.6.m3.4.4.2.2"><csymbol cd="ambiguous" id="S5.p2.6.m3.4.4.2.2.2.1.cmml" xref="S5.p2.6.m3.4.4.2.2">subscript</csymbol><ci id="S5.p2.6.m3.4.4.2.2.2.2.cmml" xref="S5.p2.6.m3.4.4.2.2.2.2">𝑦</ci><ci id="S5.p2.6.m3.4.4.2.2.2.3.cmml" xref="S5.p2.6.m3.4.4.2.2.2.3">𝑛</ci></apply><ci id="S5.p2.6.m3.2.2.1.1.cmml" xref="S5.p2.6.m3.2.2.1.1">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m3.4c">\big{(}x_{n}^{(k)},y_{n}^{(k)}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m3.4d">( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT , italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math>, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.F3" title="Figure 3 ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">3</span></a>. Denoting the position of radar <math alttext="k" class="ltx_Math" display="inline" id="S5.p2.7.m4.1"><semantics id="S5.p2.7.m4.1a"><mi id="S5.p2.7.m4.1.1" xref="S5.p2.7.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.p2.7.m4.1b"><ci id="S5.p2.7.m4.1.1.cmml" xref="S5.p2.7.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.7.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.p2.7.m4.1d">italic_k</annotation></semantics></math> in the global reference frame as <math alttext="\bm{p}^{(k)}=[x_{\text{radar}}^{(k)},y_{\text{radar}}^{(k)}]^{\top}" class="ltx_Math" display="inline" id="S5.p2.8.m5.5"><semantics id="S5.p2.8.m5.5a"><mrow id="S5.p2.8.m5.5.5" xref="S5.p2.8.m5.5.5.cmml"><msup id="S5.p2.8.m5.5.5.4" xref="S5.p2.8.m5.5.5.4.cmml"><mi id="S5.p2.8.m5.5.5.4.2" xref="S5.p2.8.m5.5.5.4.2.cmml">𝒑</mi><mrow id="S5.p2.8.m5.1.1.1.3" xref="S5.p2.8.m5.5.5.4.cmml"><mo id="S5.p2.8.m5.1.1.1.3.1" stretchy="false" xref="S5.p2.8.m5.5.5.4.cmml">(</mo><mi id="S5.p2.8.m5.1.1.1.1" xref="S5.p2.8.m5.1.1.1.1.cmml">k</mi><mo id="S5.p2.8.m5.1.1.1.3.2" stretchy="false" xref="S5.p2.8.m5.5.5.4.cmml">)</mo></mrow></msup><mo id="S5.p2.8.m5.5.5.3" xref="S5.p2.8.m5.5.5.3.cmml">=</mo><msup id="S5.p2.8.m5.5.5.2" xref="S5.p2.8.m5.5.5.2.cmml"><mrow id="S5.p2.8.m5.5.5.2.2.2" xref="S5.p2.8.m5.5.5.2.2.3.cmml"><mo id="S5.p2.8.m5.5.5.2.2.2.3" stretchy="false" xref="S5.p2.8.m5.5.5.2.2.3.cmml">[</mo><msubsup id="S5.p2.8.m5.4.4.1.1.1.1" xref="S5.p2.8.m5.4.4.1.1.1.1.cmml"><mi id="S5.p2.8.m5.4.4.1.1.1.1.2.2" xref="S5.p2.8.m5.4.4.1.1.1.1.2.2.cmml">x</mi><mtext id="S5.p2.8.m5.4.4.1.1.1.1.2.3" xref="S5.p2.8.m5.4.4.1.1.1.1.2.3a.cmml">radar</mtext><mrow id="S5.p2.8.m5.2.2.1.3" xref="S5.p2.8.m5.4.4.1.1.1.1.cmml"><mo id="S5.p2.8.m5.2.2.1.3.1" stretchy="false" xref="S5.p2.8.m5.4.4.1.1.1.1.cmml">(</mo><mi id="S5.p2.8.m5.2.2.1.1" xref="S5.p2.8.m5.2.2.1.1.cmml">k</mi><mo id="S5.p2.8.m5.2.2.1.3.2" stretchy="false" xref="S5.p2.8.m5.4.4.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.p2.8.m5.5.5.2.2.2.4" xref="S5.p2.8.m5.5.5.2.2.3.cmml">,</mo><msubsup id="S5.p2.8.m5.5.5.2.2.2.2" xref="S5.p2.8.m5.5.5.2.2.2.2.cmml"><mi id="S5.p2.8.m5.5.5.2.2.2.2.2.2" xref="S5.p2.8.m5.5.5.2.2.2.2.2.2.cmml">y</mi><mtext id="S5.p2.8.m5.5.5.2.2.2.2.2.3" xref="S5.p2.8.m5.5.5.2.2.2.2.2.3a.cmml">radar</mtext><mrow id="S5.p2.8.m5.3.3.1.3" xref="S5.p2.8.m5.5.5.2.2.2.2.cmml"><mo id="S5.p2.8.m5.3.3.1.3.1" stretchy="false" xref="S5.p2.8.m5.5.5.2.2.2.2.cmml">(</mo><mi id="S5.p2.8.m5.3.3.1.1" xref="S5.p2.8.m5.3.3.1.1.cmml">k</mi><mo id="S5.p2.8.m5.3.3.1.3.2" stretchy="false" xref="S5.p2.8.m5.5.5.2.2.2.2.cmml">)</mo></mrow></msubsup><mo id="S5.p2.8.m5.5.5.2.2.2.5" stretchy="false" xref="S5.p2.8.m5.5.5.2.2.3.cmml">]</mo></mrow><mo id="S5.p2.8.m5.5.5.2.4" xref="S5.p2.8.m5.5.5.2.4.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.8.m5.5b"><apply id="S5.p2.8.m5.5.5.cmml" xref="S5.p2.8.m5.5.5"><eq id="S5.p2.8.m5.5.5.3.cmml" xref="S5.p2.8.m5.5.5.3"></eq><apply id="S5.p2.8.m5.5.5.4.cmml" xref="S5.p2.8.m5.5.5.4"><csymbol cd="ambiguous" id="S5.p2.8.m5.5.5.4.1.cmml" xref="S5.p2.8.m5.5.5.4">superscript</csymbol><ci id="S5.p2.8.m5.5.5.4.2.cmml" xref="S5.p2.8.m5.5.5.4.2">𝒑</ci><ci id="S5.p2.8.m5.1.1.1.1.cmml" xref="S5.p2.8.m5.1.1.1.1">𝑘</ci></apply><apply id="S5.p2.8.m5.5.5.2.cmml" xref="S5.p2.8.m5.5.5.2"><csymbol cd="ambiguous" id="S5.p2.8.m5.5.5.2.3.cmml" xref="S5.p2.8.m5.5.5.2">superscript</csymbol><interval closure="closed" id="S5.p2.8.m5.5.5.2.2.3.cmml" xref="S5.p2.8.m5.5.5.2.2.2"><apply id="S5.p2.8.m5.4.4.1.1.1.1.cmml" xref="S5.p2.8.m5.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.8.m5.4.4.1.1.1.1.1.cmml" xref="S5.p2.8.m5.4.4.1.1.1.1">superscript</csymbol><apply id="S5.p2.8.m5.4.4.1.1.1.1.2.cmml" xref="S5.p2.8.m5.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.8.m5.4.4.1.1.1.1.2.1.cmml" xref="S5.p2.8.m5.4.4.1.1.1.1">subscript</csymbol><ci id="S5.p2.8.m5.4.4.1.1.1.1.2.2.cmml" xref="S5.p2.8.m5.4.4.1.1.1.1.2.2">𝑥</ci><ci id="S5.p2.8.m5.4.4.1.1.1.1.2.3a.cmml" xref="S5.p2.8.m5.4.4.1.1.1.1.2.3"><mtext id="S5.p2.8.m5.4.4.1.1.1.1.2.3.cmml" mathsize="70%" xref="S5.p2.8.m5.4.4.1.1.1.1.2.3">radar</mtext></ci></apply><ci id="S5.p2.8.m5.2.2.1.1.cmml" xref="S5.p2.8.m5.2.2.1.1">𝑘</ci></apply><apply id="S5.p2.8.m5.5.5.2.2.2.2.cmml" xref="S5.p2.8.m5.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S5.p2.8.m5.5.5.2.2.2.2.1.cmml" xref="S5.p2.8.m5.5.5.2.2.2.2">superscript</csymbol><apply id="S5.p2.8.m5.5.5.2.2.2.2.2.cmml" xref="S5.p2.8.m5.