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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Reconstructing the LISA massive black hole binary population via iterative kernel density estimation</title>   <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2410.17056v2/"/></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/2410.17056v2#S1" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span> Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span>Simulated data</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS1" title="In II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.1 </span>Astrophysical population model and detectability</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS2" title="In II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.2 </span>Parameter Estimation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS3" title="In II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.3 </span>Population reconstruction via iterative KDE</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS4" title="In II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.4 </span>Selection effects and validation of PE</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS5" title="In II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.5 </span>Application to simulated data</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Results</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S4" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A1" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Results from adaptive KDE reweighting without selection factor</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A2" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Results from adaptive KDE using simple 1/<math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline"><semantics><msub><mi>p</mi><mi>det</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑝</ci><ci>det</ci></apply></annotation-xml><annotation encoding="application/x-tex">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> reweighting</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A3" title="In Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Results from weighted KDE using simple 1/<math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline"><semantics><msub><mi>p</mi><mi>det</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑝</ci><ci>det</ci></apply></annotation-xml><annotation encoding="application/x-tex">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> weights</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"><span class="ltx_note ltx_role_thanks" id="id1"><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>Corresponding author, email: jsadiq@sissa.it</span></span></span> <h1 class="ltx_title ltx_title_document">Reconstructing the LISA massive black hole binary population via iterative kernel density estimation</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Jam Sadiq<span class="ltx_ERROR undefined" id="id1.1.id1">\orcidlink</span>0000-0001-5931-3624<sup class="ltx_sup" id="id2.2.id2">1, 2</sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Kallol Dey<span class="ltx_ERROR undefined" id="id3.1.id1">\orcidlink</span>0000-0003-2145-1145<sup class="ltx_sup" id="id4.2.id2">3</sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Thomas Dent<span class="ltx_ERROR undefined" id="id5.1.id1">\orcidlink</span>0000-0003-1354-7809<sup class="ltx_sup" id="id6.2.id2">4</sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Enrico Barausse<span class="ltx_ERROR undefined" id="id7.1.id1">\orcidlink</span>0000-0001-6499-6263<sup class="ltx_sup" id="id8.2.id2">1, 2</sup> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id9.3.id1">1</sup> SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id10.4.id1">2</sup> IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id11.5.id1">3</sup> School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, Maruthamala PO, Vithura, Thiruvananthapuram 695551, Kerala, India </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id12.6.id1">4</sup> IGFAE, University of Santiago de Compostela, E-15782 Spain </span></span></span> </div> <div class="ltx_dates">(March 20, 2025)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id13.id1">Reconstructing the properties of the astrophysical population of binary compact objects in the universe is a key science goal of gravitational wave detectors. This goal is hindered by the finite strain, frequency sensitivity and observing time of current and future detectors. This implies that we can in general observe only a selected subset of the underlying population, with limited event statistics, and also nontrivial observational uncertainties in the parameters of each event. In this work, we will focus on observations of massive black hole binaries with the Laser Interferometer Space Antenna (LISA). If these black holes grow from population III star remnants (“light seeds”), a significant fraction of the binary population at low masses and high redshift will be beyond LISA’s observational reach; thus, selection effects have to be accounted for when reconstructing the underlying population. Here we propose an iterative, kernel density estimation (KDE)-based non-parametric method, in order to tackle these statistical challenges in reconstructing the astrophysical population distribution from a finite number of observed signals over total mass and redshift. We test the method against a set of simulated LISA observations in a light seed formation scenario. We find that our approach is successful at reconstructing the underlying astrophysical distribution in mass and redshift, except in parameter regions where zero or order(1) signals are observed.</p> </div> <span class="ltx_note ltx_note_frontmatter ltx_role_preprint" id="id2"><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">preprint: </span>APS/123-QED</span></span></span> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span> Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">The birth of gravitational wave (GW) astronomy with the first detection of the binary black hole merger GW150914 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib1" title="">1</a>]</cite> and the subsequent more than a hundred detections <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib5" title="">5</a>]</cite> by the LIGO-Virgo-KAGRA (LVK) collaboration have opened a novel perspective on the cosmos, providing not only evidence that compact objects coalesce, but also a way to extract the statistical properties of their astrophysical population <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib6" title="">6</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Because GW interferometers have intrinsically limited sensitivity in strain and frequency, they typically access only a fraction of the compact object binaries in our past light cone. On the ground, these selection effects are the main reason of the uncertainties in the reconstruction of the mass function of stellar origin black hole binaries <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib9" title="">9</a>]</cite>. Previous methods, such as those outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib11" title="">11</a>]</cite>, typically factor out the dependence of the extrinsic angles on the event signal-to-noise ratio (SNR). Events can be subsequently thresholded based on the SNR of an optimal source with the same intrinsic parameters. Recent state-of-the-art GW population studies include selection biases in some approximate form <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib14" title="">14</a>]</cite>, and have been further improved with different approaches <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib17" title="">17</a>]</cite>. Sensitivity estimates based on SNR thresholding have been applied in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib6" title="">6</a>]</cite>; a recent comparison with search injection (simulated signal) campaigns in real LIGO-Virgo noise is presented in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib18" title="">18</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">While these uncertainties will be mitigated with updates to the current facilities and with next-generation ground based detectors such as the Einstein Telescope <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib21" title="">21</a>]</cite> or Cosmic Explorer <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib23" title="">23</a>]</cite>, it is expected that even the latter may miss a significant fraction of the black hole population at high redshift (see e.g. Fig. 3 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib24" title="">24</a>]</cite>). Moreover, even when high-redshift binaries are observed, their distance determination will be significantly degraded, due to statistical errors (caused by the low signal-to-noise ratios) and weak-lensing systematics, which are the dominant source of error on the distance at high <math alttext="z" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_z</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib26" title="">26</a>]</cite>. Another potential source of systematics is given by environmental effects. However, those are expected to be negligible for massive black hole binaries in the LISA band <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib28" title="">28</a>]</cite>, which are the focus of this paper. This makes the reconstruction of the distribution of the underlying population in mass and redshift even more challenging.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.2">It should also be noted that the black hole population of the universe does not only consist of remnants of stellar evolution, such as those observed by the LVK collaboration. Massive black holes (MBHs) with masses <math alttext="10^{5}" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><msup id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mn id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">10</mn><mn id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml">5</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1">superscript</csymbol><cn id="S1.p4.1.m1.1.1.2.cmml" type="integer" xref="S1.p4.1.m1.1.1.2">10</cn><cn id="S1.p4.1.m1.1.1.3.cmml" type="integer" xref="S1.p4.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">10^{5}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math>–<math alttext="10^{9}\,M_{\odot}" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mrow id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml"><msup id="S1.p4.2.m2.1.1.2" xref="S1.p4.2.m2.1.1.2.cmml"><mn id="S1.p4.2.m2.1.1.2.2" xref="S1.p4.2.m2.1.1.2.2.cmml">10</mn><mn id="S1.p4.2.m2.1.1.2.3" xref="S1.p4.2.m2.1.1.2.3.cmml">9</mn></msup><mo id="S1.p4.2.m2.1.1.1" lspace="0.170em" xref="S1.p4.2.m2.1.1.1.cmml">⁢</mo><msub id="S1.p4.2.m2.1.1.3" xref="S1.p4.2.m2.1.1.3.cmml"><mi id="S1.p4.2.m2.1.1.3.2" xref="S1.p4.2.m2.1.1.3.2.cmml">M</mi><mo id="S1.p4.2.m2.1.1.3.3" xref="S1.p4.2.m2.1.1.3.3.cmml">⊙</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><apply id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"><times id="S1.p4.2.m2.1.1.1.cmml" xref="S1.p4.2.m2.1.1.1"></times><apply id="S1.p4.2.m2.1.1.2.cmml" xref="S1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.2.1.cmml" xref="S1.p4.2.m2.1.1.2">superscript</csymbol><cn id="S1.p4.2.m2.1.1.2.2.cmml" type="integer" xref="S1.p4.2.m2.1.1.2.2">10</cn><cn id="S1.p4.2.m2.1.1.2.3.cmml" type="integer" xref="S1.p4.2.m2.1.1.2.3">9</cn></apply><apply id="S1.p4.2.m2.1.1.3.cmml" xref="S1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.3.1.cmml" xref="S1.p4.2.m2.1.1.3">subscript</csymbol><ci id="S1.p4.2.m2.1.1.3.2.cmml" xref="S1.p4.2.m2.1.1.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p4.2.m2.1.1.3.3.cmml" xref="S1.p4.2.m2.1.1.3.3">direct-product</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">10^{9}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> have been observed at the center of most large elliptical galaxies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib30" title="">30</a>]</cite>, and also in some lower mass systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib33" title="">33</a>]</cite>. These MBHs, when they accrete, are also the engine of Active Galactic Nuclei (AGNs) and quasars <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib34" title="">34</a>]</cite>, and as such they constitute a crucial ingredient of galaxy formation and evolution, since they are believed to exert feedback (via radiation, disk winds or jets) on their galactic hosts <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib37" title="">37</a>]</cite>. Because galaxies form hierarchically, through the merger of smaller systems into bigger ones, the MBHs at the center of galaxies are also expected to merge, emitting GW signals of frequencies lower than those accessible from the ground (where interferometers are limited by seismic noise).</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.3">The origin of MBHs is still to some extent a mystery. Several models for the black hole “seeds” of the MBH population at high redshift <math alttext="z\gtrsim 10-15" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">z</mi><mo id="S1.p5.1.m1.1.1.1" xref="S1.p5.1.m1.1.1.1.cmml">≳</mo><mrow id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml"><mn id="S1.p5.1.m1.1.1.3.2" xref="S1.p5.1.m1.1.1.3.2.cmml">10</mn><mo id="S1.p5.1.m1.1.1.3.1" xref="S1.p5.1.m1.1.1.3.1.cmml">−</mo><mn id="S1.p5.1.m1.1.1.3.3" xref="S1.p5.1.m1.1.1.3.3.cmml">15</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="latexml" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">𝑧</ci><apply id="S1.p5.1.m1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.3"><minus id="S1.p5.1.m1.1.1.3.1.cmml" xref="S1.p5.1.m1.1.1.3.1"></minus><cn id="S1.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S1.p5.1.m1.1.1.3.2">10</cn><cn id="S1.p5.1.m1.1.1.3.3.cmml" type="integer" xref="S1.p5.1.m1.1.1.3.3">15</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">z\gtrsim 10-15</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_z ≳ 10 - 15</annotation></semantics></math> have been proposed, but can be broadly divided into two classes, “heavy seeds” of masses <math alttext="\sim 10^{5}\,M_{\odot}" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mrow id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mi id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml"></mi><mo id="S1.p5.2.m2.1.1.1" xref="S1.p5.2.m2.1.1.1.cmml">∼</mo><mrow id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml"><msup id="S1.p5.2.m2.1.1.3.2" xref="S1.p5.2.m2.1.1.3.2.cmml"><mn id="S1.p5.2.m2.1.1.3.2.2" xref="S1.p5.2.m2.1.1.3.2.2.cmml">10</mn><mn id="S1.p5.2.m2.1.1.3.2.3" xref="S1.p5.2.m2.1.1.3.2.3.cmml">5</mn></msup><mo id="S1.p5.2.m2.1.1.3.1" xref="S1.p5.2.m2.1.1.3.1.cmml">⁢</mo><msub id="S1.p5.2.m2.1.1.3.3" xref="S1.p5.2.m2.1.1.3.3.cmml"><mi id="S1.p5.2.m2.1.1.3.3.2" xref="S1.p5.2.m2.1.1.3.3.2.cmml">M</mi><mo id="S1.p5.2.m2.1.1.3.3.3" xref="S1.p5.2.m2.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><csymbol cd="latexml" id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p5.2.m2.1.1.2.cmml" xref="S1.p5.2.m2.1.1.2">absent</csymbol><apply id="S1.p5.2.m2.1.1.3.cmml" xref="S1.p5.2.m2.1.1.3"><times id="S1.p5.2.m2.1.1.3.1.cmml" xref="S1.p5.2.m2.1.1.3.1"></times><apply id="S1.p5.2.m2.1.1.3.2.cmml" xref="S1.p5.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S1.p5.2.m2.1.1.3.2.1.cmml" xref="S1.p5.2.m2.1.1.3.2">superscript</csymbol><cn id="S1.p5.2.m2.1.1.3.2.2.cmml" type="integer" xref="S1.p5.2.m2.1.1.3.2.2">10</cn><cn id="S1.p5.2.m2.1.1.3.2.3.cmml" type="integer" xref="S1.p5.2.m2.1.1.3.2.3">5</cn></apply><apply id="S1.p5.2.m2.1.1.3.3.cmml" xref="S1.p5.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S1.p5.2.m2.1.1.3.3.1.cmml" xref="S1.p5.2.m2.1.1.3.3">subscript</csymbol><ci id="S1.p5.2.m2.1.1.3.3.2.cmml" xref="S1.p5.2.m2.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p5.2.m2.1.1.3.3.3.cmml" xref="S1.p5.2.m2.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">\sim 10^{5}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">∼ 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> or “light seeds” of masses <math alttext="10^{2}-10^{3}\,M_{\odot}" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mrow id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><msup id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2.cmml"><mn id="S1.p5.3.m3.1.1.2.2" xref="S1.p5.3.m3.1.1.2.2.cmml">10</mn><mn id="S1.p5.3.m3.1.1.2.3" xref="S1.p5.3.m3.1.1.2.3.cmml">2</mn></msup><mo id="S1.p5.3.m3.1.1.1" xref="S1.p5.3.m3.1.1.1.cmml">−</mo><mrow id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml"><msup id="S1.p5.3.m3.1.1.3.2" xref="S1.p5.3.m3.1.1.3.2.cmml"><mn id="S1.p5.3.m3.1.1.3.2.2" xref="S1.p5.3.m3.1.1.3.2.2.cmml">10</mn><mn id="S1.p5.3.m3.1.1.3.2.3" xref="S1.p5.3.m3.1.1.3.2.3.cmml">3</mn></msup><mo id="S1.p5.3.m3.1.1.3.1" xref="S1.p5.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S1.p5.3.m3.1.1.3.3" xref="S1.p5.3.m3.1.1.3.3.cmml"><mi id="S1.p5.3.m3.1.1.3.3.2" xref="S1.p5.3.m3.1.1.3.3.2.cmml">M</mi><mo id="S1.p5.3.m3.1.1.3.3.3" xref="S1.p5.3.m3.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1"><minus id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.1.1"></minus><apply id="S1.p5.3.m3.1.1.2.cmml" xref="S1.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.p5.3.m3.1.1.2.1.cmml" xref="S1.p5.3.m3.1.1.2">superscript</csymbol><cn id="S1.p5.3.m3.1.1.2.2.cmml" type="integer" xref="S1.p5.3.m3.1.1.2.2">10</cn><cn id="S1.p5.3.m3.1.1.2.3.cmml" type="integer" xref="S1.p5.3.m3.1.1.2.3">2</cn></apply><apply id="S1.p5.3.m3.1.1.3.cmml" xref="S1.p5.3.m3.1.1.3"><times id="S1.p5.3.m3.1.1.3.1.cmml" xref="S1.p5.3.m3.1.1.3.1"></times><apply id="S1.p5.3.m3.1.1.3.2.cmml" xref="S1.p5.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S1.p5.3.m3.1.1.3.2.1.cmml" xref="S1.p5.3.m3.1.1.3.2">superscript</csymbol><cn id="S1.p5.3.m3.1.1.3.2.2.cmml" type="integer" xref="S1.p5.3.m3.1.1.3.2.2">10</cn><cn id="S1.p5.3.m3.1.1.3.2.3.cmml" type="integer" xref="S1.p5.3.m3.1.1.3.2.3">3</cn></apply><apply id="S1.p5.3.m3.1.1.3.3.cmml" xref="S1.p5.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S1.p5.3.m3.1.1.3.3.1.cmml" xref="S1.p5.3.m3.1.1.3.3">subscript</csymbol><ci id="S1.p5.3.m3.1.1.3.3.2.cmml" xref="S1.p5.3.m3.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p5.3.m3.1.1.3.3.3.cmml" xref="S1.p5.3.m3.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">10^{2}-10^{3}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib38" title="">38</a>]</cite> for a review). The latter may even survive to low redshift to provide a population of intermediate mass black holes between MBHs and stellar-origin black holes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib39" title="">39</a>]</cite>. A mixing <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib41" title="">41</a>]</cite> of these light and heavy populations is also possible, if not probable. The growth of the MBH seeds, via accretion and mergers, is also subject to considerable uncertainties. On the theory side, MBHs are tiny compared to galactic scales, and their sphere of influence is difficult to resolve in simulations. Also, many processes that are crucial for their evolution, such as star formation, supernova feedback, AGN feedback and accretion itself, are not yet fully understood from first principles. For this reason, MBH evolution, especially at high redshift, is heavily affected by the “sub-grid” prescriptions used to describe these processes in hydrodynamic simulations (cf. e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib44" title="">44</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib48" title="">48</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib50" title="">50</a>]</cite>), or is followed by using semi-analytic galaxy evolution models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib51" title="">51</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib52" title="">52</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib56" title="">56</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib57" title="">57</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib59" title="">59</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite>. On the observational side, with electromagnetic telescopes, MBHs are hard to observe at high redshift, when their growth and AGN activity are expected to be strongest.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.7">These difficulties open a significant potential discovery window for space-borne detectors of GWs and for pulsar-timing arrays. Regarding the latter, there is now evidence of a common correlated signal in the data of several experiments <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib62" title="">62</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib63" title="">63</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib64" title="">64</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib65" title="">65</a>]</cite>, which could be the long-expected stochastic background of GWs in the nHz band from a population of MBHs of masses <math alttext="\gtrsim 10^{8}\,M_{\odot}" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mrow id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml"><mi id="S1.p6.1.m1.1.1.2" xref="S1.p6.1.m1.1.1.2.cmml"></mi><mo id="S1.p6.1.m1.1.1.1" xref="S1.p6.1.m1.1.1.1.cmml">≳</mo><mrow id="S1.p6.1.m1.1.1.3" xref="S1.p6.1.m1.1.1.3.cmml"><msup id="S1.p6.1.m1.1.1.3.2" xref="S1.p6.1.m1.1.1.3.2.cmml"><mn id="S1.p6.1.m1.1.1.3.2.2" xref="S1.p6.1.m1.1.1.3.2.2.cmml">10</mn><mn id="S1.p6.1.m1.1.1.3.2.3" xref="S1.p6.1.m1.1.1.3.2.3.cmml">8</mn></msup><mo id="S1.p6.1.m1.1.1.3.1" xref="S1.p6.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S1.p6.1.m1.1.1.3.3" xref="S1.p6.1.m1.1.1.3.3.cmml"><mi id="S1.p6.1.m1.1.1.3.3.2" xref="S1.p6.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S1.p6.1.m1.1.1.3.3.3" xref="S1.p6.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><apply id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1"><csymbol cd="latexml" id="S1.p6.1.m1.1.1.1.cmml" xref="S1.p6.1.m1.1.1.1">greater-than-or-equivalent-to</csymbol><csymbol cd="latexml" id="S1.p6.1.m1.1.1.2.cmml" xref="S1.p6.1.m1.1.1.2">absent</csymbol><apply id="S1.p6.1.m1.1.1.3.cmml" xref="S1.p6.1.m1.1.1.3"><times id="S1.p6.1.m1.1.1.3.1.cmml" xref="S1.p6.1.m1.1.1.3.1"></times><apply id="S1.p6.1.m1.1.1.3.2.cmml" xref="S1.p6.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.p6.1.m1.1.1.3.2.1.cmml" xref="S1.p6.1.m1.1.1.3.2">superscript</csymbol><cn id="S1.p6.1.m1.1.1.3.2.2.cmml" type="integer" xref="S1.p6.1.m1.1.1.3.2.2">10</cn><cn id="S1.p6.1.m1.1.1.3.2.3.cmml" type="integer" xref="S1.p6.1.m1.1.1.3.2.3">8</cn></apply><apply id="S1.p6.1.m1.1.1.3.3.cmml" xref="S1.p6.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p6.1.m1.1.1.3.3.1.cmml" xref="S1.p6.1.m1.1.1.3.3">subscript</csymbol><ci id="S1.p6.1.m1.1.1.3.3.2.cmml" xref="S1.p6.1.m1.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p6.1.m1.1.1.3.3.3.cmml" xref="S1.p6.1.m1.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">\gtrsim 10^{8}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">≳ 10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib66" title="">66</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib67" title="">67</a>]</cite>. At higher frequencies, the Laser Interferometer Space Antenna (LISA) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib68" title="">68</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib69" title="">69</a>]</cite> is scheduled to be launched in 2034 and will observe MBH binaries of masses <math alttext="10^{4}-10^{7}\,M_{\odot}" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mrow id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml"><msup id="S1.p6.2.m2.1.1.2" xref="S1.p6.2.m2.1.1.2.cmml"><mn id="S1.p6.2.m2.1.1.2.2" xref="S1.p6.2.m2.1.1.2.2.cmml">10</mn><mn id="S1.p6.2.m2.1.1.2.3" xref="S1.p6.2.m2.1.1.2.3.cmml">4</mn></msup><mo id="S1.p6.2.m2.1.1.1" xref="S1.p6.2.m2.1.1.1.cmml">−</mo><mrow id="S1.p6.2.m2.1.1.3" xref="S1.p6.2.m2.1.1.3.cmml"><msup id="S1.p6.2.m2.1.1.3.2" xref="S1.p6.2.m2.1.1.3.2.cmml"><mn id="S1.p6.2.m2.1.1.3.2.2" xref="S1.p6.2.m2.1.1.3.2.2.cmml">10</mn><mn id="S1.p6.2.m2.1.1.3.2.3" xref="S1.p6.2.m2.1.1.3.2.3.cmml">7</mn></msup><mo id="S1.p6.2.m2.1.1.3.1" xref="S1.p6.2.m2.1.1.3.1.cmml">⁢</mo><msub id="S1.p6.2.m2.1.1.3.3" xref="S1.p6.2.m2.1.1.3.3.cmml"><mi id="S1.p6.2.m2.1.1.3.3.2" xref="S1.p6.2.m2.1.1.3.3.2.cmml">M</mi><mo id="S1.p6.2.m2.1.1.3.3.3" xref="S1.p6.2.m2.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><apply id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1"><minus id="S1.p6.2.m2.1.1.1.cmml" xref="S1.p6.2.m2.1.1.1"></minus><apply id="S1.p6.2.m2.1.1.2.cmml" xref="S1.p6.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p6.2.m2.1.1.2.1.cmml" xref="S1.p6.2.m2.1.1.2">superscript</csymbol><cn id="S1.p6.2.m2.1.1.2.2.cmml" type="integer" xref="S1.p6.2.m2.1.1.2.2">10</cn><cn id="S1.p6.2.m2.1.1.2.3.cmml" type="integer" xref="S1.p6.2.m2.1.1.2.3">4</cn></apply><apply id="S1.p6.2.m2.1.1.3.cmml" xref="S1.p6.2.m2.1.1.3"><times id="S1.p6.2.m2.1.1.3.1.cmml" xref="S1.p6.2.m2.1.1.3.1"></times><apply id="S1.p6.2.m2.1.1.3.2.cmml" xref="S1.p6.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S1.p6.2.m2.1.1.3.2.1.cmml" xref="S1.p6.2.m2.1.1.3.2">superscript</csymbol><cn id="S1.p6.2.m2.1.1.3.2.2.cmml" type="integer" xref="S1.p6.2.m2.1.1.3.2.2">10</cn><cn id="S1.p6.2.m2.1.1.3.2.3.cmml" type="integer" xref="S1.p6.2.m2.1.1.3.2.3">7</cn></apply><apply id="S1.p6.2.m2.1.1.3.3.cmml" xref="S1.p6.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S1.p6.2.m2.1.1.3.3.1.cmml" xref="S1.p6.2.m2.1.1.3.3">subscript</csymbol><ci id="S1.p6.2.m2.1.1.3.3.2.cmml" xref="S1.p6.2.m2.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p6.2.m2.1.1.3.3.3.cmml" xref="S1.p6.2.m2.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">10^{4}-10^{7}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> in the band from <math alttext="\sim 0.01" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><mrow id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml"><mi id="S1.p6.3.m3.1.1.2" xref="S1.p6.3.m3.1.1.2.cmml"></mi><mo id="S1.p6.3.m3.1.1.1" xref="S1.p6.3.m3.1.1.1.cmml">∼</mo><mn id="S1.p6.3.m3.1.1.3" xref="S1.p6.3.m3.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><apply id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1"><csymbol cd="latexml" id="S1.p6.3.m3.1.1.1.cmml" xref="S1.p6.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p6.3.m3.1.1.2.cmml" xref="S1.p6.3.m3.1.1.2">absent</csymbol><cn id="S1.p6.3.m3.1.1.3.cmml" type="float" xref="S1.p6.3.m3.