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S5.p2.8.m5.5.5.2.2.2.2.2.1.cmml" xref="S5.p2.8.m5.5.5.2.2.2.2">subscript</csymbol><ci id="S5.p2.8.m5.5.5.2.2.2.2.2.2.cmml" xref="S5.p2.8.m5.5.5.2.2.2.2.2.2">𝑦</ci><ci id="S5.p2.8.m5.5.5.2.2.2.2.2.3a.cmml" xref="S5.p2.8.m5.5.5.2.2.2.2.2.3"><mtext id="S5.p2.8.m5.5.5.2.2.2.2.2.3.cmml" mathsize="70%" xref="S5.p2.8.m5.5.5.2.2.2.2.2.3">radar</mtext></ci></apply><ci id="S5.p2.8.m5.3.3.1.1.cmml" xref="S5.p2.8.m5.3.3.1.1">𝑘</ci></apply></interval><csymbol cd="latexml" id="S5.p2.8.m5.5.5.2.4.cmml" xref="S5.p2.8.m5.5.5.2.4">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.8.m5.5c">\bm{p}^{(k)}=[x_{\text{radar}}^{(k)},y_{\text{radar}}^{(k)}]^{\top}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.8.m5.5d">bold_italic_p start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = [ italic_x start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT , italic_y start_POSTSUBSCRIPT radar end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, the position may be translated into the same frame as</p> <table class="ltx_equation ltx_eqn_table" id="S5.E39"> <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{bmatrix}x_{n}^{(k)}\\ y_{n}^{(k)}\end{bmatrix}=\bm{R}\left(\psi^{(k)}\right)\left(\begin{bmatrix}u^{% (k)}_{n}\\ v_{n}^{(k)}\end{bmatrix}-\bm{p}^{(k)}\right)," class="ltx_Math" display="block" id="S5.E39.m1.5"><semantics id="S5.E39.m1.5a"><mrow id="S5.E39.m1.5.5.1" xref="S5.E39.m1.5.5.1.1.cmml"><mrow id="S5.E39.m1.5.5.1.1" xref="S5.E39.m1.5.5.1.1.cmml"><mrow id="S5.E39.m1.1.1.3" xref="S5.E39.m1.1.1.2.cmml"><mo id="S5.E39.m1.1.1.3.1" xref="S5.E39.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S5.E39.m1.1.1.1.1" rowspacing="0pt" xref="S5.E39.m1.1.1.1.1.cmml"><mtr id="S5.E39.m1.1.1.1.1a" xref="S5.E39.m1.1.1.1.1.cmml"><mtd id="S5.E39.m1.1.1.1.1b" xref="S5.E39.m1.1.1.1.1.cmml"><msubsup id="S5.E39.m1.1.1.1.1.1.1.1.1" xref="S5.E39.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.E39.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.E39.m1.1.1.1.1.1.1.1.1.3.2.cmml">x</mi><mi id="S5.E39.m1.1.1.1.1.1.1.1.1.3.3" xref="S5.E39.m1.1.1.1.1.1.1.1.1.3.3.cmml">n</mi><mrow id="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.E39.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.3.1" stretchy="false" xref="S5.E39.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">k</mi><mo id="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.3.2" stretchy="false" xref="S5.E39.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr><mtr id="S5.E39.m1.1.1.1.1c" xref="S5.E39.m1.1.1.1.1.cmml"><mtd id="S5.E39.m1.1.1.1.1d" xref="S5.E39.m1.1.1.1.1.cmml"><msubsup id="S5.E39.m1.1.1.1.1.2.2.1.1" xref="S5.E39.m1.1.1.1.1.2.2.1.1.cmml"><mi id="S5.E39.m1.1.1.1.1.2.2.1.1.3.2" xref="S5.E39.m1.1.1.1.1.2.2.1.1.3.2.cmml">y</mi><mi id="S5.E39.m1.1.1.1.1.2.2.1.1.3.3" xref="S5.E39.m1.1.1.1.1.2.2.1.1.3.3.cmml">n</mi><mrow id="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.3" xref="S5.E39.m1.1.1.1.1.2.2.1.1.cmml"><mo id="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.3.1" stretchy="false" xref="S5.E39.m1.1.1.1.1.2.2.1.1.cmml">(</mo><mi id="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.1" xref="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.1.cmml">k</mi><mo id="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.3.2" stretchy="false" xref="S5.E39.m1.1.1.1.1.2.2.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr></mtable><mo id="S5.E39.m1.1.1.3.2" xref="S5.E39.m1.1.1.2.1.cmml">]</mo></mrow><mo id="S5.E39.m1.5.5.1.1.3" xref="S5.E39.m1.5.5.1.1.3.cmml">=</mo><mrow id="S5.E39.m1.5.5.1.1.2" xref="S5.E39.m1.5.5.1.1.2.cmml"><mi id="S5.E39.m1.5.5.1.1.2.4" xref="S5.E39.m1.5.5.1.1.2.4.cmml">𝑹</mi><mo id="S5.E39.m1.5.5.1.1.2.3" xref="S5.E39.m1.5.5.1.1.2.3.cmml">⁢</mo><mrow id="S5.E39.m1.5.5.1.1.1.1.1" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml"><mo id="S5.E39.m1.5.5.1.1.1.1.1.2" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml">(</mo><msup id="S5.E39.m1.5.5.1.1.1.1.1.1" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml"><mi id="S5.E39.m1.5.5.1.1.1.1.1.1.2" xref="S5.E39.m1.5.5.1.1.1.1.1.1.2.cmml">ψ</mi><mrow id="S5.E39.m1.3.3.1.3" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml"><mo id="S5.E39.m1.3.3.1.3.1" stretchy="false" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mi id="S5.E39.m1.3.3.1.1" xref="S5.E39.m1.3.3.1.1.cmml">k</mi><mo id="S5.E39.m1.3.3.1.3.2" stretchy="false" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S5.E39.m1.5.5.1.1.1.1.1.3" xref="S5.E39.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.E39.m1.5.5.1.1.2.3a" xref="S5.E39.m1.5.5.1.1.2.3.cmml">⁢</mo><mrow id="S5.E39.m1.5.5.1.1.2.2.1" xref="S5.E39.m1.5.5.1.1.2.2.1.1.cmml"><mo id="S5.E39.m1.5.5.1.1.2.2.1.2" xref="S5.E39.m1.5.5.1.1.2.2.1.1.cmml">(</mo><mrow id="S5.E39.m1.5.5.1.1.2.2.1.1" xref="S5.E39.m1.5.5.1.1.2.2.1.1.cmml"><mrow id="S5.E39.m1.2.2.3" xref="S5.E39.m1.2.2.2.cmml"><mo id="S5.E39.m1.2.2.3.1" xref="S5.E39.m1.2.2.2.1.cmml">[</mo><mtable displaystyle="true" id="S5.E39.m1.2.2.1.1" rowspacing="0pt" xref="S5.E39.m1.2.2.1.1.cmml"><mtr id="S5.E39.m1.2.2.1.1a" xref="S5.E39.m1.2.2.1.1.cmml"><mtd id="S5.E39.m1.2.2.1.1b" xref="S5.E39.m1.2.2.1.1.cmml"><msubsup id="S5.E39.m1.2.2.1.1.1.1.1.1" xref="S5.E39.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S5.E39.m1.2.2.1.1.1.1.1.1.3.2" xref="S5.E39.m1.2.2.1.1.1.1.1.1.3.2.cmml">u</mi><mi id="S5.E39.m1.2.2.1.1.1.1.1.1.4" xref="S5.E39.m1.2.2.1.1.1.1.1.1.4.cmml">n</mi><mrow id="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.3" xref="S5.E39.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.3.1" stretchy="false" xref="S5.E39.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mi id="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">k</mi><mo id="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.3.2" stretchy="false" xref="S5.E39.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr><mtr id="S5.E39.m1.2.2.1.1c" xref="S5.E39.m1.2.2.1.1.cmml"><mtd id="S5.E39.m1.2.2.1.1d" xref="S5.E39.m1.2.2.1.1.cmml"><msubsup id="S5.E39.m1.2.2.1.1.2.2.1.1" xref="S5.E39.m1.2.2.1.1.2.2.1.1.cmml"><mi id="S5.E39.m1.2.2.1.1.2.2.1.1.3.2" xref="S5.E39.m1.2.2.1.1.2.2.1.1.3.2.cmml">v</mi><mi id="S5.E39.m1.2.2.1.1.2.2.1.1.3.3" xref="S5.E39.m1.2.2.1.1.2.2.1.1.3.3.cmml">n</mi><mrow id="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.3" xref="S5.E39.m1.2.2.1.1.2.2.1.1.cmml"><mo id="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.3.1" stretchy="false" xref="S5.E39.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mi id="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.1" xref="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.1.cmml">k</mi><mo id="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.3.2" stretchy="false" xref="S5.E39.m1.2.2.1.1.2.2.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr></mtable><mo id="S5.E39.m1.2.2.3.2" xref="S5.E39.m1.2.2.2.1.cmml">]</mo></mrow><mo id="S5.E39.m1.5.5.1.1.2.2.1.1.1" xref="S5.E39.m1.5.5.1.1.2.2.1.1.1.cmml">−</mo><msup id="S5.E39.m1.5.5.1.1.2.2.1.1.2" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2.cmml"><mi id="S5.E39.m1.5.5.1.1.2.2.1.1.2.2" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2.2.cmml">𝒑</mi><mrow id="S5.E39.m1.4.4.1.3" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2.cmml"><mo id="S5.E39.m1.4.4.1.3.1" stretchy="false" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2.cmml">(</mo><mi id="S5.E39.m1.4.4.1.1" xref="S5.E39.m1.4.4.1.1.cmml">k</mi><mo id="S5.E39.m1.4.4.1.3.2" stretchy="false" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2.cmml">)</mo></mrow></msup></mrow><mo id="S5.E39.m1.5.5.1.1.2.2.1.3" xref="S5.E39.m1.5.5.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E39.m1.5.5.1.2" xref="S5.E39.m1.5.5.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E39.m1.5b"><apply id="S5.E39.m1.5.5.1.1.cmml" xref="S5.E39.m1.5.5.1"><eq id="S5.E39.m1.5.5.1.1.3.cmml" xref="S5.E39.m1.5.5.1.1.3"></eq><apply id="S5.E39.m1.1.1.2.cmml" xref="S5.E39.m1.1.1.3"><csymbol cd="latexml" id="S5.E39.m1.1.1.2.1.cmml" xref="S5.E39.m1.1.1.3.1">matrix</csymbol><matrix id="S5.E39.m1.1.1.1.1.cmml" xref="S5.E39.m1.1.1.1.1"><matrixrow id="S5.E39.m1.1.1.1.1a.cmml" xref="S5.E39.m1.1.1.1.1"><apply id="S5.E39.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.E39.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.E39.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1.3.2">𝑥</ci><ci id="S5.E39.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1.3.3">𝑛</ci></apply><ci id="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E39.m1.1.1.1.1.1.1.1.1.1.1.1">𝑘</ci></apply></matrixrow><matrixrow id="S5.E39.m1.1.1.1.1b.cmml" xref="S5.E39.m1.1.1.1.1"><apply id="S5.E39.m1.1.1.1.1.2.2.1.1.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.1.1.1.1.2.2.1.1.2.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1">superscript</csymbol><apply id="S5.E39.m1.1.1.1.1.2.2.1.1.3.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.1.1.1.1.2.2.1.1.3.1.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1">subscript</csymbol><ci id="S5.E39.m1.1.1.1.1.2.2.1.1.3.2.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1.3.2">𝑦</ci><ci id="S5.E39.m1.1.1.1.1.2.2.1.1.3.3.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1.3.3">𝑛</ci></apply><ci id="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.1.cmml" xref="S5.E39.m1.1.1.1.1.2.2.1.1.1.1.1">𝑘</ci></apply></matrixrow></matrix></apply><apply id="S5.E39.m1.5.5.1.1.2.cmml" xref="S5.E39.m1.5.5.1.1.2"><times id="S5.E39.m1.5.5.1.1.2.3.cmml" xref="S5.E39.m1.5.5.1.1.2.3"></times><ci id="S5.E39.m1.5.5.1.1.2.4.cmml" xref="S5.E39.m1.5.5.1.1.2.4">𝑹</ci><apply id="S5.E39.m1.5.5.1.1.1.1.1.1.cmml" xref="S5.E39.m1.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S5.E39.m1.5.5.1.1.1.1.1">superscript</csymbol><ci id="S5.E39.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S5.E39.m1.5.5.1.1.1.1.1.1.2">𝜓</ci><ci id="S5.E39.m1.3.3.1.1.cmml" xref="S5.E39.m1.3.3.1.1">𝑘</ci></apply><apply id="S5.E39.m1.5.5.1.1.2.2.1.1.cmml" xref="S5.E39.m1.5.5.1.1.2.2.1"><minus id="S5.E39.m1.5.5.1.1.2.2.1.1.1.cmml" xref="S5.E39.m1.5.5.1.1.2.2.1.1.1"></minus><apply id="S5.E39.m1.2.2.2.cmml" xref="S5.E39.m1.2.2.3"><csymbol cd="latexml" id="S5.E39.m1.2.2.2.1.cmml" xref="S5.E39.m1.2.2.3.1">matrix</csymbol><matrix id="S5.E39.m1.2.2.1.1.cmml" xref="S5.E39.m1.2.2.1.1"><matrixrow id="S5.E39.m1.2.2.1.1a.cmml" xref="S5.E39.m1.2.2.1.1"><apply id="S5.E39.m1.2.2.1.1.1.1.1.1.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1">subscript</csymbol><apply id="S5.E39.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1">superscript</csymbol><ci id="S5.E39.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1.3.2">𝑢</ci><ci id="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1.1.1.1">𝑘</ci></apply><ci id="S5.E39.m1.2.2.1.1.1.1.1.1.4.cmml" xref="S5.E39.m1.2.2.1.1.1.1.1.1.4">𝑛</ci></apply></matrixrow><matrixrow id="S5.E39.m1.2.2.1.1b.cmml" xref="S5.E39.m1.2.2.1.1"><apply id="S5.E39.m1.2.2.1.1.2.2.1.1.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.2.2.1.1.2.2.1.1.2.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1">superscript</csymbol><apply id="S5.E39.m1.2.2.1.1.2.2.1.1.3.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S5.E39.m1.2.2.1.1.2.2.1.1.3.1.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1">subscript</csymbol><ci id="S5.E39.m1.2.2.1.1.2.2.1.1.3.2.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1.3.2">𝑣</ci><ci id="S5.E39.m1.2.2.1.1.2.2.1.1.3.3.