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">\sim 0.01</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">∼ 0.01</annotation></semantics></math> mHz to <math alttext="\sim 0.1" class="ltx_Math" display="inline" id="S1.p6.4.m4.1"><semantics id="S1.p6.4.m4.1a"><mrow id="S1.p6.4.m4.1.1" xref="S1.p6.4.m4.1.1.cmml"><mi id="S1.p6.4.m4.1.1.2" xref="S1.p6.4.m4.1.1.2.cmml"></mi><mo id="S1.p6.4.m4.1.1.1" xref="S1.p6.4.m4.1.1.1.cmml">∼</mo><mn id="S1.p6.4.m4.1.1.3" xref="S1.p6.4.m4.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.4.m4.1b"><apply id="S1.p6.4.m4.1.1.cmml" xref="S1.p6.4.m4.1.1"><csymbol cd="latexml" id="S1.p6.4.m4.1.1.1.cmml" xref="S1.p6.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p6.4.m4.1.1.2.cmml" xref="S1.p6.4.m4.1.1.2">absent</csymbol><cn id="S1.p6.4.m4.1.1.3.cmml" type="float" xref="S1.p6.4.m4.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.4.m4.1c">\sim 0.1</annotation><annotation encoding="application/x-llamapun" id="S1.p6.4.m4.1d">∼ 0.1</annotation></semantics></math> Hz. The exact redshift reach of LISA depends on the binary mass, with masses <math alttext="10^{5}-10^{6}M_{\odot}" class="ltx_Math" display="inline" id="S1.p6.5.m5.1"><semantics id="S1.p6.5.m5.1a"><mrow id="S1.p6.5.m5.1.1" xref="S1.p6.5.m5.1.1.cmml"><msup id="S1.p6.5.m5.1.1.2" xref="S1.p6.5.m5.1.1.2.cmml"><mn id="S1.p6.5.m5.1.1.2.2" xref="S1.p6.5.m5.1.1.2.2.cmml">10</mn><mn id="S1.p6.5.m5.1.1.2.3" xref="S1.p6.5.m5.1.1.2.3.cmml">5</mn></msup><mo id="S1.p6.5.m5.1.1.1" xref="S1.p6.5.m5.1.1.1.cmml">−</mo><mrow id="S1.p6.5.m5.1.1.3" xref="S1.p6.5.m5.1.1.3.cmml"><msup id="S1.p6.5.m5.1.1.3.2" xref="S1.p6.5.m5.1.1.3.2.cmml"><mn id="S1.p6.5.m5.1.1.3.2.2" xref="S1.p6.5.m5.1.1.3.2.2.cmml">10</mn><mn id="S1.p6.5.m5.1.1.3.2.3" xref="S1.p6.5.m5.1.1.3.2.3.cmml">6</mn></msup><mo id="S1.p6.5.m5.1.1.3.1" xref="S1.p6.5.m5.1.1.3.1.cmml">⁢</mo><msub id="S1.p6.5.m5.1.1.3.3" xref="S1.p6.5.m5.1.1.3.3.cmml"><mi id="S1.p6.5.m5.1.1.3.3.2" xref="S1.p6.5.m5.1.1.3.3.2.cmml">M</mi><mo id="S1.p6.5.m5.1.1.3.3.3" xref="S1.p6.5.m5.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.5.m5.1b"><apply id="S1.p6.5.m5.1.1.cmml" xref="S1.p6.5.m5.1.1"><minus id="S1.p6.5.m5.1.1.1.cmml" xref="S1.p6.5.m5.1.1.1"></minus><apply id="S1.p6.5.m5.1.1.2.cmml" xref="S1.p6.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.p6.5.m5.1.1.2.1.cmml" xref="S1.p6.5.m5.1.1.2">superscript</csymbol><cn id="S1.p6.5.m5.1.1.2.2.cmml" type="integer" xref="S1.p6.5.m5.1.1.2.2">10</cn><cn id="S1.p6.5.m5.1.1.2.3.cmml" type="integer" xref="S1.p6.5.m5.1.1.2.3">5</cn></apply><apply id="S1.p6.5.m5.1.1.3.cmml" xref="S1.p6.5.m5.1.1.3"><times id="S1.p6.5.m5.1.1.3.1.cmml" xref="S1.p6.5.m5.1.1.3.1"></times><apply id="S1.p6.5.m5.1.1.3.2.cmml" xref="S1.p6.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S1.p6.5.m5.1.1.3.2.1.cmml" xref="S1.p6.5.m5.1.1.3.2">superscript</csymbol><cn id="S1.p6.5.m5.1.1.3.2.2.cmml" type="integer" xref="S1.p6.5.m5.1.1.3.2.2">10</cn><cn id="S1.p6.5.m5.1.1.3.2.3.cmml" type="integer" xref="S1.p6.5.m5.1.1.3.2.3">6</cn></apply><apply id="S1.p6.5.m5.1.1.3.3.cmml" xref="S1.p6.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S1.p6.5.m5.1.1.3.3.1.cmml" xref="S1.p6.5.m5.1.1.3.3">subscript</csymbol><ci id="S1.p6.5.m5.1.1.3.3.2.cmml" xref="S1.p6.5.m5.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p6.5.m5.1.1.3.3.3.cmml" xref="S1.p6.5.m5.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.5.m5.1c">10^{5}-10^{6}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.5.m5.1d">10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> observable up to <math alttext="z\sim 20" class="ltx_Math" display="inline" id="S1.p6.6.m6.1"><semantics id="S1.p6.6.m6.1a"><mrow id="S1.p6.6.m6.1.1" xref="S1.p6.6.m6.1.1.cmml"><mi id="S1.p6.6.m6.1.1.2" xref="S1.p6.6.m6.1.1.2.cmml">z</mi><mo id="S1.p6.6.m6.1.1.1" xref="S1.p6.6.m6.1.1.1.cmml">∼</mo><mn id="S1.p6.6.m6.1.1.3" xref="S1.p6.6.m6.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.6.m6.1b"><apply id="S1.p6.6.m6.1.1.cmml" xref="S1.p6.6.m6.1.1"><csymbol cd="latexml" id="S1.p6.6.m6.1.1.1.cmml" xref="S1.p6.6.m6.1.1.1">similar-to</csymbol><ci id="S1.p6.6.m6.1.1.2.cmml" xref="S1.p6.6.m6.1.1.2">𝑧</ci><cn id="S1.p6.6.m6.1.1.3.cmml" type="integer" xref="S1.p6.6.m6.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.6.m6.1c">z\sim 20</annotation><annotation encoding="application/x-llamapun" id="S1.p6.6.m6.1d">italic_z ∼ 20</annotation></semantics></math> or more. However, binaries with masses <math alttext="\lesssim 10^{4}\,M_{\odot}" class="ltx_Math" display="inline" id="S1.p6.7.m7.1"><semantics id="S1.p6.7.m7.1a"><mrow id="S1.p6.7.m7.1.1" xref="S1.p6.7.m7.1.1.cmml"><mi id="S1.p6.7.m7.1.1.2" xref="S1.p6.7.m7.1.1.2.cmml"></mi><mo id="S1.p6.7.m7.1.1.1" xref="S1.p6.7.m7.1.1.1.cmml">≲</mo><mrow id="S1.p6.7.m7.1.1.3" xref="S1.p6.7.m7.1.1.3.cmml"><msup id="S1.p6.7.m7.1.1.3.2" xref="S1.p6.7.m7.1.1.3.2.cmml"><mn id="S1.p6.7.m7.1.1.3.2.2" xref="S1.p6.7.m7.1.1.3.2.2.cmml">10</mn><mn id="S1.p6.7.m7.1.1.3.2.3" xref="S1.p6.7.m7.1.1.3.2.3.cmml">4</mn></msup><mo id="S1.p6.7.m7.1.1.3.1" xref="S1.p6.7.m7.1.1.3.1.cmml">⁢</mo><msub id="S1.p6.7.m7.1.1.3.3" xref="S1.p6.7.m7.1.1.3.3.cmml"><mi id="S1.p6.7.m7.1.1.3.3.2" xref="S1.p6.7.m7.1.1.3.3.2.cmml">M</mi><mo id="S1.p6.7.m7.1.1.3.3.3" xref="S1.p6.7.m7.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.7.m7.1b"><apply id="S1.p6.7.m7.1.1.cmml" xref="S1.p6.7.m7.1.1"><csymbol cd="latexml" id="S1.p6.7.m7.1.1.1.cmml" xref="S1.p6.7.m7.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="S1.p6.7.m7.1.1.2.cmml" xref="S1.p6.7.m7.1.1.2">absent</csymbol><apply id="S1.p6.7.m7.1.1.3.cmml" xref="S1.p6.7.m7.1.1.3"><times id="S1.p6.7.m7.1.1.3.1.cmml" xref="S1.p6.7.m7.1.1.3.1"></times><apply id="S1.p6.7.m7.1.1.3.2.cmml" xref="S1.p6.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S1.p6.7.m7.1.1.3.2.1.cmml" xref="S1.p6.7.m7.1.1.3.2">superscript</csymbol><cn id="S1.p6.7.m7.1.1.3.2.2.cmml" type="integer" xref="S1.p6.7.m7.1.1.3.2.2">10</cn><cn id="S1.p6.7.m7.1.1.3.2.3.cmml" type="integer" xref="S1.p6.7.m7.1.1.3.2.3">4</cn></apply><apply id="S1.p6.7.m7.1.1.3.3.cmml" xref="S1.p6.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S1.p6.7.m7.1.1.3.3.1.cmml" xref="S1.p6.7.m7.1.1.3.3">subscript</csymbol><ci id="S1.p6.7.m7.1.1.3.3.2.cmml" xref="S1.p6.7.m7.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p6.7.m7.1.1.3.3.3.cmml" xref="S1.p6.7.m7.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.7.m7.1c">\lesssim 10^{4}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.7.m7.1d">≲ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> are harder to observe at high redshift, and the inference of their distance is more challenging (due to the lower signal-to-noise ratio and to weak lensing <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib26" title="">26</a>]</cite>). This is exactly the mass range of light seeds/intermediate mass black holes, whose existence is still elusive <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib39" title="">39</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">While we expect LISA to offer unique insights into this black hole population, if it exists, the reconstruction of its distribution in mass and redshift will therefore be subject to various statistical challenges. First, the finite number of observed events; second, the presence of significant selection effects, because many such systems will be undetected (both from space and from the ground-based detector network); third, the significant uncertainties in the parameters of any given merger event due to detector noise, particularly affecting distance and redshift measurements.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1">In this paper, we propose a method to deal with these statistical issues and correct or reduce associated biases in the reconstructed population. In more detail, we will use the results of a semi-analytic galaxy formation model in which MBHs evolve from light seeds <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite>, forming as the remnants of the explosion of Population III stars at high <math alttext="z" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><mi id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><ci id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">italic_z</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib70" title="">70</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib71" title="">71</a>]</cite>, to simulate a population of MBH binary mergers. For each merger in our past light cone, we will assess its detectability with LISA, and for each detectable system we will obtain posterior samples for all source parameters. These samples include the effect of weak lensing on the distance estimation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib26" title="">26</a>]</cite>. Based on these simulated samples, we will then attempt to reproduce the distribution of the astrophysical population in a two-dimensional mass-redshift parameter space, using an adaptive kernel density estimation (KDE) method and correcting for selection effects due to the finite LISA sensitivity. We will show that our method provides an unbiased reconstruction of the MBH mass and redshift distribution function, allowing for shedding light on the astrophysical formation channels of MBH binaries and on the prevalence of intermediate mass black holes in the universe.</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.1">This paper is organized as follows: in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2" title="II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II</span></a>, we describe the generation of mock (simulated) data and our analysis methods for this work; in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3" title="III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">III</span></a>, we report and discuss our results, and in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S4" title="IV Conclusion ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">IV</span></a> we describe conclusion and future work.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Simulated data</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.1 </span>Astrophysical population model and detectability</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">To describe the astrophysical population of MBH binaries, we consider the light-seed semi-analytic model presented in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib72" title="">72</a>]</cite> (and based in turn on Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib73" title="">73</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib74" title="">74</a>]</cite>, to which we refer for more details on the model’s physical content). This model was referred to as “popIII-d (K+16)” in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite>, where it was shown to agree with the amplitude estimate for the stochastic background of GWs in the band of pulsar-timing arrays <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib62" title="">62</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib63" title="">63</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib64" title="">64</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib65" title="">65</a>]</cite>. Note that the “popIII-d (K+16)” of Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite> does include delays between galaxy and MBH mergers. We stress, however, that our goal here is not to reconstruct the distribution of the binaries at formation, but the distribution at merger. For this task, the delays need only to be included in the simulated data used for the KDE reconstruction.</p> </div> <figure class="ltx_figure" id="S2.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="S2.F1.g1" src="extracted/6297058/Figure_1aMsrchistogram.png" width="287"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="S2.F1.g2" src="extracted/6297058/Figure_1bRedshifthistogram.png" width="287"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>The differential rate of all mergers (light blue line) and of detected mergers (orange line), for the light-seed “popIII-d (K+16)” model that we consider in this work. Top: density over source frame total mass. Bottom: density over redshift.</figcaption> </figure> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.11">From the astrophysical population, we obtain simulated catalogs of merger events in the band of the LISA mission, for which we assume a duration of 4 years. To simulate the GW signal, we follow Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite> and use the non-precessing IMRPhenomHM model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib75" title="">75</a>]</cite>, which describes GW signals from binaries with aligned spins and also includes contributions from higher modes. 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start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="M_{2}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><msub id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml"><mi id="S2.SS1.p2.2.m2.1.1.2" xref="S2.SS1.p2.2.m2.1.1.2.cmml">M</mi><mn id="S2.SS1.p2.2.m2.1.1.3" xref="S2.SS1.p2.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.1.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p2.2.m2.1.1.2.cmml" xref="S2.SS1.p2.2.m2.1.1.2">𝑀</ci><cn id="S2.SS1.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">M_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>), the spins components along the orbital axis (<math alttext="\chi_{1}" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><msub id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml">χ</mi><mn id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2">𝜒</ci><cn id="S2.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">\chi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\chi_{2}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><msub id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml"><mi id="S2.SS1.p2.4.m4.1.1.2" xref="S2.SS1.p2.4.m4.1.1.2.cmml">χ</mi><mn id="S2.SS1.p2.4.m4.1.1.3" xref="S2.SS1.p2.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2">𝜒</ci><cn id="S2.SS1.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p2.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">\chi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>), the time of coalescence (<math alttext="t_{c}" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><msub id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml"><mi id="S2.SS1.p2.5.m5.1.1.2" xref="S2.SS1.p2.5.m5.1.1.2.cmml">t</mi><mi id="S2.SS1.p2.5.m5.1.1.3" xref="S2.SS1.p2.5.m5.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><apply id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.1.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p2.5.m5.1.1.2.cmml" xref="S2.SS1.p2.5.m5.1.1.2">𝑡</ci><ci id="S2.SS1.p2.5.m5.1.1.3.cmml" xref="S2.SS1.p2.5.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">t_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>), the luminosity distance (<math alttext="D_{L}" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.1"><semantics id="S2.SS1.p2.6.m6.1a"><msub id="S2.SS1.p2.6.m6.1.1" xref="S2.SS1.p2.6.m6.1.1.cmml"><mi id="S2.SS1.p2.6.m6.1.1.2" xref="S2.SS1.p2.6.m6.1.1.2.cmml">D</mi><mi id="S2.SS1.p2.6.m6.1.1.3" xref="S2.SS1.p2.6.m6.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m6.1b"><apply id="S2.SS1.p2.6.m6.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.1.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1">subscript</csymbol><ci id="S2.SS1.p2.6.m6.1.1.2.cmml" xref="S2.SS1.p2.6.m6.1.1.2">𝐷</ci><ci id="S2.SS1.p2.6.m6.1.1.3.cmml" xref="S2.SS1.p2.6.m6.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.1c">D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>), the inclination angle (<math alttext="\iota" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.1"><semantics id="S2.SS1.p2.7.m7.1a"><mi id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml">ι</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.1b"><ci id="S2.SS1.p2.7.m7.1.1.cmml" xref="S2.SS1.p2.7.m7.1.1">𝜄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.1c">\iota</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.1d">italic_ι</annotation></semantics></math>), the sky location described by the longitude (<math alttext="\lambda" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.1"><semantics id="S2.SS1.p2.8.m8.1a"><mi id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.1b"><ci id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.1d">italic_λ</annotation></semantics></math>) and latitude (<math alttext="\zeta" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m9.1"><semantics id="S2.SS1.p2.9.m9.1a"><mi id="S2.SS1.p2.9.m9.1.1" xref="S2.SS1.p2.9.m9.1.1.cmml">ζ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m9.1b"><ci id="S2.SS1.p2.9.m9.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1">𝜁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m9.1c">\zeta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m9.1d">italic_ζ</annotation></semantics></math>), the polarization angle (<math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m10.1"><semantics id="S2.SS1.p2.10.m10.1a"><mi id="S2.SS1.p2.10.m10.1.1" xref="S2.SS1.p2.10.m10.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m10.1b"><ci id="S2.SS1.p2.10.m10.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m10.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m10.1d">italic_ψ</annotation></semantics></math>) and the phase at coalescence (<math alttext="\phi" class="ltx_Math" display="inline" id="S2.SS1.p2.11.m11.1"><semantics id="S2.SS1.p2.11.m11.1a"><mi id="S2.SS1.p2.11.m11.1.1" xref="S2.SS1.p2.11.m11.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.11.m11.1b"><ci id="S2.SS1.p2.11.m11.1.1.cmml" xref="S2.SS1.p2.11.m11.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.11.m11.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.11.m11.1d">italic_ϕ</annotation></semantics></math>). The masses, spins and the distance are obtained from the astrophysical model, the merger time is drawn uniformly between 0 and 4 yr (the mission duration), while the angles are drawn from an isotropic distribution.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1">For the instrumental response, we use the full LISA response function, following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib76" title="">76</a>]</cite>. The Time Delay Interferometry (TDI) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib77" title="">77</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib78" title="">78</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib79" title="">79</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib80" title="">80</a>]</cite> observables A, E, and T are calculated for each merger event. As the T channel has very low signal content, it can be excluded, and we use the A and E channels for all further analyses. Once the signal of an event has been simulated, we estimate the optimal SNR <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mi id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_ρ</annotation></semantics></math> to assess detectability, using</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="\rho^{2}=(h|h)\,," class="ltx_Math" display="block" id="S2.E1.m1.1"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><mrow id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><msup id="S2.E1.m1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.3.cmml"><mi id="S2.E1.m1.1.1.1.1.3.2" xref="S2.E1.m1.1.1.1.1.3.2.cmml">ρ</mi><mn id="S2.E1.m1.1.1.1.1.3.3" xref="S2.E1.m1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E1.m1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.2.cmml">=</mo><mrow id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.1.2.cmml">h</mi><mo fence="false" id="S2.E1.m1.1.1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.1.cmml">|</mo><mi id="S2.E1.m1.1.1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.1.1.1.3.cmml">h</mi></mrow><mo id="S2.E1.m1.1.1.1.1.1.1.3" rspace="0.170em" stretchy="false" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.1.1.1.2" 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id="S2.E1.m1.1c">\rho^{2}=(h|h)\,,</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( italic_h | italic_h ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.3">where <math alttext="h(t)" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m1.1"><semantics id="S2.SS1.p3.2.m1.1a"><mrow id="S2.SS1.p3.2.m1.1.2" xref="S2.SS1.p3.2.m1.1.2.cmml"><mi id="S2.SS1.p3.2.m1.1.2.2" xref="S2.SS1.p3.2.m1.1.2.2.cmml">h</mi><mo id="S2.SS1.p3.2.m1.1.2.1" xref="S2.SS1.p3.2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.2.m1.1.2.3.2" xref="S2.SS1.p3.2.m1.1.2.cmml"><mo id="S2.SS1.p3.2.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.2.m1.1.2.cmml">(</mo><mi id="S2.SS1.p3.2.m1.1.1" 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end_POSTSUPERSCRIPT ( italic_f ) over~ start_ARG italic_b end_ARG ( italic_f ) end_ARG start_ARG italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f ) end_ARG italic_d italic_f ,</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.SS1.p3.6">where <math alttext="S_{n}(f)" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m1.1"><semantics id="S2.SS1.p3.4.m1.1a"><mrow id="S2.SS1.p3.4.m1.1.2" xref="S2.SS1.p3.4.m1.1.2.cmml"><msub id="S2.SS1.p3.4.m1.1.2.2" xref="S2.SS1.p3.4.m1.1.2.2.cmml"><mi id="S2.SS1.p3.4.m1.1.2.2.2" xref="S2.SS1.p3.4.m1.1.2.2.2.cmml">S</mi><mi id="S2.SS1.p3.4.m1.1.2.2.3" xref="S2.SS1.p3.4.m1.1.2.2.3.cmml">n</mi></msub><mo id="S2.SS1.p3.4.m1.1.2.1" xref="S2.SS1.p3.4.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.4.m1.1.2.3.2" xref="S2.SS1.p3.4.m1.1.2.cmml"><mo id="S2.SS1.p3.4.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.4.m1.1.2.cmml">(</mo><mi id="S2.SS1.p3.4.m1.1.1" xref="S2.SS1.p3.4.m1.1.1.cmml">f</mi><mo id="S2.SS1.p3.4.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.4.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m1.1b"><apply id="S2.SS1.p3.4.m1.1.2.cmml" xref="S2.SS1.p3.4.m1.1.2"><times id="S2.SS1.p3.4.m1.1.2.1.cmml" xref="S2.SS1.p3.4.m1.1.2.1"></times><apply id="S2.SS1.p3.4.m1.1.2.2.cmml" xref="S2.SS1.p3.4.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m1.1.2.2.1.cmml" xref="S2.SS1.p3.4.m1.1.2.2">subscript</csymbol><ci id="S2.SS1.p3.4.m1.1.2.2.2.cmml" xref="S2.SS1.p3.4.m1.1.2.2.2">𝑆</ci><ci id="S2.SS1.p3.4.m1.1.2.2.3.cmml" xref="S2.SS1.p3.4.m1.1.2.2.3">𝑛</ci></apply><ci id="S2.SS1.p3.4.m1.1.1.cmml" xref="S2.SS1.p3.4.m1.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m1.1c">S_{n}(f)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m1.1d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f )</annotation></semantics></math> is the noise power spectral density. We employ the SciRDv1 noise model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib81" title="">81</a>]</cite>, augmented with an unresolved white dwarf background from Galactic binaries <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib82" title="">82</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib83" title="">83</a>]</cite>. Our computation of <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m2.1"><semantics id="S2.SS1.p3.5.m2.1a"><mi id="S2.SS1.p3.5.m2.1.1" xref="S2.SS1.p3.5.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m2.1b"><ci id="S2.SS1.p3.5.m2.1.1.cmml" xref="S2.SS1.p3.5.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m2.1d">italic_ρ</annotation></semantics></math> accounts for the significant cross-terms between different emission modes (multipoles) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib84" title="">84</a>]</cite>. For the events with <math alttext="\rho>8" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m3.1"><semantics id="S2.SS1.p3.6.m3.1a"><mrow id="S2.SS1.p3.6.m3.1.1" xref="S2.SS1.p3.6.m3.1.1.cmml"><mi id="S2.SS1.p3.6.m3.1.1.2" xref="S2.SS1.p3.6.m3.1.1.2.cmml">ρ</mi><mo id="S2.SS1.p3.6.m3.1.1.1" xref="S2.SS1.p3.6.m3.1.1.1.cmml">></mo><mn id="S2.SS1.p3.6.m3.1.1.3" xref="S2.SS1.p3.6.m3.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m3.1b"><apply id="S2.SS1.p3.6.m3.1.1.cmml" xref="S2.SS1.p3.6.m3.1.1"><gt id="S2.SS1.p3.6.m3.1.1.1.cmml" xref="S2.SS1.p3.6.m3.1.1.1"></gt><ci id="S2.SS1.p3.6.m3.1.1.2.cmml" xref="S2.SS1.p3.6.m3.1.1.2">𝜌</ci><cn id="S2.SS1.p3.6.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p3.6.m3.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m3.1c">\rho>8</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m3.1d">italic_ρ > 8</annotation></semantics></math>, we perform detailed parameter estimation (PE) as described in the next section.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.1">Figure <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.F1" title="Figure 1 ‣ II.1 Astrophysical population model and detectability ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">1</span></a> presents the expected number of mergers predicted by the model over 4 years, plotted as distributions over source mass (top) and redshift (bottom). The light blue line represents the total number of mergers (<em class="ltx_emph ltx_font_italic" id="S2.SS1.p4.1.1">astrophysical</em> distribution), while the orange line indicates the mergers detectable by LISA (<span class="ltx_text ltx_font_italic" id="S2.SS1.p4.1.2">detected</span> distribution), assuming a 4-year mission and SNR threshold of 8. Because of the way the semi-analytic model’s results are post-processed to produce simulated catalogs of black hole mergers, the latter occur at discrete redshift values. To avoid unrealistic distribution artifacts due to this, when plotting the <math alttext="z" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mi id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">italic_z</annotation></semantics></math> distribution we apply a small Gaussian scatter (negligible compared to measurement errors). The expected total number of BBH mergers taking place in 4 years is 1425, as compared to 338 detected events. As can be seen, LISA is expected to miss a significant fraction of the merger population, making it a nontrivial task to reconstruct the astrophysical population. By contrast, this is typically not the case for heavy-seed population models, where virtually the entire merger population is detectable by LISA <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib72" title="">72</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite>. The <span class="ltx_text ltx_font_italic" id="S2.SS1.p4.1.3">actual</span> number of mergers taking place, or detected, in the LISA observation time is a stochastic variable: in this work we will simulate one realization of 4 years of LISA data, and attempt to reconstruct underlying properties of the astrophysical model from these observations.