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1.3.3">𝑛</ci></apply><ci id="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.1.cmml" xref="S5.E39.m1.2.2.1.1.2.2.1.1.1.1.1">𝑘</ci></apply></matrixrow></matrix></apply><apply id="S5.E39.m1.5.5.1.1.2.2.1.1.2.cmml" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2"><csymbol cd="ambiguous" id="S5.E39.m1.5.5.1.1.2.2.1.1.2.1.cmml" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2">superscript</csymbol><ci id="S5.E39.m1.5.5.1.1.2.2.1.1.2.2.cmml" xref="S5.E39.m1.5.5.1.1.2.2.1.1.2.2">𝒑</ci><ci id="S5.E39.m1.4.4.1.1.cmml" xref="S5.E39.m1.4.4.1.1">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E39.m1.5c">\begin{bmatrix}x_{n}^{(k)}\\ y_{n}^{(k)}\end{bmatrix}=\bm{R}\left(\psi^{(k)}\right)\left(\begin{bmatrix}u^{% (k)}_{n}\\ v_{n}^{(k)}\end{bmatrix}-\bm{p}^{(k)}\right),</annotation><annotation encoding="application/x-llamapun" id="S5.E39.m1.5d">[ start_ARG start_ROW start_CELL italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] = bold_italic_R ( italic_ψ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) ( [ start_ARG start_ROW start_CELL italic_u start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] - bold_italic_p 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">(39)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.11">where <math alttext="\bm{R}" class="ltx_Math" display="inline" id="S5.p2.9.m1.1"><semantics id="S5.p2.9.m1.1a"><mi id="S5.p2.9.m1.1.1" xref="S5.p2.9.m1.1.1.cmml">𝑹</mi><annotation-xml encoding="MathML-Content" id="S5.p2.9.m1.1b"><ci id="S5.p2.9.m1.1.1.cmml" xref="S5.p2.9.m1.1.1">𝑹</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.9.m1.1c">\bm{R}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.9.m1.1d">bold_italic_R</annotation></semantics></math> is the rotation matrix evaluated in <math alttext="\psi^{(k)}" class="ltx_Math" display="inline" id="S5.p2.10.m2.1"><semantics id="S5.p2.10.m2.1a"><msup id="S5.p2.10.m2.1.2" xref="S5.p2.10.m2.1.2.cmml"><mi id="S5.p2.10.m2.1.2.2" xref="S5.p2.10.m2.1.2.2.cmml">ψ</mi><mrow id="S5.p2.10.m2.1.1.1.3" xref="S5.p2.10.m2.1.2.cmml"><mo id="S5.p2.10.m2.1.1.1.3.1" stretchy="false" xref="S5.p2.10.m2.1.2.cmml">(</mo><mi id="S5.p2.10.m2.1.1.1.1" xref="S5.p2.10.m2.1.1.1.1.cmml">k</mi><mo id="S5.p2.10.m2.1.1.1.3.2" stretchy="false" xref="S5.p2.10.m2.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.p2.10.m2.1b"><apply id="S5.p2.10.m2.1.2.cmml" xref="S5.p2.10.m2.1.2"><csymbol cd="ambiguous" id="S5.p2.10.m2.1.2.1.cmml" xref="S5.p2.10.m2.1.2">superscript</csymbol><ci id="S5.p2.10.m2.1.2.2.cmml" xref="S5.p2.10.m2.1.2.2">𝜓</ci><ci id="S5.p2.10.m2.1.1.1.1.cmml" xref="S5.p2.10.m2.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.10.m2.1c">\psi^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.10.m2.1d">italic_ψ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> which is the angle between the second axis of the global coordinate system and the boresight of radar <math alttext="k" class="ltx_Math" display="inline" id="S5.p2.11.m3.1"><semantics id="S5.p2.11.m3.1a"><mi id="S5.p2.11.m3.1.1" xref="S5.p2.11.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.p2.11.m3.1b"><ci id="S5.p2.11.m3.1.1.cmml" xref="S5.p2.11.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.11.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.p2.11.m3.1d">italic_k</annotation></semantics></math>. These global coordinates are then passed as inputs to the KF.</p> </div> <figure class="ltx_figure" id="S5.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="139" id="S5.F3.g1" src="x3.png" width="304"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>The relation between the local coordinate system <math alttext="(u^{(k)},v^{(k)})" class="ltx_Math" display="inline" id="S5.F3.3.m1.4"><semantics id="S5.F3.3.m1.4b"><mrow id="S5.F3.3.m1.4.4.2" xref="S5.F3.3.m1.4.4.3.cmml"><mo id="S5.F3.3.m1.4.4.2.3" stretchy="false" xref="S5.F3.3.m1.4.4.3.cmml">(</mo><msup id="S5.F3.3.m1.3.3.1.1" xref="S5.F3.3.m1.3.3.1.1.cmml"><mi id="S5.F3.3.m1.3.3.1.1.2" xref="S5.F3.3.m1.3.3.1.1.2.cmml">u</mi><mrow id="S5.F3.3.m1.1.1.1.3" xref="S5.F3.3.m1.3.3.1.1.cmml"><mo id="S5.F3.3.m1.1.1.1.3.1" stretchy="false" xref="S5.F3.3.m1.3.3.1.1.cmml">(</mo><mi id="S5.F3.3.m1.1.1.1.1" xref="S5.F3.3.m1.1.1.1.1.cmml">k</mi><mo id="S5.F3.3.m1.1.1.1.3.2" stretchy="false" xref="S5.F3.3.m1.3.3.1.1.cmml">)</mo></mrow></msup><mo id="S5.F3.3.m1.4.4.2.4" xref="S5.F3.3.m1.4.4.3.cmml">,</mo><msup id="S5.F3.3.m1.4.4.2.2" xref="S5.F3.3.m1.4.4.2.2.cmml"><mi id="S5.F3.3.m1.4.4.2.2.2" xref="S5.F3.3.m1.4.4.2.2.2.cmml">v</mi><mrow id="S5.F3.3.m1.2.2.1.3" xref="S5.F3.3.m1.4.4.2.2.cmml"><mo id="S5.F3.3.m1.2.2.1.3.1" stretchy="false" xref="S5.F3.3.m1.4.4.2.2.cmml">(</mo><mi id="S5.F3.3.m1.2.2.1.1" xref="S5.F3.3.m1.2.2.1.1.cmml">k</mi><mo id="S5.F3.3.m1.2.2.1.3.2" stretchy="false" xref="S5.F3.3.m1.4.4.2.2.cmml">)</mo></mrow></msup><mo id="S5.F3.3.m1.4.4.2.5" stretchy="false" xref="S5.F3.3.m1.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.F3.3.m1.4c"><interval closure="open" id="S5.F3.3.m1.4.4.3.cmml" xref="S5.F3.3.m1.4.4.2"><apply id="S5.F3.3.m1.3.3.1.1.cmml" xref="S5.F3.3.m1.3.3.1.1"><csymbol cd="ambiguous" id="S5.F3.3.m1.3.3.1.1.1.cmml" xref="S5.F3.3.m1.3.3.1.1">superscript</csymbol><ci id="S5.F3.3.m1.3.3.1.1.2.cmml" xref="S5.F3.3.m1.3.3.1.1.2">𝑢</ci><ci id="S5.F3.3.m1.1.1.1.1.cmml" xref="S5.F3.3.m1.1.1.1.1">𝑘</ci></apply><apply id="S5.F3.3.m1.4.4.2.2.cmml" xref="S5.F3.3.m1.4.4.2.2"><csymbol cd="ambiguous" id="S5.F3.3.m1.4.4.2.2.1.cmml" xref="S5.F3.3.m1.4.4.2.2">superscript</csymbol><ci id="S5.F3.3.m1.4.4.2.2.2.cmml" xref="S5.F3.3.m1.4.4.2.2.2">𝑣</ci><ci id="S5.F3.3.m1.2.2.1.1.cmml" xref="S5.F3.3.m1.2.2.1.1">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.F3.3.m1.4d">(u^{(k)},v^{(k)})</annotation><annotation encoding="application/x-llamapun" id="S5.F3.3.m1.4e">( italic_u start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math> and the global coordinate system <math alttext="(x,y)" class="ltx_Math" display="inline" id="S5.F3.4.m2.2"><semantics id="S5.F3.4.m2.2b"><mrow id="S5.F3.4.m2.2.3.2" xref="S5.F3.4.m2.2.3.1.cmml"><mo id="S5.F3.4.m2.2.3.2.1" stretchy="false" xref="S5.F3.4.m2.2.3.1.cmml">(</mo><mi id="S5.F3.4.m2.1.1" xref="S5.F3.4.m2.1.1.cmml">x</mi><mo id="S5.F3.4.m2.2.3.2.2" xref="S5.F3.4.m2.2.3.1.cmml">,</mo><mi id="S5.F3.4.m2.2.2" xref="S5.F3.4.m2.2.2.cmml">y</mi><mo id="S5.F3.4.m2.2.3.2.3" stretchy="false" xref="S5.F3.4.