</p> </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="180" id="S2.F2.g1" src="extracted/6297058/Figure_2scatter_Trueparams.png" width="287"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>True values of source-frame mass and redshift for one realization of the mergers occurring in 4 years within the light-seed model “popIII-d (K+16)” employed in this work. The contours show optimal LISA SNR values assuming equal masses, no spins, 4 years of observations and averaging over the extrinsic angle parameters, while the color code denotes the optimal SNR computed with the actual source parameters.</figcaption> </figure> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.6">As another way to visualize the astrophysical population and its detectability, we show in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.F2" title="Figure 2 ‣ II.1 Astrophysical population model and detectability ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">2</span></a> the true parameters of one realization of the mergers for 4 years of observation, within the “popIII-d (K+16)” light-seed model considered in this work. Also shown are contours of constant sky-, inclination- and polarization-averaged optimal SNR in the parameter plane of source-frame total mass <math alttext="M" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><mi id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><ci id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.1d">italic_M</annotation></semantics></math> and redshift <math alttext="z" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.1"><semantics id="S2.SS1.p5.2.m2.1a"><mi id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><ci id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">italic_z</annotation></semantics></math>, assuming a 4-year observation period, zero component spins and a mass ratio <math alttext="q=1" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><mrow id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml"><mi id="S2.SS1.p5.3.m3.1.1.2" xref="S2.SS1.p5.3.m3.1.1.2.cmml">q</mi><mo id="S2.SS1.p5.3.m3.1.1.1" xref="S2.SS1.p5.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS1.p5.3.m3.1.1.3" xref="S2.SS1.p5.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><apply id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1"><eq id="S2.SS1.p5.3.m3.1.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1.1"></eq><ci id="S2.SS1.p5.3.m3.1.1.2.cmml" xref="S2.SS1.p5.3.m3.1.1.2">𝑞</ci><cn id="S2.SS1.p5.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">q=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_q = 1</annotation></semantics></math>. For each source, we also show by the color code the actual optimal SNR, computed with the mass ratio and spins predicted by the astrophysical model, with the sky position, inclination and polarization angle and observation time values of the catalog realization, and with the luminosity distance predicted by the astrophysical model with the addition of a weak-lensing contribution as detailed in the next section. With this realization of a detected event set, we can hope to reconstruct the astrophysical distribution only between total masses of <math alttext="\sim" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.1"><semantics id="S2.SS1.p5.4.m4.1a"><mo id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.1b"><csymbol cd="latexml" id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.1d">∼</annotation></semantics></math><math alttext="10^{3}-\mathrm{few}\times 10^{6}\,M_{\odot}" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m5.1"><semantics id="S2.SS1.p5.5.m5.1a"><mrow id="S2.SS1.p5.5.m5.1.1" xref="S2.SS1.p5.5.m5.1.1.cmml"><msup id="S2.SS1.p5.5.m5.1.1.2" xref="S2.SS1.p5.5.m5.1.1.2.cmml"><mn id="S2.SS1.p5.5.m5.1.1.2.2" xref="S2.SS1.p5.5.m5.1.1.2.2.cmml">10</mn><mn id="S2.SS1.p5.5.m5.1.1.2.3" xref="S2.SS1.p5.5.m5.1.1.2.3.cmml">3</mn></msup><mo id="S2.SS1.p5.5.m5.1.1.1" xref="S2.SS1.p5.5.m5.1.1.1.cmml">−</mo><mrow id="S2.SS1.p5.5.m5.1.1.3" xref="S2.SS1.p5.5.m5.1.1.3.cmml"><mrow id="S2.SS1.p5.5.m5.1.1.3.2" xref="S2.SS1.p5.5.m5.1.1.3.2.cmml"><mi id="S2.SS1.p5.5.m5.1.1.3.2.2" xref="S2.SS1.p5.5.m5.1.1.3.2.2.cmml">few</mi><mo id="S2.SS1.p5.5.m5.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p5.5.m5.1.1.3.2.1.cmml">×</mo><msup id="S2.SS1.p5.5.m5.1.1.3.2.3" xref="S2.SS1.p5.5.m5.1.1.3.2.3.cmml"><mn id="S2.SS1.p5.5.m5.1.1.3.2.3.2" xref="S2.SS1.p5.5.m5.1.1.3.2.3.2.cmml">10</mn><mn id="S2.SS1.p5.5.m5.1.1.3.2.3.3" xref="S2.SS1.p5.5.m5.1.1.3.2.3.3.cmml">6</mn></msup></mrow><mo id="S2.SS1.p5.5.m5.1.1.3.1" xref="S2.SS1.p5.5.m5.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p5.5.m5.1.1.3.3" xref="S2.SS1.p5.5.m5.1.1.3.3.cmml"><mi id="S2.SS1.p5.5.m5.1.1.3.3.2" xref="S2.SS1.p5.5.m5.1.1.3.3.2.cmml">M</mi><mo id="S2.SS1.p5.5.m5.1.1.3.3.3" xref="S2.SS1.p5.5.m5.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m5.1b"><apply id="S2.SS1.p5.5.m5.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1"><minus id="S2.SS1.p5.5.m5.1.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1.1"></minus><apply id="S2.SS1.p5.5.m5.1.1.2.cmml" xref="S2.SS1.p5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m5.1.1.2.1.cmml" xref="S2.SS1.p5.5.m5.1.1.2">superscript</csymbol><cn id="S2.SS1.p5.5.m5.1.1.2.2.cmml" type="integer" xref="S2.SS1.p5.5.m5.1.1.2.2">10</cn><cn id="S2.SS1.p5.5.m5.1.1.2.3.cmml" type="integer" xref="S2.SS1.p5.5.m5.1.1.2.3">3</cn></apply><apply id="S2.SS1.p5.5.m5.1.1.3.cmml" xref="S2.SS1.p5.5.m5.1.1.3"><times id="S2.SS1.p5.5.m5.1.1.3.1.cmml" xref="S2.SS1.p5.5.m5.1.1.3.1"></times><apply id="S2.SS1.p5.5.m5.1.1.3.2.cmml" xref="S2.SS1.p5.5.m5.1.1.3.2"><times id="S2.SS1.p5.5.m5.1.1.3.2.1.cmml" xref="S2.SS1.p5.5.m5.1.1.3.2.1"></times><ci id="S2.SS1.p5.5.m5.1.1.3.2.2.cmml" xref="S2.SS1.p5.5.m5.1.1.3.2.2">few</ci><apply id="S2.SS1.p5.5.m5.1.1.3.2.3.cmml" xref="S2.SS1.p5.5.m5.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m5.1.1.3.2.3.1.cmml" xref="S2.SS1.p5.5.m5.1.1.3.2.3">superscript</csymbol><cn id="S2.SS1.p5.5.m5.1.1.3.2.3.2.cmml" type="integer" xref="S2.SS1.p5.5.m5.1.1.3.2.3.2">10</cn><cn id="S2.SS1.p5.5.m5.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS1.p5.5.m5.1.1.3.2.3.3">6</cn></apply></apply><apply id="S2.SS1.p5.5.m5.1.1.3.3.cmml" xref="S2.SS1.p5.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m5.1.1.3.3.1.cmml" xref="S2.SS1.p5.5.m5.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p5.5.m5.1.1.3.3.2.cmml" xref="S2.SS1.p5.5.m5.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S2.SS1.p5.5.m5.1.1.3.3.3.cmml" xref="S2.SS1.p5.5.m5.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m5.1c">10^{3}-\mathrm{few}\times 10^{6}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m5.1d">10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - roman_few × 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> and redshifts <math alttext="\sim 0-15" class="ltx_Math" display="inline" id="S2.SS1.p5.6.m6.1"><semantics id="S2.SS1.p5.6.m6.1a"><mrow id="S2.SS1.p5.6.m6.1.1" xref="S2.SS1.p5.6.m6.1.1.cmml"><mi id="S2.SS1.p5.6.m6.1.1.2" xref="S2.SS1.p5.6.m6.1.1.2.cmml"></mi><mo id="S2.SS1.p5.6.m6.1.1.1" xref="S2.SS1.p5.6.m6.1.1.1.cmml">∼</mo><mrow id="S2.SS1.p5.6.m6.1.1.3" xref="S2.SS1.p5.6.m6.1.1.3.cmml"><mn id="S2.SS1.p5.6.m6.1.1.3.2" xref="S2.SS1.p5.6.m6.1.1.3.2.cmml">0</mn><mo id="S2.SS1.p5.6.m6.1.1.3.1" xref="S2.SS1.p5.6.m6.1.1.3.1.cmml">−</mo><mn id="S2.SS1.p5.6.m6.1.1.3.3" xref="S2.SS1.p5.6.m6.1.1.3.3.cmml">15</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.6.m6.1b"><apply id="S2.SS1.p5.6.m6.1.1.cmml" xref="S2.SS1.p5.6.m6.1.1"><csymbol cd="latexml" id="S2.SS1.p5.6.m6.1.1.1.cmml" xref="S2.SS1.p5.6.m6.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS1.p5.6.m6.1.1.2.cmml" xref="S2.SS1.p5.6.m6.1.1.2">absent</csymbol><apply id="S2.SS1.p5.6.m6.1.1.3.cmml" xref="S2.SS1.p5.6.m6.1.1.3"><minus id="S2.SS1.p5.6.m6.1.1.3.1.cmml" xref="S2.SS1.p5.6.m6.1.1.3.1"></minus><cn id="S2.SS1.p5.6.m6.1.1.3.2.cmml" type="integer" xref="S2.SS1.p5.6.m6.1.1.3.2">0</cn><cn id="S2.SS1.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS1.p5.6.m6.1.1.3.3">15</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.6.m6.1c">\sim 0-15</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.6.m6.1d">∼ 0 - 15</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.2 </span>Parameter Estimation</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">We use the GW signals described in the previous section as injections on which we perform Bayesian PE, following Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite>. Although the detected GW signals from MBH binaries are expected to overlap in the LISA band, we consider each signal in isolation to simplify the analysis. We also neglect the noise contribution to the injection (noiseless limit) and ignore possible data artifacts such as gaps and glitches <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib85" title="">85</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib86" title="">86</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib87" title="">87</a>]</cite>. The posteriors are expected to only undergo a shift up to the statistical error with the inclusion of Gaussian noise in the injection, while the variance measurements are expected to be largely unaffected <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib88" title="">88</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib89" title="">89</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">We also consider the effect of weak lensing, which is expected to have an impact on the measurement of <math alttext="D_{L}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><msub id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml">D</mi><mi id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">𝐷</ci><ci id="S2.SS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> by introducing a random change in the GW signal amplitude. The weak-lensing error is estimated as <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib26" title="">26</a>]</cite></p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{\rm wl}(z)=D_{L}\times 0.066\left(\frac{1-\left(1+z\right)^{-0.25}}{0.% 25}\right)^{1.8}," class="ltx_Math" display="block" id="S2.E3.m1.3"><semantics id="S2.E3.m1.3a"><mrow id="S2.E3.m1.3.3.1" xref="S2.E3.m1.3.3.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1" xref="S2.E3.m1.3.3.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1.2" xref="S2.E3.m1.3.3.1.1.2.cmml"><msub id="S2.E3.m1.3.3.1.1.2.2" xref="S2.E3.m1.3.3.1.1.2.2.cmml"><mi id="S2.E3.m1.3.3.1.1.2.2.2" xref="S2.E3.m1.3.3.1.1.2.2.2.cmml">σ</mi><mi id="S2.E3.m1.3.3.1.1.2.2.3" xref="S2.E3.m1.3.3.1.1.2.2.3.cmml">wl</mi></msub><mo id="S2.E3.m1.3.3.1.1.2.1" xref="S2.E3.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.1.1.2.3.2" xref="S2.E3.m1.3.3.1.1.2.cmml"><mo id="S2.E3.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S2.E3.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2.cmml">z</mi><mo id="S2.E3.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S2.E3.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.3.3.1.1.1" xref="S2.E3.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.3.3.1.1.3" xref="S2.E3.m1.3.3.1.1.3.cmml"><mrow id="S2.E3.m1.3.3.1.1.3.2" xref="S2.E3.m1.3.3.1.1.3.2.cmml"><msub id="S2.E3.m1.3.3.1.1.3.2.2" xref="S2.E3.m1.3.3.1.1.3.2.2.cmml"><mi id="S2.E3.m1.3.3.1.1.3.2.2.2" xref="S2.E3.m1.3.3.1.1.3.2.2.2.cmml">D</mi><mi id="S2.E3.m1.3.3.1.1.3.2.2.3" xref="S2.E3.m1.3.3.1.1.3.2.2.3.cmml">L</mi></msub><mo id="S2.E3.m1.3.3.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E3.m1.3.3.1.1.3.2.1.cmml">×</mo><mn id="S2.E3.m1.3.3.1.1.3.2.3" xref="S2.E3.m1.3.3.1.1.3.2.3.cmml">0.066</mn></mrow><mo 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id="S2.E3.m1.1.1.1.1.1.1.3" xref="S2.E3.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.E3.m1.1.1.1.1.3" xref="S2.E3.m1.1.1.1.1.3.cmml"><mo id="S2.E3.m1.1.1.1.1.3a" xref="S2.E3.m1.1.1.1.1.3.cmml">−</mo><mn id="S2.E3.m1.1.1.1.1.3.2" xref="S2.E3.m1.1.1.1.1.3.2.cmml">0.25</mn></mrow></msup></mrow><mn id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml">0.25</mn></mfrac><mo id="S2.E3.m1.3.3.1.1.3.3.2.2.2" xref="S2.E3.m1.1.1.cmml">)</mo></mrow><mn id="S2.E3.m1.3.3.1.1.3.3.3" xref="S2.E3.m1.3.3.1.1.3.3.3.cmml">1.8</mn></msup></mrow></mrow><mo id="S2.E3.m1.3.3.1.2" xref="S2.E3.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.3b"><apply id="S2.E3.m1.3.3.1.1.cmml" xref="S2.E3.m1.3.3.1"><eq id="S2.E3.m1.3.3.1.1.1.cmml" xref="S2.E3.m1.3.3.1.1.1"></eq><apply id="S2.E3.m1.3.3.1.1.2.cmml" xref="S2.E3.m1.3.3.1.1.2"><times id="S2.E3.m1.3.3.1.1.2.1.cmml" xref="S2.E3.m1.3.3.1.1.2.1"></times><apply id="S2.E3.m1.3.3.1.1.2.2.cmml" xref="S2.E3.m1.3.3.1.1.2.2"><csymbol 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xref="S2.E3.m1.3.3.1.1.3.2.3">0.066</cn></apply><apply id="S2.E3.m1.3.3.1.1.3.3.cmml" xref="S2.E3.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.3.3.1.1.3.3.1.cmml" xref="S2.E3.m1.3.3.1.1.3.3">superscript</csymbol><apply id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.3.3.1.1.3.3.2.2"><divide id="S2.E3.m1.1.1.2.cmml" xref="S2.E3.m1.3.3.1.1.3.3.2.2"></divide><apply id="S2.E3.m1.1.1.1.cmml" xref="S2.E3.m1.1.1.1"><minus id="S2.E3.m1.1.1.1.2.cmml" xref="S2.E3.m1.1.1.1.2"></minus><cn id="S2.E3.m1.1.1.1.3.cmml" type="integer" xref="S2.E3.m1.1.1.1.3">1</cn><apply id="S2.E3.m1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.2.cmml" xref="S2.E3.m1.1.1.1.1">superscript</csymbol><apply id="S2.E3.m1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1.1.1"><plus id="S2.E3.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1.1.1.1.1"></plus><cn id="S2.E3.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.E3.m1.1.1.1.1.1.1.1.2">1</cn><ci id="S2.E3.m1.1.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.1.1.1.1.1.1.1.3">𝑧</ci></apply><apply id="S2.E3.m1.1.1.1.1.3.cmml" xref="S2.E3.m1.1.1.1.1.3"><minus id="S2.E3.m1.1.1.1.1.3.1.cmml" xref="S2.E3.m1.1.1.1.1.3"></minus><cn id="S2.E3.m1.1.1.1.1.3.2.cmml" type="float" xref="S2.E3.m1.1.1.1.1.3.2">0.25</cn></apply></apply></apply><cn id="S2.E3.m1.1.1.3.cmml" type="float" xref="S2.E3.m1.1.1.3">0.25</cn></apply><cn id="S2.E3.m1.3.3.1.1.3.3.3.cmml" type="float" xref="S2.E3.m1.3.3.1.1.3.3.3">1.8</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.3c">\sigma_{\rm wl}(z)=D_{L}\times 0.066\left(\frac{1-\left(1+z\right)^{-0.25}}{0.% 25}\right)^{1.8},</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.3d">italic_σ start_POSTSUBSCRIPT roman_wl end_POSTSUBSCRIPT ( italic_z ) = italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT × 0.066 ( divide start_ARG 1 - ( 1 + italic_z ) start_POSTSUPERSCRIPT - 0.25 end_POSTSUPERSCRIPT end_ARG start_ARG 0.25 end_ARG ) start_POSTSUPERSCRIPT 1.8 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.6">assuming a Planck 2015 <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m1.1"><semantics id="S2.SS2.p2.2.m1.1a"><mi id="S2.SS2.p2.2.m1.1.1" mathvariant="normal" xref="S2.SS2.p2.2.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m1.1b"><ci id="S2.SS2.p2.2.m1.1.1.cmml" xref="S2.SS2.p2.2.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m1.1d">roman_Λ</annotation></semantics></math>CDM cosmology. Considering <math alttext="\sigma_{\rm wl}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m2.1"><semantics id="S2.SS2.p2.3.m2.1a"><msubsup id="S2.SS2.p2.3.m2.1.1" xref="S2.SS2.p2.3.m2.1.1.cmml"><mi id="S2.SS2.p2.3.m2.1.1.2.2" xref="S2.SS2.p2.3.m2.1.1.2.2.cmml">σ</mi><mi id="S2.SS2.p2.3.m2.1.1.2.3" xref="S2.SS2.p2.3.m2.1.1.2.3.cmml">wl</mi><mn id="S2.SS2.p2.3.m2.1.1.3" xref="S2.SS2.p2.3.m2.1.1.3.cmml">2</mn></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m2.1b"><apply id="S2.SS2.p2.3.m2.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1">superscript</csymbol><apply id="S2.SS2.p2.3.m2.1.1.2.cmml" xref="S2.SS2.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.2.1.cmml" xref="S2.SS2.p2.3.m2.1.1">subscript</csymbol><ci id="S2.SS2.p2.3.m2.1.1.2.2.cmml" xref="S2.SS2.p2.3.m2.1.1.2.2">𝜎</ci><ci id="S2.SS2.p2.3.m2.1.1.2.3.cmml" xref="S2.SS2.p2.3.m2.1.1.2.3">wl</ci></apply><cn id="S2.SS2.p2.3.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p2.3.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m2.1c">\sigma_{\rm wl}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m2.1d">italic_σ start_POSTSUBSCRIPT roman_wl end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> as the variance, we draw <math alttext="\delta D_{L}" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m3.1"><semantics id="S2.SS2.p2.4.m3.1a"><mrow id="S2.SS2.p2.4.m3.1.1" xref="S2.SS2.p2.4.m3.1.1.cmml"><mi id="S2.SS2.p2.4.m3.1.1.2" xref="S2.SS2.p2.4.m3.1.1.2.cmml">δ</mi><mo id="S2.SS2.p2.4.m3.1.1.1" xref="S2.SS2.p2.4.m3.1.1.1.cmml">⁢</mo><msub id="S2.SS2.p2.4.m3.1.1.3" xref="S2.SS2.p2.4.m3.1.1.3.cmml"><mi id="S2.SS2.p2.4.m3.1.1.3.2" xref="S2.SS2.p2.4.m3.1.1.3.2.cmml">D</mi><mi id="S2.SS2.p2.4.m3.1.1.3.3" xref="S2.SS2.p2.4.m3.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m3.1b"><apply id="S2.SS2.p2.4.m3.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1"><times id="S2.SS2.p2.4.m3.1.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1.1"></times><ci id="S2.SS2.p2.4.m3.1.1.2.cmml" xref="S2.SS2.p2.4.m3.1.1.2">𝛿</ci><apply id="S2.SS2.p2.4.m3.1.1.3.cmml" xref="S2.SS2.p2.4.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m3.1.1.3.1.cmml" xref="S2.SS2.p2.4.m3.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.4.m3.1.1.3.2.cmml" xref="S2.SS2.p2.4.m3.1.1.3.2">𝐷</ci><ci id="S2.SS2.p2.4.m3.1.1.3.3.cmml" xref="S2.SS2.p2.4.m3.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m3.1c">\delta D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m3.1d">italic_δ italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> from a normal distribution <math alttext="\mathcal{N}(0,\sigma_{\rm wl}^{2})" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m4.2"><semantics id="S2.SS2.p2.5.m4.2a"><mrow id="S2.SS2.p2.5.m4.2.2" xref="S2.SS2.p2.5.m4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.5.m4.2.2.3" xref="S2.SS2.p2.5.m4.2.2.3.cmml">𝒩</mi><mo id="S2.SS2.p2.5.m4.2.2.2" xref="S2.SS2.p2.5.m4.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p2.5.m4.2.2.1.1" xref="S2.SS2.p2.5.m4.2.2.1.2.cmml"><mo id="S2.SS2.p2.5.m4.2.2.1.1.2" stretchy="false" xref="S2.SS2.p2.5.m4.2.2.1.2.cmml">(</mo><mn id="S2.SS2.p2.5.m4.1.1" xref="S2.SS2.p2.5.m4.1.1.cmml">0</mn><mo id="S2.SS2.p2.5.m4.2.2.1.1.3" xref="S2.SS2.p2.5.m4.2.2.1.2.cmml">,</mo><msubsup id="S2.SS2.p2.5.m4.2.2.1.1.1" xref="S2.SS2.p2.5.m4.2.2.1.1.1.cmml"><mi id="S2.SS2.p2.5.m4.2.2.1.1.1.2.2" xref="S2.SS2.p2.5.m4.2.2.1.1.1.2.2.cmml">σ</mi><mi id="S2.SS2.p2.5.m4.2.2.1.1.1.2.3" xref="S2.SS2.p2.5.m4.2.2.1.1.1.2.3.cmml">wl</mi><mn id="S2.SS2.p2.5.m4.2.2.1.1.1.3" xref="S2.SS2.p2.5.m4.2.2.1.1.1.3.cmml">2</mn></msubsup><mo id="S2.SS2.p2.5.m4.2.2.1.1.4" stretchy="false" xref="S2.SS2.p2.5.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m4.2b"><apply id="S2.SS2.p2.5.m4.2.2.cmml" xref="S2.SS2.p2.5.m4.2.2"><times id="S2.SS2.p2.5.m4.2.2.2.cmml" xref="S2.SS2.p2.5.m4.2.2.2"></times><ci id="S2.SS2.p2.5.m4.2.2.3.cmml" xref="S2.SS2.p2.5.m4.2.2.3">𝒩</ci><interval closure="open" id="S2.SS2.p2.5.m4.2.2.1.2.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1"><cn id="S2.SS2.p2.5.m4.1.1.cmml" type="integer" xref="S2.SS2.p2.5.m4.1.1">0</cn><apply id="S2.SS2.p2.5.m4.2.2.1.1.1.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.2.2.1.1.1.1.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1.1">superscript</csymbol><apply id="S2.SS2.p2.5.m4.2.2.1.1.1.2.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.2.2.1.1.1.2.1.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2.p2.5.m4.2.2.1.1.1.2.2.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1.1.2.2">𝜎</ci><ci id="S2.SS2.p2.5.m4.2.2.1.1.1.2.3.cmml" xref="S2.SS2.p2.5.m4.2.2.1.1.1.2.3">wl</ci></apply><cn id="S2.SS2.p2.5.m4.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS2.p2.5.m4.2.2.1.1.1.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m4.2c">\mathcal{N}(0,\sigma_{\rm wl}^{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m4.2d">caligraphic_N ( 0 , italic_σ start_POSTSUBSCRIPT roman_wl end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> and add this term to the <math alttext="D_{L}" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m5.1"><semantics id="S2.SS2.p2.6.m5.1a"><msub id="S2.SS2.p2.6.m5.1.1" xref="S2.SS2.p2.6.m5.1.1.cmml"><mi id="S2.SS2.p2.6.m5.1.1.2" xref="S2.SS2.p2.6.m5.1.1.2.cmml">D</mi><mi id="S2.SS2.p2.6.m5.1.1.3" xref="S2.SS2.p2.6.m5.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m5.1b"><apply id="S2.SS2.p2.6.m5.1.1.cmml" xref="S2.SS2.p2.6.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.6.m5.1.1.1.cmml" xref="S2.SS2.p2.6.m5.1.1">subscript</csymbol><ci id="S2.SS2.p2.6.m5.1.1.2.cmml" xref="S2.SS2.p2.6.m5.1.1.2">𝐷</ci><ci id="S2.SS2.p2.6.m5.1.1.3.cmml" xref="S2.SS2.p2.6.m5.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m5.1c">D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m5.1d">italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> predicted by our astrophysical model, so that the simulated LISA signal includes the effect of weak lensing.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.21">We infer the posterior parameter distribution for every source with optimal SNR larger than 8, using the <span class="ltx_text ltx_font_italic" id="S2.SS2.p3.21.1">LISAbeta</span> code <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib90" title="">90</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib91" title="">91</a>]</cite>. The sampled parameters are total mass (<math alttext="M" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mi id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><ci id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_M</annotation></semantics></math>), mass ratio (<math alttext="q" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><mi id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><ci id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">q</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">italic_q</annotation></semantics></math>), <math alttext="\chi_{1}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.1"><semantics id="S2.SS2.p3.3.m3.1a"><msub id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml"><mi id="S2.SS2.p3.3.m3.1.1.2" xref="S2.SS2.p3.3.m3.1.1.2.cmml">χ</mi><mn id="S2.SS2.p3.3.m3.1.1.3" xref="S2.SS2.p3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m3.1b"><apply id="S2.SS2.p3.3.m3.1.1.cmml" xref="S2.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.3.m3.1.1.1.cmml" xref="S2.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.p3.3.m3.1.1.2.cmml" xref="S2.SS2.p3.3.m3.1.1.2">𝜒</ci><cn id="S2.SS2.p3.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m3.1c">\chi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m3.1d">italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\chi_{2}" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m4.1"><semantics id="S2.SS2.p3.4.m4.1a"><msub id="S2.SS2.p3.4.m4.1.1" xref="S2.SS2.p3.4.m4.1.1.cmml"><mi id="S2.SS2.p3.4.m4.1.1.2" xref="S2.SS2.p3.4.m4.1.1.2.cmml">χ</mi><mn id="S2.SS2.p3.4.m4.1.1.3" xref="S2.SS2.p3.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m4.1b"><apply id="S2.SS2.p3.4.m4.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.4.m4.1.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.p3.4.m4.1.1.2.cmml" xref="S2.SS2.p3.4.m4.1.1.2">𝜒</ci><cn id="S2.SS2.p3.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.p3.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m4.1c">\chi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m4.1d">italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="t_{c}" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m5.1"><semantics id="S2.SS2.p3.5.m5.1a"><msub id="S2.SS2.p3.5.m5.1.1" xref="S2.SS2.p3.5.m5.1.1.cmml"><mi id="S2.SS2.p3.5.m5.1.1.2" xref="S2.SS2.p3.5.m5.1.1.2.cmml">t</mi><mi id="S2.SS2.p3.5.m5.1.1.3" xref="S2.SS2.p3.5.m5.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m5.1b"><apply id="S2.SS2.p3.5.m5.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.5.m5.1.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1">subscript</csymbol><ci id="S2.SS2.p3.5.m5.1.1.2.cmml" xref="S2.SS2.p3.5.m5.1.1.2">𝑡</ci><ci id="S2.SS2.p3.5.m5.1.1.3.cmml" xref="S2.SS2.p3.5.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m5.1c">t_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m5.1d">italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="D_{L}" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m6.1"><semantics id="S2.SS2.p3.6.m6.1a"><msub id="S2.SS2.p3.6.m6.1.1" xref="S2.SS2.p3.6.m6.1.1.cmml"><mi id="S2.SS2.p3.6.m6.1.1.2" xref="S2.SS2.p3.6.m6.1.1.2.cmml">D</mi><mi id="S2.SS2.p3.6.m6.1.1.3" xref="S2.SS2.p3.6.m6.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.6.m6.1b"><apply id="S2.SS2.p3.6.m6.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.6.m6.1.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1">subscript</csymbol><ci id="S2.SS2.p3.6.m6.1.1.2.cmml" xref="S2.SS2.p3.6.m6.1.1.2">𝐷</ci><ci id="S2.SS2.p3.6.m6.1.1.3.cmml" xref="S2.SS2.p3.6.m6.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m6.1c">D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\iota" class="ltx_Math" display="inline" id="S2.SS2.p3.7.m7.1"><semantics id="S2.SS2.p3.7.m7.1a"><mi id="S2.SS2.p3.7.m7.1.1" xref="S2.SS2.p3.7.m7.1.1.cmml">ι</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.7.m7.1b"><ci id="S2.SS2.p3.7.m7.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1">𝜄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.7.m7.1c">\iota</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.7.m7.1d">italic_ι</annotation></semantics></math>, <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.SS2.p3.8.m8.1"><semantics id="S2.SS2.p3.8.m8.1a"><mi id="S2.SS2.p3.8.m8.1.1" xref="S2.SS2.p3.8.m8.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.8.m8.1b"><ci id="S2.SS2.p3.8.m8.1.1.cmml" xref="S2.SS2.p3.8.m8.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.8.m8.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.8.m8.1d">italic_λ</annotation></semantics></math>, <math alttext="\zeta" class="ltx_Math" display="inline" id="S2.SS2.p3.9.m9.1"><semantics id="S2.SS2.p3.9.m9.1a"><mi id="S2.SS2.p3.9.m9.1.1" xref="S2.SS2.p3.9.m9.1.1.cmml">ζ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.9.m9.1b"><ci id="S2.SS2.p3.9.m9.1.1.cmml" xref="S2.SS2.p3.9.m9.1.1">𝜁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.9.m9.1c">\zeta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.9.m9.1d">italic_ζ</annotation></semantics></math>, <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS2.p3.10.m10.1"><semantics id="S2.SS2.p3.10.m10.1a"><mi id="S2.SS2.p3.10.m10.1.1" xref="S2.SS2.p3.10.m10.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.10.m10.1b"><ci id="S2.SS2.p3.10.m10.1.1.cmml" xref="S2.SS2.p3.10.m10.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.10.m10.