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.F3.4.m2.2c"><interval closure="open" id="S5.F3.4.m2.2.3.1.cmml" xref="S5.F3.4.m2.2.3.2"><ci id="S5.F3.4.m2.1.1.cmml" xref="S5.F3.4.m2.1.1">𝑥</ci><ci id="S5.F3.4.m2.2.2.cmml" xref="S5.F3.4.m2.2.2">𝑦</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.F3.4.m2.2d">(x,y)</annotation><annotation encoding="application/x-llamapun" id="S5.F3.4.m2.2e">( italic_x , italic_y )</annotation></semantics></math> which the algorithm runs in.</figcaption> </figure> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.4">When running MRBLaT, we minimize the KL divergence in (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S4.E27" title="In IV Variational Message Passing ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">27</span></a>) numerically. We further simplify this task by assuming the covariance matrix <math alttext="\bar{\bar{\bm{\epsilon}}}_{n}^{(k)}" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><msubsup id="S5.p3.1.m1.1.2" xref="S5.p3.1.m1.1.2.cmml"><mover accent="true" id="S5.p3.1.m1.1.2.2.2" xref="S5.p3.1.m1.1.2.2.2.cmml"><mover accent="true" id="S5.p3.1.m1.1.2.2.2.2" xref="S5.p3.1.m1.1.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.p3.1.m1.1.2.2.2.2.2" mathvariant="bold-italic" xref="S5.p3.1.m1.1.2.2.2.2.2.cmml">ϵ</mi><mo id="S5.p3.1.m1.1.2.2.2.2.1" xref="S5.p3.1.m1.1.2.2.2.2.1.cmml">¯</mo></mover><mo id="S5.p3.1.m1.1.2.2.2.1" xref="S5.p3.1.m1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S5.p3.1.m1.1.2.2.3" xref="S5.p3.1.m1.1.2.2.3.cmml">n</mi><mrow id="S5.p3.1.m1.1.1.1.3" xref="S5.p3.1.m1.1.2.cmml"><mo id="S5.p3.1.m1.1.1.1.3.1" stretchy="false" xref="S5.p3.1.m1.1.2.cmml">(</mo><mi id="S5.p3.1.m1.1.1.1.1" xref="S5.p3.1.m1.1.1.1.1.cmml">k</mi><mo id="S5.p3.1.m1.1.1.1.3.2" stretchy="false" xref="S5.p3.1.m1.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><apply id="S5.p3.1.m1.1.2.cmml" xref="S5.p3.1.m1.1.2"><csymbol cd="ambiguous" id="S5.p3.1.m1.1.2.1.cmml" xref="S5.p3.1.m1.1.2">superscript</csymbol><apply id="S5.p3.1.m1.1.2.2.cmml" xref="S5.p3.1.m1.1.2"><csymbol cd="ambiguous" id="S5.p3.1.m1.1.2.2.1.cmml" xref="S5.p3.1.m1.1.2">subscript</csymbol><apply id="S5.p3.1.m1.1.2.2.2.cmml" xref="S5.p3.1.m1.1.2.2.2"><ci id="S5.p3.1.m1.1.2.2.2.1.cmml" xref="S5.p3.1.m1.1.2.2.2.1">¯</ci><apply id="S5.p3.1.m1.1.2.2.2.2.cmml" xref="S5.p3.1.m1.1.2.2.2.2"><ci id="S5.p3.1.m1.1.2.2.2.2.1.cmml" xref="S5.p3.1.m1.1.2.2.2.2.1">¯</ci><ci id="S5.p3.1.m1.1.2.2.2.2.2.cmml" xref="S5.p3.1.m1.1.2.2.2.2.2">bold-italic-ϵ</ci></apply></apply><ci id="S5.p3.1.m1.1.2.2.3.cmml" xref="S5.p3.1.m1.1.2.2.3">𝑛</ci></apply><ci id="S5.p3.1.m1.1.1.1.1.cmml" xref="S5.p3.1.m1.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">\bar{\bar{\bm{\epsilon}}}_{n}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> to be diagonal. For initialization of <math alttext="N=0" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mrow id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml">N</mi><mo id="S5.p3.2.m2.1.1.1" xref="S5.p3.2.m2.1.1.1.cmml">=</mo><mn id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><eq id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1"></eq><ci id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2">𝑁</ci><cn id="S5.p3.2.m2.1.1.3.cmml" type="integer" xref="S5.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">N=0</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">italic_N = 0</annotation></semantics></math> we use the same input as was given to the KF and for each subsequent iteration we use <math alttext="\bm{T}\bar{\bm{\phi}}_{N-1}" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mrow id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml"><mi id="S5.p3.3.m3.1.1.2" xref="S5.p3.3.m3.1.1.2.cmml">𝑻</mi><mo id="S5.p3.3.m3.1.1.1" xref="S5.p3.3.m3.1.1.1.cmml">⁢</mo><msub id="S5.p3.3.m3.1.1.3" xref="S5.p3.3.m3.1.1.3.cmml"><mover accent="true" id="S5.p3.3.m3.1.1.3.2" xref="S5.p3.3.m3.1.1.3.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.p3.3.m3.1.1.3.2.2" mathvariant="bold-italic" xref="S5.p3.3.m3.1.1.3.2.2.cmml">ϕ</mi><mo id="S5.p3.3.m3.1.1.3.2.1" xref="S5.p3.3.m3.1.1.3.2.1.cmml">¯</mo></mover><mrow id="S5.p3.3.m3.1.1.3.3" xref="S5.p3.3.m3.1.1.3.3.cmml"><mi id="S5.p3.3.m3.1.1.3.3.2" xref="S5.p3.3.m3.1.1.3.3.2.cmml">N</mi><mo id="S5.p3.3.m3.1.1.3.3.1" xref="S5.p3.3.m3.1.1.3.3.1.cmml">−</mo><mn id="S5.p3.3.m3.1.1.3.3.3" xref="S5.p3.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><apply id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1"><times id="S5.p3.3.m3.1.1.1.cmml" xref="S5.p3.3.m3.1.1.1"></times><ci id="S5.p3.3.m3.1.1.2.cmml" xref="S5.p3.3.m3.1.1.2">𝑻</ci><apply id="S5.p3.3.m3.1.1.3.cmml" xref="S5.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.p3.3.m3.1.1.3.1.cmml" xref="S5.p3.3.m3.1.1.3">subscript</csymbol><apply id="S5.p3.3.m3.1.1.3.2.cmml" xref="S5.p3.3.m3.1.1.3.2"><ci id="S5.p3.3.m3.1.1.3.2.1.cmml" xref="S5.p3.3.m3.1.1.3.2.1">¯</ci><ci id="S5.p3.3.m3.1.1.3.2.2.cmml" xref="S5.p3.3.m3.1.1.3.2.2">bold-italic-ϕ</ci></apply><apply id="S5.p3.3.m3.1.1.3.3.cmml" xref="S5.p3.3.m3.1.1.3.3"><minus id="S5.p3.3.m3.1.1.3.3.1.cmml" xref="S5.p3.3.m3.1.1.3.3.1"></minus><ci id="S5.p3.3.m3.1.1.3.3.2.cmml" xref="S5.p3.3.m3.1.1.3.3.2">𝑁</ci><cn id="S5.p3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S5.p3.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">\bm{T}\bar{\bm{\phi}}_{N-1}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">bold_italic_T over¯ start_ARG bold_italic_ϕ end_ARG start_POSTSUBSCRIPT italic_N - 1 end_POSTSUBSCRIPT</annotation></semantics></math> translated into the local coordinate system. Each radar uses the data to determine the statistics, i.e. <math alttext="\big{(}\bar{\bm{\epsilon}}_{n}^{(k)},\bar{\bar{\bm{\epsilon}}}_{n}^{(k)}\big{)}" class="ltx_Math" display="inline" id="S5.p3.4.m4.4"><semantics id="S5.p3.4.m4.4a"><mrow id="S5.p3.4.m4.4.4.2" xref="S5.p3.4.m4.4.4.3.cmml"><mo id="S5.p3.4.m4.4.4.2.3" maxsize="120%" minsize="120%" xref="S5.p3.4.m4.4.4.3.cmml">(</mo><msubsup id="S5.p3.4.m4.3.3.1.1" xref="S5.p3.4.m4.3.3.1.1.cmml"><mover accent="true" id="S5.p3.4.m4.3.3.1.1.2.2" xref="S5.p3.4.m4.3.3.1.1.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.p3.4.m4.3.3.1.1.2.2.2" mathvariant="bold-italic" xref="S5.p3.4.m4.3.3.1.1.2.2.2.cmml">ϵ</mi><mo id="S5.p3.4.m4.3.3.1.1.2.2.1" xref="S5.p3.4.m4.3.3.1.1.2.2.1.cmml">¯</mo></mover><mi id="S5.p3.4.m4.3.3.1.1.2.3" xref="S5.p3.4.m4.3.3.1.1.2.3.cmml">n</mi><mrow id="S5.p3.4.m4.1.1.1.3" xref="S5.p3.4.m4.3.3.1.1.cmml"><mo id="S5.p3.4.m4.1.1.1.3.1" stretchy="false" xref="S5.p3.4.m4.3.3.1.1.cmml">(</mo><mi id="S5.p3.4.m4.1.1.1.1" xref="S5.p3.4.m4.1.1.1.1.cmml">k</mi><mo id="S5.p3.4.m4.1.1.1.3.2" stretchy="false" xref="S5.p3.4.m4.3.3.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.p3.4.m4.4.4.2.4" xref="S5.p3.4.m4.4.4.3.cmml">,</mo><msubsup id="S5.p3.4.m4.4.4.2.2" xref="S5.p3.4.m4.4.4.2.2.cmml"><mover accent="true" id="S5.p3.4.m4.4.4.2.2.2.2" xref="S5.p3.4.m4.4.4.2.2.2.2.cmml"><mover accent="true" id="S5.p3.4.m4.4.4.2.2.2.2.2" xref="S5.p3.4.m4.4.4.2.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold-italic" id="S5.p3.4.m4.4.4.2.2.2.2.2.2" mathvariant="bold-italic" xref="S5.p3.4.m4.4.4.2.2.2.2.2.2.cmml">ϵ</mi><mo id="S5.p3.4.m4.4.4.2.2.2.2.2.1" xref="S5.p3.4.m4.4.4.2.2.2.2.2.1.cmml">¯</mo></mover><mo id="S5.p3.4.m4.4.4.2.2.2.2.1" xref="S5.p3.4.m4.4.4.2.2.2.2.1.cmml">¯</mo></mover><mi id="S5.p3.4.m4.4.4.2.2.2.3" xref="S5.p3.4.m4.4.4.2.2.2.3.cmml">n</mi><mrow id="S5.p3.4.m4.2.2.1.3" xref="S5.p3.4.m4.4.4.2.2.cmml"><mo id="S5.p3.4.m4.2.2.1.3.1" stretchy="false" xref="S5.p3.4.m4.4.4.2.2.cmml">(</mo><mi id="S5.p3.4.m4.2.2.1.1" xref="S5.p3.4.m4.2.2.1.1.cmml">k</mi><mo id="S5.p3.4.m4.2.2.1.3.2" stretchy="false" xref="S5.p3.4.m4.4.4.2.2.cmml">)</mo></mrow></msubsup><mo id="S5.p3.4.m4.4.4.2.5" maxsize="120%" minsize="120%" xref="S5.p3.4.m4.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.4.m4.4b"><interval closure="open" id="S5.p3.4.m4.4.4.3.cmml" xref="S5.p3.4.m4.4.4.2"><apply id="S5.p3.4.m4.3.3.1.1.cmml" xref="S5.p3.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="S5.p3.4.m4.3.3.1.1.1.cmml" xref="S5.p3.4.m4.3.3.1.1">superscript</csymbol><apply id="S5.p3.4.m4.3.3.1.1.2.cmml" xref="S5.p3.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="S5.p3.4.m4.3.3.1.1.2.1.cmml" xref="S5.p3.4.m4.3.3.1.1">subscript</csymbol><apply id="S5.p3.4.m4.3.3.1.1.2.2.cmml" xref="S5.p3.4.m4.3.3.1.1.2.2"><ci id="S5.p3.4.m4.3.3.1.1.2.2.1.cmml" xref="S5.p3.4.m4.3.3.1.1.2.2.1">¯</ci><ci id="S5.p3.4.m4.3.3.1.1.2.2.2.cmml" xref="S5.p3.4.m4.3.3.1.1.2.2.2">bold-italic-ϵ</ci></apply><ci id="S5.p3.4.m4.3.3.1.1.2.3.cmml" xref="S5.p3.4.m4.3.3.1.1.2.3">𝑛</ci></apply><ci id="S5.p3.4.m4.1.1.1.1.cmml" xref="S5.p3.4.m4.1.1.1.1">𝑘</ci></apply><apply id="S5.p3.4.m4.4.4.2.2.cmml" xref="S5.p3.4.m4.4.4.2.2"><csymbol cd="ambiguous" id="S5.p3.4.m4.4.4.2.2.1.cmml" xref="S5.p3.4.m4.4.4.2.2">superscript</csymbol><apply id="S5.p3.4.m4.4.4.2.2.2.cmml" xref="S5.p3.4.m4.4.4.2.2"><csymbol cd="ambiguous" id="S5.p3.4.m4.4.4.2.2.2.1.cmml" xref="S5.p3.4.m4.4.4.2.2">subscript</csymbol><apply id="S5.p3.4.m4.4.4.2.2.2.2.cmml" xref="S5.p3.4.m4.4.4.2.2.2.2"><ci id="S5.p3.4.m4.4.4.2.2.2.2.1.cmml" xref="S5.p3.4.m4.4.4.2.2.2.2.1">¯</ci><apply id="S5.p3.4.m4.4.4.2.2.2.2.2.cmml" xref="S5.p3.4.m4.4.4.2.2.2.2.2"><ci id="S5.p3.4.m4.4.4.2.2.2.2.2.1.cmml" xref="S5.p3.4.m4.4.4.2.2.2.2.2.1">¯</ci><ci id="S5.p3.4.m4.4.4.2.2.2.2.2.2.cmml" xref="S5.p3.4.m4.4.4.2.2.2.2.2.2">bold-italic-ϵ</ci></apply></apply><ci id="S5.p3.4.m4.4.4.2.2.2.3.cmml" xref="S5.p3.4.m4.4.4.2.2.2.3">𝑛</ci></apply><ci id="S5.p3.4.m4.2.2.1.1.cmml" xref="S5.p3.4.m4.2.2.1.1">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.4.m4.4c">\big{(}\bar{\bm{\epsilon}}_{n}^{(k)},\bar{\bar{\bm{\epsilon}}}_{n}^{(k)}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.4.m4.4d">( over¯ start_ARG bold_italic_ϵ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT , over¯ start_ARG over¯ start_ARG bold_italic_ϵ end_ARG end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT )</annotation></semantics></math>, and translates it back into the global frame using (<a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.E39" title="In V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">39</span></a>) which all radars run MRBLaT in.</p> </div> <figure class="ltx_figure" id="S5.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="859" id="S5.F4.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Track (A): Example of track estimates. The yellow dots represents the radars positions and the arrow represents the boresight direction. The red shaded area represents the <span class="ltx_ERROR undefined" id="S5.F4.2.1">\qty</span>95 confidence interval for MRBLaT.</figcaption> </figure> <figure class="ltx_figure" id="S5.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="835" id="S5.F5.g1" src="x5.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Track (B): Example of track estimates. The yellow dots represents the radars positions and the arrow represents the boresight direction. The red shaded area represents the <span class="ltx_ERROR undefined" id="S5.F5.2.1">\qty</span>95 confidence interval for MRBLaT.</figcaption> </figure> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">In the simulations, we evaluated tracks (A) and (B), shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.F4" title="Figure 4 ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">4</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.F5" title="Figure 5 ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">5</span></a>, respectively. Along with the SNR contours, and 95% point wise confidence intervals. Both tracks consist of circle segments in differing directions at a constant velocity of <span class="ltx_ERROR undefined" id="S5.p4.1.1">\qty</span>10m/s. When changing direction, the circle segments are separated by a linear motion where the drone comes to a full stop with a constant acceleration of <span class="ltx_ERROR undefined" id="S5.p4.1.2">\qty</span>10m/s^2.