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.10.m10.1d">italic_ψ</annotation></semantics></math> and <math alttext="\phi" class="ltx_Math" display="inline" id="S2.SS2.p3.11.m11.1"><semantics id="S2.SS2.p3.11.m11.1a"><mi id="S2.SS2.p3.11.m11.1.1" xref="S2.SS2.p3.11.m11.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.11.m11.1b"><ci id="S2.SS2.p3.11.m11.1.1.cmml" xref="S2.SS2.p3.11.m11.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.11.m11.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.11.m11.1d">italic_ϕ</annotation></semantics></math>. The priors for the parameters <math alttext="M" class="ltx_Math" display="inline" id="S2.SS2.p3.12.m12.1"><semantics id="S2.SS2.p3.12.m12.1a"><mi id="S2.SS2.p3.12.m12.1.1" xref="S2.SS2.p3.12.m12.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.12.m12.1b"><ci id="S2.SS2.p3.12.m12.1.1.cmml" xref="S2.SS2.p3.12.m12.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.12.m12.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.12.m12.1d">italic_M</annotation></semantics></math>, <math alttext="q" class="ltx_Math" display="inline" id="S2.SS2.p3.13.m13.1"><semantics id="S2.SS2.p3.13.m13.1a"><mi id="S2.SS2.p3.13.m13.1.1" xref="S2.SS2.p3.13.m13.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.13.m13.1b"><ci id="S2.SS2.p3.13.m13.1.1.cmml" xref="S2.SS2.p3.13.m13.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.13.m13.1c">q</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.13.m13.1d">italic_q</annotation></semantics></math>, <math alttext="\chi_{1}" class="ltx_Math" display="inline" id="S2.SS2.p3.14.m14.1"><semantics id="S2.SS2.p3.14.m14.1a"><msub id="S2.SS2.p3.14.m14.1.1" xref="S2.SS2.p3.14.m14.1.1.cmml"><mi id="S2.SS2.p3.14.m14.1.1.2" xref="S2.SS2.p3.14.m14.1.1.2.cmml">χ</mi><mn id="S2.SS2.p3.14.m14.1.1.3" xref="S2.SS2.p3.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.14.m14.1b"><apply id="S2.SS2.p3.14.m14.1.1.cmml" xref="S2.SS2.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.14.m14.1.1.1.cmml" xref="S2.SS2.p3.14.m14.1.1">subscript</csymbol><ci id="S2.SS2.p3.14.m14.1.1.2.cmml" xref="S2.SS2.p3.14.m14.1.1.2">𝜒</ci><cn id="S2.SS2.p3.14.m14.1.1.3.cmml" type="integer" xref="S2.SS2.p3.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.14.m14.1c">\chi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.14.m14.1d">italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\chi_{2}" class="ltx_Math" display="inline" id="S2.SS2.p3.15.m15.1"><semantics id="S2.SS2.p3.15.m15.1a"><msub id="S2.SS2.p3.15.m15.1.1" xref="S2.SS2.p3.15.m15.1.1.cmml"><mi id="S2.SS2.p3.15.m15.1.1.2" xref="S2.SS2.p3.15.m15.1.1.2.cmml">χ</mi><mn id="S2.SS2.p3.15.m15.1.1.3" xref="S2.SS2.p3.15.m15.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.15.m15.1b"><apply id="S2.SS2.p3.15.m15.1.1.cmml" xref="S2.SS2.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.15.m15.1.1.1.cmml" xref="S2.SS2.p3.15.m15.1.1">subscript</csymbol><ci id="S2.SS2.p3.15.m15.1.1.2.cmml" xref="S2.SS2.p3.15.m15.1.1.2">𝜒</ci><cn id="S2.SS2.p3.15.m15.1.1.3.cmml" type="integer" xref="S2.SS2.p3.15.m15.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.15.m15.1c">\chi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.15.m15.1d">italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="t_{c}" class="ltx_Math" display="inline" id="S2.SS2.p3.16.m16.1"><semantics id="S2.SS2.p3.16.m16.1a"><msub id="S2.SS2.p3.16.m16.1.1" xref="S2.SS2.p3.16.m16.1.1.cmml"><mi id="S2.SS2.p3.16.m16.1.1.2" xref="S2.SS2.p3.16.m16.1.1.2.cmml">t</mi><mi id="S2.SS2.p3.16.m16.1.1.3" xref="S2.SS2.p3.16.m16.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.16.m16.1b"><apply id="S2.SS2.p3.16.m16.1.1.cmml" xref="S2.SS2.p3.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.16.m16.1.1.1.cmml" xref="S2.SS2.p3.16.m16.1.1">subscript</csymbol><ci id="S2.SS2.p3.16.m16.1.1.2.cmml" xref="S2.SS2.p3.16.m16.1.1.2">𝑡</ci><ci id="S2.SS2.p3.16.m16.1.1.3.cmml" xref="S2.SS2.p3.16.m16.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.16.m16.1c">t_{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.16.m16.1d">italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="D_{L}" class="ltx_Math" display="inline" id="S2.SS2.p3.17.m17.1"><semantics id="S2.SS2.p3.17.m17.1a"><msub id="S2.SS2.p3.17.m17.1.1" xref="S2.SS2.p3.17.m17.1.1.cmml"><mi id="S2.SS2.p3.17.m17.1.1.2" xref="S2.SS2.p3.17.m17.1.1.2.cmml">D</mi><mi id="S2.SS2.p3.17.m17.1.1.3" xref="S2.SS2.p3.17.m17.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.17.m17.1b"><apply id="S2.SS2.p3.17.m17.1.1.cmml" xref="S2.SS2.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.17.m17.1.1.1.cmml" xref="S2.SS2.p3.17.m17.1.1">subscript</csymbol><ci id="S2.SS2.p3.17.m17.1.1.2.cmml" xref="S2.SS2.p3.17.m17.1.1.2">𝐷</ci><ci id="S2.SS2.p3.17.m17.1.1.3.cmml" xref="S2.SS2.p3.17.m17.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.17.m17.1c">D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.17.m17.1d">italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> are chosen to be uniform. For <math alttext="\iota" class="ltx_Math" display="inline" id="S2.SS2.p3.18.m18.1"><semantics id="S2.SS2.p3.18.m18.1a"><mi id="S2.SS2.p3.18.m18.1.1" xref="S2.SS2.p3.18.m18.1.1.cmml">ι</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.18.m18.1b"><ci id="S2.SS2.p3.18.m18.1.1.cmml" xref="S2.SS2.p3.18.m18.1.1">𝜄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.18.m18.1c">\iota</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.18.m18.1d">italic_ι</annotation></semantics></math> and the sky location, we consider uniform priors on <math alttext="\cos\iota" class="ltx_Math" display="inline" id="S2.SS2.p3.19.m19.1"><semantics id="S2.SS2.p3.19.m19.1a"><mrow id="S2.SS2.p3.19.m19.1.1" xref="S2.SS2.p3.19.m19.1.1.cmml"><mi id="S2.SS2.p3.19.m19.1.1.1" xref="S2.SS2.p3.19.m19.1.1.1.cmml">cos</mi><mo id="S2.SS2.p3.19.m19.1.1a" lspace="0.167em" xref="S2.SS2.p3.19.m19.1.1.cmml">⁡</mo><mi id="S2.SS2.p3.19.m19.1.1.2" xref="S2.SS2.p3.19.m19.1.1.2.cmml">ι</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.19.m19.1b"><apply id="S2.SS2.p3.19.m19.1.1.cmml" xref="S2.SS2.p3.19.m19.1.1"><cos id="S2.SS2.p3.19.m19.1.1.1.cmml" xref="S2.SS2.p3.19.m19.1.1.1"></cos><ci id="S2.SS2.p3.19.m19.1.1.2.cmml" xref="S2.SS2.p3.19.m19.1.1.2">𝜄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.19.m19.1c">\cos\iota</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.19.m19.1d">roman_cos italic_ι</annotation></semantics></math>, <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.SS2.p3.20.m20.1"><semantics id="S2.SS2.p3.20.m20.1a"><mi id="S2.SS2.p3.20.m20.1.1" xref="S2.SS2.p3.20.m20.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.20.m20.1b"><ci id="S2.SS2.p3.20.m20.1.1.cmml" xref="S2.SS2.p3.20.m20.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.20.m20.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.20.m20.1d">italic_λ</annotation></semantics></math> and <math alttext="\sin\zeta" class="ltx_Math" display="inline" id="S2.SS2.p3.21.m21.1"><semantics id="S2.SS2.p3.21.m21.1a"><mrow id="S2.SS2.p3.21.m21.1.1" xref="S2.SS2.p3.21.m21.1.1.cmml"><mi id="S2.SS2.p3.21.m21.1.1.1" xref="S2.SS2.p3.21.m21.1.1.1.cmml">sin</mi><mo id="S2.SS2.p3.21.m21.1.1a" lspace="0.167em" xref="S2.SS2.p3.21.m21.1.1.cmml">⁡</mo><mi id="S2.SS2.p3.21.m21.1.1.2" xref="S2.SS2.p3.21.m21.1.1.2.cmml">ζ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.21.m21.1b"><apply id="S2.SS2.p3.21.m21.1.1.cmml" xref="S2.SS2.p3.21.m21.1.1"><sin id="S2.SS2.p3.21.m21.1.1.1.cmml" xref="S2.SS2.p3.21.m21.1.1.1"></sin><ci id="S2.SS2.p3.21.m21.1.1.2.cmml" xref="S2.SS2.p3.21.m21.1.1.2">𝜁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.21.m21.1c">\sin\zeta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.21.m21.1d">roman_sin italic_ζ</annotation></semantics></math>. For the sampling, we use <span class="ltx_text ltx_font_italic" id="S2.SS2.p3.21.2">ptemcee</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib92" title="">92</a>]</cite>, an implementation of the Parallel Tempering Markov Chain Monte Carlo algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib93" title="">93</a>]</cite>. In order to reduce computation times, we initiate the sampler at the ground truth of the sources, with an initial covariance matrix for proposal distributions coming from the Fisher matrix. This allows us to directly sample regions with high posterior probabilities while reducing the computation cost. Therefore, the Fisher matrix is never used for approximating the posteriors, which are instead fully Bayesian. The proposal distributions are also tuned to include secondary peaks in the sky locations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib94" title="">94</a>]</cite>, which may have been missed due to our Fisher-based initialization.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.10">The sampler is configured to run with 64 walkers and 10 “temperatures”, and generates 8000 samples per source. The frequency bounds for the likelihood integral are taken as <math alttext="f_{\rm min}=10^{-4}" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mrow id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml"><msub id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml"><mi id="S2.SS2.p4.1.m1.1.1.2.2" xref="S2.SS2.p4.1.m1.1.1.2.2.cmml">f</mi><mi id="S2.SS2.p4.1.m1.1.1.2.3" xref="S2.SS2.p4.1.m1.1.1.2.3.cmml">min</mi></msub><mo id="S2.SS2.p4.1.m1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.cmml">=</mo><msup id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3.cmml"><mn id="S2.SS2.p4.1.m1.1.1.3.2" xref="S2.SS2.p4.1.m1.1.1.3.2.cmml">10</mn><mrow id="S2.SS2.p4.1.m1.1.1.3.3" xref="S2.SS2.p4.1.m1.1.1.3.3.cmml"><mo id="S2.SS2.p4.1.m1.1.1.3.3a" xref="S2.SS2.p4.1.m1.1.1.3.3.cmml">−</mo><mn id="S2.SS2.p4.1.m1.1.1.3.3.2" xref="S2.SS2.p4.1.m1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><apply id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1"><eq id="S2.SS2.p4.1.m1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1"></eq><apply id="S2.SS2.p4.1.m1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.2.1.cmml" xref="S2.SS2.p4.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS2.p4.1.m1.1.1.2.2.cmml" xref="S2.SS2.p4.1.m1.1.1.2.2">𝑓</ci><ci id="S2.SS2.p4.1.m1.1.1.2.3.cmml" xref="S2.SS2.p4.1.m1.1.1.2.3">min</ci></apply><apply id="S2.SS2.p4.1.m1.1.1.3.cmml" xref="S2.SS2.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.3.1.cmml" xref="S2.SS2.p4.1.m1.1.1.3">superscript</csymbol><cn id="S2.SS2.p4.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1.3.2">10</cn><apply id="S2.SS2.p4.1.m1.1.1.3.3.cmml" xref="S2.SS2.p4.1.m1.1.1.3.3"><minus id="S2.SS2.p4.1.m1.1.1.3.3.1.cmml" xref="S2.SS2.p4.1.m1.1.1.3.3"></minus><cn id="S2.SS2.p4.1.m1.1.1.3.3.2.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">f_{\rm min}=10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">italic_f start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> Hz and <math alttext="f_{\rm max}=0.5" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mrow id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml"><msub id="S2.SS2.p4.2.m2.1.1.2" xref="S2.SS2.p4.2.m2.1.1.2.cmml"><mi id="S2.SS2.p4.2.m2.1.1.2.2" xref="S2.SS2.p4.2.m2.1.1.2.2.cmml">f</mi><mi id="S2.SS2.p4.2.m2.1.1.2.3" xref="S2.SS2.p4.2.m2.1.1.2.3.cmml">max</mi></msub><mo id="S2.SS2.p4.2.m2.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS2.p4.2.m2.1.1.3" xref="S2.SS2.p4.2.m2.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><apply id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1"><eq id="S2.SS2.p4.2.m2.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1"></eq><apply id="S2.SS2.p4.2.m2.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m2.1.1.2.1.cmml" xref="S2.SS2.p4.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.p4.2.m2.1.1.2.2.cmml" xref="S2.SS2.p4.2.m2.1.1.2.2">𝑓</ci><ci id="S2.SS2.p4.2.m2.1.1.2.3.cmml" xref="S2.SS2.p4.2.m2.1.1.2.3">max</ci></apply><cn id="S2.SS2.p4.2.m2.1.1.3.cmml" type="float" xref="S2.SS2.p4.2.m2.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">f_{\rm max}=0.5</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">italic_f start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = 0.5</annotation></semantics></math> Hz. Typical resulting fractional uncertainties in the redshifted mass <math alttext="M_{z}\equiv M(1+z)" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><mrow id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml"><msub id="S2.SS2.p4.3.m3.1.1.3" xref="S2.SS2.p4.3.m3.1.1.3.cmml"><mi id="S2.SS2.p4.3.m3.1.1.3.2" xref="S2.SS2.p4.3.m3.1.1.3.2.cmml">M</mi><mi id="S2.SS2.p4.3.m3.1.1.3.3" xref="S2.SS2.p4.3.m3.1.1.3.3.cmml">z</mi></msub><mo id="S2.SS2.p4.3.m3.1.1.2" xref="S2.SS2.p4.3.m3.1.1.2.cmml">≡</mo><mrow id="S2.SS2.p4.3.m3.1.1.1" xref="S2.SS2.p4.3.m3.1.1.1.cmml"><mi id="S2.SS2.p4.3.m3.1.1.1.3" xref="S2.SS2.p4.3.m3.1.1.1.3.cmml">M</mi><mo id="S2.SS2.p4.3.m3.1.1.1.2" xref="S2.SS2.p4.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.3.m3.1.1.1.1.1" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p4.3.m3.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p4.3.m3.1.1.1.1.1.1" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.cmml"><mn id="S2.SS2.p4.3.m3.1.1.1.1.1.1.2" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS2.p4.3.m3.1.1.1.1.1.1.1" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.1.cmml">+</mo><mi id="S2.SS2.p4.3.m3.1.1.1.1.1.1.3" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="S2.SS2.p4.3.m3.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><apply id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1"><equivalent id="S2.SS2.p4.3.m3.