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">Figs. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.F4" title="Figure 4 ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.F5" title="Figure 5 ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">5</span></a> clearly shows that the MRBLaT algorithm outperforms the KF with backwards smoothing in tracking the target in low SNR conditions. Moreover, MRBLaT can track maneuvers deviating from the assumed kinematics. For track (A), the ground truth is covered by the confidence interval 84% of the time. For track (B) this number is 93%. When it does fall outside, the SNR is below <span class="ltx_ERROR undefined" id="S5.p5.1.1">\qty</span>1dB suggesting the estimator underestimates the variance in this region. Contrary, the KF appears to completely lose track of the target when the SNR falls below <span class="ltx_ERROR undefined" id="S5.p5.1.2">\qty</span>5dB for one or more radars as the variance of the estimates passed to the filter increases. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16236v1#S5.F6" title="Figure 6 ‣ V Simulations ‣ Distributed Algorithm for Cooperative Joint Localization and Tracking Using Multiple-Input Multiple-Output Radars ∗ These authors contributed equally. This work is funded by the Thomas B. Thriges Foundation grant 7538-1806."><span class="ltx_text ltx_ref_tag">6</span></a> shows that the average RMSE for MRBLaT remains stable throughout the maneuvers for both tracks. For the KF, the RMSE of track (A) remains quite stable until index 200 where after it increases dramatically. For track (B), the RMSE of the KF is significantly larger than MRBLaT in index 110-270 and 280-450, which is in areas where the SNR is lower than <span class="ltx_ERROR undefined" id="S5.p5.1.3">\qty</span>5dB for one or more radars. For both tracks, the RMSE generally fluctuates more when the target reaches zero velocity, but the impact is greater on the KF.</p> </div> <figure class="ltx_figure" id="S5.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="617" id="S5.F6.g1" src="x6.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>RMSE of track (A) and (B) calculated based on 512 Monte-Carlo simulations for two different <math alttext="y" class="ltx_Math" display="inline" id="S5.F6.2.m1.1"><semantics id="S5.F6.2.m1.1b"><mi id="S5.F6.2.m1.1.1" xref="S5.F6.2.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S5.F6.2.m1.1c"><ci id="S5.F6.2.m1.1.1.cmml" xref="S5.F6.2.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F6.2.m1.1d">y</annotation><annotation encoding="application/x-llamapun" id="S5.F6.2.m1.1e">italic_y</annotation></semantics></math>-axes. For track (A), the maximum RMSE is <span class="ltx_ERROR undefined" id="S5.F6.7.1">\qty</span>69m for the KF and <span class="ltx_ERROR undefined" id="S5.F6.8.2">\qty</span>1.0m for MRBLaT and for track (B), the maximum RMSE is <span class="ltx_ERROR undefined" id="S5.F6.9.3">\qty</span>59m for the KF and <span class="ltx_ERROR undefined" id="S5.F6.10.4">\qty</span>0.9m for MRBLaT. The vertical lines shows where the target has a velocity of zero.</figcaption> </figure> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span><span class="ltx_text ltx_font_smallcaps" id="S6.1.1">Conclusion</span> </h2> <div class="ltx_para ltx_noindent" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">We propose a distributed localization and tracking algorithm with low communication overhead which scales linearly with the number of radars in the system. The algorithm builds on a Bayesian variational message passing framework, and outperforms a multi input KF in low SNR conditions. The algorithm tracks maneuvers differing from the assumed kinematics well across a wide range of SNR values, with the ground truth falling within the 95% confidence interval of the estimator <span class="ltx_ERROR undefined" id="S6.p1.1.1">\qty</span>84 and <span class="ltx_ERROR undefined" id="S6.p1.1.2">\qty</span>93 of the time for track (A) and (B), respectively.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> M. J. Miantezila and B. Guo, “Sensing spectrum sharing based massive MIMO radar for drone tracking and interception,” <em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">PloS one</em>, vol. 17, no. 5, pp. e0 268 834–e0 268 834, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> S. V. Shvetsova and A. V. Shvetsov, “Ensuring safety and security in employing drones at airports,” <em class="ltx_emph ltx_font_italic" id="bib.bib2.1.1">Journal of transportation security</em>, vol. 14, no. 1-2, pp. 41–53, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> FT REPORTERS, “Drones force closure of UK’s Gatwick airport: Travel chaos [Europe Region],” <em class="ltx_emph ltx_font_italic" id="bib.bib3.1.1">The Financial times (London ed.)</em>, 2018. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> R. A. Zitar, A. Mohsen, A. E. Seghrouchni, F. Barbaresco, and N. A. Al-Dmour, “Intensive Review of Drones Detection and Tracking: Linear Kalman Filter Versus Nonlinear Regression, an Analysis Case,” <em class="ltx_emph ltx_font_italic" id="bib.bib4.1.1">Archives of computational methods in engineering</em>, pp. 2811–2830, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> C. Clemente, F. Fioranelli, F. Colone, G. Li <em class="ltx_emph ltx_font_italic" id="bib.bib5.1.1">et al.</em>, <em class="ltx_emph ltx_font_italic" id="bib.bib5.2.2">Radar countermeasures for unmanned aerial vehicles</em>. Institution of Engineering and Technology, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> Q. Yu, T. B. Dinh, and G. Medioni, “Online Tracking and Reacquisition Using Co-trained Generative and Discriminative Trackers,” in <em class="ltx_emph ltx_font_italic" id="bib.bib6.1.1">Computer Vision – ECCV 2008</em>, vol. 5303, no. 2. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, pp. 678–691. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> M. Uzair, R. S. Brinkworth, and A. Finn, “Bio-Inspired Video Enhancement for Small Moving Target Detection,” <em class="ltx_emph ltx_font_italic" id="bib.bib7.1.1">IEEE transactions on image processing</em>, vol. 30, pp. 1232–1244, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> D. Huang, Z. Zhang, X. Fang, M. He, H. Lai, and B. Mi, “STIF: A Spatial-Temporal Integrated Framework for End-to-End Micro-UAV Trajectory Tracking and Prediction With 4-D MIMO Radar,” <em class="ltx_emph ltx_font_italic" id="bib.bib8.1.1">IEEE internet of things journal</em>, vol. 10, no. 21, pp. 18 821–18 836, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> X. Fang, J. Zhu, D. Huang, Z. Zhang, and G. Xiao, “E2DTF: An End-to-End Detection and Tracking Framework for Multiple Micro-UAVs with FMCW-MIMO Radar,” <em class="ltx_emph ltx_font_italic" id="bib.bib9.1.1">IEEE transactions on geoscience and remote sensing</em>, vol. 61, pp. 1–1, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> N. Knudde, B. Vandersmissen, K. Parashar, I. Couckuyt, A. Jalalvand, A. Bourdoux, W. De Neve, and T. Dhaene, “Indoor tracking of multiple persons with a 77 GHz MIMO FMCW radar,” in <em class="ltx_emph ltx_font_italic" id="bib.bib10.1.1">2017 European Radar Conference (EURAD)</em>, 2017, pp. 61–64. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> A. M. Westerkam, C. N. Manchón, P. Mogensen, and T. Pedersen, “Bayesian Joint Localization and Tracking Algorithm Using Multiple-Input Multiple-Output Radar,” in <em class="ltx_emph ltx_font_italic" id="bib.bib11.1.1">2023 IEEE 9th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)</em>. IEEE, 2023, pp. 316–320. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> T. Haarnoja, A. Ajay, S. Levine, and P. Abbeel, “Backprop KF: Learning Discriminative Deterministic State Estimators,” <em class="ltx_emph ltx_font_italic" id="bib.bib12.1.1">Advances in neural information processing systems</em>, vol. 29, 2016. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> K. Zarai, I. Ben Abdallah, and A. Cherif, “Improved multi-target tracking crossing paths in MIMO FMCW 8X16 radar system using a new hybrid AMC-JPDAF algorithm,” <em class="ltx_emph ltx_font_italic" id="bib.bib13.1.1">Aerospace systems (Online)</em>, vol. 6, no. 2, pp. 343–351, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> F. Lehmann, “Recursive Bayesian Filtering for Multitarget Track-Before-Detect in Passive Radars,” <em class="ltx_emph ltx_font_italic" id="bib.bib14.1.1">IEEE transactions on aerospace and electronic systems</em>, vol. 48, no. 3, pp. 2458–2480, 2012. </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> Y. Lai, L. Venturino, E. Grossi, and W. Yi, “Joint Detection and Localization in Distributed MIMO Radars Employing Waveforms with Imperfect Auto- and Cross-Correlation,” <em class="ltx_emph ltx_font_italic" id="bib.bib15.1.1">IEEE transactions on vehicular technology</em>, vol. 72, no. 12, pp. 1–13, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> W. Yi, T. Zhou, Y. Ai, and R. S. Blum, “Suboptimal Low Complexity Joint Multi-Target Detection and Localization for Non-Coherent MIMO Radar With Widely Separated Antennas,” <em class="ltx_emph ltx_font_italic" id="bib.bib16.1.1">IEEE transactions on signal processing</em>, vol. 68, pp. 901–916, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Spatial Diversity in Radars-Models and Detection Performance,” <em class="ltx_emph ltx_font_italic" id="bib.bib17.1.1">IEEE transactions on signal processing</em>, vol. 54, no. 3, pp. 823–838, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> R. Niu, R. S. Blum, P. K. Varshney, and A. L. Drozd, “Target Localization and Tracking in Noncoherent Multiple-Input Multiple-Output Radar Systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib18.1.1">IEEE transactions on aerospace and electronic systems</em>, vol. 48, no. 2, pp. 1466–1489, 2012. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> H. Godrich, A. M. Haimovich, and R. S. Blum, “Target Localization Accuracy Gain in MIMO Radar-Based Systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib19.1.1">IEEE transactions on information theory</em>, vol. 56, no. 6, pp. 2783–2803, 2010. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> G. Zhang, W. Yi, P. K. Varshney, and L. Kong, “Direct Target Localization With Quantized Measurements in Noncoherent Distributed MIMO Radar Systems,” <em class="ltx_emph ltx_font_italic" id="bib.bib20.1.1">IEEE transactions on geoscience and remote sensing</em>, vol. 61, pp. 1–18, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> S. Yang, W. Yi, and A. Jakobsson, “Multitarget Detection Strategy for Distributed MIMO Radar With Widely Separated Antennas,” <em class="ltx_emph ltx_font_italic" id="bib.bib21.1.1">IEEE transactions on geoscience and remote sensing</em>, vol. 60, pp. 1–16, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> C. M. Bishop, <em class="ltx_emph ltx_font_italic" id="bib.bib22.1.1">Pattern Recognition and Machine Learning (Information Science and Statistics)</em>, 1st ed. Springer, 2007. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Mar 20 15:28:05 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" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>