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2"></equivalent><apply id="S2.SS2.p4.3.m3.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.3.1.cmml" xref="S2.SS2.p4.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.3.m3.1.1.3.2.cmml" xref="S2.SS2.p4.3.m3.1.1.3.2">𝑀</ci><ci id="S2.SS2.p4.3.m3.1.1.3.3.cmml" xref="S2.SS2.p4.3.m3.1.1.3.3">𝑧</ci></apply><apply id="S2.SS2.p4.3.m3.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1"><times id="S2.SS2.p4.3.m3.1.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.1.2"></times><ci id="S2.SS2.p4.3.m3.1.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.1.3">𝑀</ci><apply id="S2.SS2.p4.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1"><plus id="S2.SS2.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.1"></plus><cn id="S2.SS2.p4.3.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.2">1</cn><ci id="S2.SS2.p4.3.m3.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1.1.3">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">M_{z}\equiv M(1+z)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">italic_M start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≡ italic_M ( 1 + italic_z )</annotation></semantics></math> and in the redshift <math alttext="z" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mi id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><ci id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">italic_z</annotation></semantics></math> are <math alttext="\sim 10^{-3}" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.1"><semantics id="S2.SS2.p4.5.m5.1a"><mrow id="S2.SS2.p4.5.m5.1.1" xref="S2.SS2.p4.5.m5.1.1.cmml"><mi id="S2.SS2.p4.5.m5.1.1.2" xref="S2.SS2.p4.5.m5.1.1.2.cmml"></mi><mo id="S2.SS2.p4.5.m5.1.1.1" xref="S2.SS2.p4.5.m5.1.1.1.cmml">∼</mo><msup id="S2.SS2.p4.5.m5.1.1.3" xref="S2.SS2.p4.5.m5.1.1.3.cmml"><mn id="S2.SS2.p4.5.m5.1.1.3.2" xref="S2.SS2.p4.5.m5.1.1.3.2.cmml">10</mn><mrow id="S2.SS2.p4.5.m5.1.1.3.3" xref="S2.SS2.p4.5.m5.1.1.3.3.cmml"><mo id="S2.SS2.p4.5.m5.1.1.3.3a" xref="S2.SS2.p4.5.m5.1.1.3.3.cmml">−</mo><mn id="S2.SS2.p4.5.m5.1.1.3.3.2" xref="S2.SS2.p4.5.m5.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.1b"><apply id="S2.SS2.p4.5.m5.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1"><csymbol cd="latexml" id="S2.SS2.p4.5.m5.1.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p4.5.m5.1.1.2.cmml" xref="S2.SS2.p4.5.m5.1.1.2">absent</csymbol><apply id="S2.SS2.p4.5.m5.1.1.3.cmml" xref="S2.SS2.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.1.1.3.1.cmml" xref="S2.SS2.p4.5.m5.1.1.3">superscript</csymbol><cn id="S2.SS2.p4.5.m5.1.1.3.2.cmml" type="integer" xref="S2.SS2.p4.5.m5.1.1.3.2">10</cn><apply id="S2.SS2.p4.5.m5.1.1.3.3.cmml" xref="S2.SS2.p4.5.m5.1.1.3.3"><minus id="S2.SS2.p4.5.m5.1.1.3.3.1.cmml" xref="S2.SS2.p4.5.m5.1.1.3.3"></minus><cn id="S2.SS2.p4.5.m5.1.1.3.3.2.cmml" type="integer" xref="S2.SS2.p4.5.m5.1.1.3.3.2">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.1c">\sim 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.1d">∼ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sim 0.25" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.1"><semantics id="S2.SS2.p4.6.m6.1a"><mrow id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml"><mi id="S2.SS2.p4.6.m6.1.1.2" xref="S2.SS2.p4.6.m6.1.1.2.cmml"></mi><mo id="S2.SS2.p4.6.m6.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.cmml">∼</mo><mn id="S2.SS2.p4.6.m6.1.1.3" xref="S2.SS2.p4.6.m6.1.1.3.cmml">0.25</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.1b"><apply id="S2.SS2.p4.6.m6.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1"><csymbol cd="latexml" id="S2.SS2.p4.6.m6.1.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p4.6.m6.1.1.2.cmml" xref="S2.SS2.p4.6.m6.1.1.2">absent</csymbol><cn id="S2.SS2.p4.6.m6.1.1.3.cmml" type="float" xref="S2.SS2.p4.6.m6.1.1.3">0.25</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.1c">\sim 0.25</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.1d">∼ 0.25</annotation></semantics></math>, respectively <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib60" title="">60</a>]</cite>. Thus, as <math alttext="M_{z}" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.1"><semantics id="S2.SS2.p4.7.m7.1a"><msub id="S2.SS2.p4.7.m7.1.1" xref="S2.SS2.p4.7.m7.1.1.cmml"><mi id="S2.SS2.p4.7.m7.1.1.2" xref="S2.SS2.p4.7.m7.1.1.2.cmml">M</mi><mi id="S2.SS2.p4.7.m7.1.1.3" xref="S2.SS2.p4.7.m7.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.1b"><apply id="S2.SS2.p4.7.m7.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.1.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1">subscript</csymbol><ci id="S2.SS2.p4.7.m7.1.1.2.cmml" xref="S2.SS2.p4.7.m7.1.1.2">𝑀</ci><ci id="S2.SS2.p4.7.m7.1.1.3.cmml" xref="S2.SS2.p4.7.m7.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.1c">M_{z}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.1d">italic_M start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> is precisely determined, errors in the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p4.10.1">source</em> total mass <math alttext="M" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m8.1"><semantics id="S2.SS2.p4.8.m8.1a"><mi id="S2.SS2.p4.8.m8.1.1" xref="S2.SS2.p4.8.m8.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m8.1b"><ci id="S2.SS2.p4.8.m8.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m8.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m8.1d">italic_M</annotation></semantics></math> are strongly correlated with errors in <math alttext="D_{L}" class="ltx_Math" display="inline" id="S2.SS2.p4.9.m9.1"><semantics id="S2.SS2.p4.9.m9.1a"><msub id="S2.SS2.p4.9.m9.1.1" xref="S2.SS2.p4.9.m9.1.1.cmml"><mi id="S2.SS2.p4.9.m9.1.1.2" xref="S2.SS2.p4.9.m9.1.1.2.cmml">D</mi><mi id="S2.SS2.p4.9.m9.1.1.3" xref="S2.SS2.p4.9.m9.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.9.m9.1b"><apply id="S2.SS2.p4.9.m9.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.9.m9.1.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1">subscript</csymbol><ci id="S2.SS2.p4.9.m9.1.1.2.cmml" xref="S2.SS2.p4.9.m9.1.1.2">𝐷</ci><ci id="S2.SS2.p4.9.m9.1.1.3.cmml" xref="S2.SS2.p4.9.m9.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m9.1c">D_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m9.1d">italic_D start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="z" class="ltx_Math" display="inline" id="S2.SS2.p4.10.m10.1"><semantics id="S2.SS2.p4.10.m10.1a"><mi id="S2.SS2.p4.10.m10.1.1" xref="S2.SS2.p4.10.m10.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.10.m10.1b"><ci id="S2.SS2.p4.10.m10.1.1.cmml" xref="S2.SS2.p4.10.m10.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.10.m10.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.10.m10.1d">italic_z</annotation></semantics></math>. These large and correlated event-level uncertainties are a challenge for accurate population reconstruction.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.3 </span>Population reconstruction via iterative KDE</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib95" title="">95</a>]</cite> demonstrated the use of adaptive Kernel Density Estimation (KDE) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib96" title="">96</a>]</cite> to study rates and populations of binary black holes observed with the current GW detector network <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib5" title="">5</a>]</cite>. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib97" title="">97</a>]</cite> we further developed the adaptive KDE method to account for measurement uncertainties in a self-consistent way. In this study, we use the methods developed in those works, with minor modifications.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.4">The basic idea of KDE is to choose a global bandwidth <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_β</annotation></semantics></math> for a choice of kernel <math alttext="K" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">italic_K</annotation></semantics></math>, typically Gaussian, and given observations <math alttext="X_{i}" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><msub id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml"><mi id="S2.SS3.p2.3.m3.1.1.2" xref="S2.SS3.p2.3.m3.1.1.2.cmml">X</mi><mi id="S2.SS3.p2.3.m3.1.1.3" xref="S2.SS3.p2.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.3.m3.1.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS3.p2.3.m3.1.1.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2">𝑋</ci><ci id="S2.SS3.p2.3.m3.1.1.3.cmml" xref="S2.SS3.p2.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> with corresponding weights <math alttext="W_{i}" class="ltx_Math" display="inline" id="S2.SS3.p2.4.m4.1"><semantics id="S2.SS3.p2.4.m4.1a"><msub id="S2.SS3.p2.4.m4.1.1" xref="S2.SS3.p2.4.m4.1.1.cmml"><mi id="S2.SS3.p2.4.m4.1.1.2" xref="S2.SS3.p2.4.m4.1.1.2.cmml">W</mi><mi id="S2.SS3.p2.4.m4.1.1.3" xref="S2.SS3.p2.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.4.m4.1b"><apply id="S2.SS3.p2.4.m4.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.4.m4.1.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS3.p2.4.m4.1.1.2.cmml" xref="S2.SS3.p2.4.m4.1.1.2">𝑊</ci><ci id="S2.SS3.p2.4.m4.1.1.3.cmml" xref="S2.SS3.p2.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.4.m4.1c">W_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.4.m4.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> estimate the density as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\hat{f}(x)=\frac{1}{\sum_{i}W_{i}}\sum_{i=1}^{n}\frac{W_{i}}{\beta\lambda_{i}}% K\left(\frac{x-X_{i}}{\beta\lambda_{i}}\right)\,." class="ltx_Math" display="block" id="S2.E4.m1.3"><semantics id="S2.E4.m1.3a"><mrow id="S2.E4.m1.3.3.1" xref="S2.E4.m1.3.3.1.1.cmml"><mrow id="S2.E4.m1.3.3.1.1" xref="S2.E4.m1.3.3.1.1.cmml"><mrow id="S2.E4.m1.3.3.1.1.2" xref="S2.E4.m1.3.3.1.1.2.cmml"><mover accent="true" id="S2.E4.m1.3.3.1.1.2.2" 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id="S2.E4.m1.3c">\hat{f}(x)=\frac{1}{\sum_{i}W_{i}}\sum_{i=1}^{n}\frac{W_{i}}{\beta\lambda_{i}}% K\left(\frac{x-X_{i}}{\beta\lambda_{i}}\right)\,.</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.3d">over^ start_ARG italic_f end_ARG ( italic_x ) = divide start_ARG 1 end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT divide start_ARG italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_β italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG italic_K ( divide start_ARG italic_x - italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_β italic_λ start_POSTSUBSCRIPT italic_i 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">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p2.5">Here, <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S2.SS3.p2.5.m1.1"><semantics id="S2.SS3.p2.5.m1.1a"><msub id="S2.SS3.p2.5.m1.1.1" xref="S2.SS3.p2.5.m1.1.1.cmml"><mi id="S2.SS3.p2.5.m1.1.1.2" xref="S2.SS3.p2.5.m1.1.1.2.cmml">λ</mi><mi id="S2.SS3.p2.5.m1.1.1.3" xref="S2.SS3.p2.5.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.5.m1.1b"><apply id="S2.SS3.p2.5.m1.1.1.cmml" xref="S2.SS3.p2.5.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.5.m1.1.1.1.cmml" xref="S2.SS3.p2.5.m1.1.1">subscript</csymbol><ci id="S2.SS3.p2.5.m1.1.1.2.cmml" xref="S2.SS3.p2.5.m1.1.1.2">𝜆</ci><ci id="S2.SS3.p2.5.m1.1.1.3.cmml" xref="S2.SS3.p2.5.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.5.m1.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.5.m1.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a local parameter multiplying the bandwidth for each observation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib98" title="">98</a>]</cite>. The expected error of the estimate has contributions from both its bias and variance: the choice of bandwidth is crucial to control these terms. Too small bandwidth, implying under-smoothing, causes the KDE to be too sensitive to small fluctuations in the data, inflating the variance, whereas over-smoothing occurs with large bandwidth, making the KDE unable to represent rapid variations in density and thus increasing its bias. Furthermore, the optimal bandwidth at a given point in parameter space will depend on the density of observations in its neighborhood: smaller bandwidth is preferred in denser regions.</p> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.3">To address this issue, a per-point adaptive bandwidth is used: this is implemented by first evaluating a pilot KDE <math alttext="\hat{f}_{0}(x)" class="ltx_Math" display="inline" id="S2.SS3.p3.1.m1.1"><semantics id="S2.SS3.p3.1.m1.1a"><mrow id="S2.SS3.p3.1.m1.1.2" xref="S2.SS3.p3.1.m1.1.2.cmml"><msub id="S2.SS3.p3.1.m1.1.2.2" xref="S2.SS3.p3.1.m1.1.2.2.cmml"><mover accent="true" id="S2.SS3.p3.1.m1.1.2.2.2" xref="S2.SS3.p3.1.m1.1.2.2.2.cmml"><mi id="S2.SS3.p3.1.m1.1.2.2.2.2" xref="S2.SS3.p3.1.m1.1.2.2.2.2.cmml">f</mi><mo id="S2.SS3.p3.1.m1.1.2.2.2.1" xref="S2.SS3.p3.1.m1.1.2.2.2.1.cmml">^</mo></mover><mn id="S2.SS3.p3.1.m1.1.2.2.3" xref="S2.SS3.p3.1.m1.1.2.2.3.cmml">0</mn></msub><mo id="S2.SS3.p3.1.m1.1.2.1" xref="S2.SS3.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.1.m1.1.2.3.2" xref="S2.SS3.p3.1.m1.1.2.cmml"><mo id="S2.SS3.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.p3.1.m1.1.1" xref="S2.SS3.p3.1.m1.1.1.cmml">x</mi><mo id="S2.SS3.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.1.m1.1b"><apply id="S2.SS3.p3.1.m1.1.2.cmml" xref="S2.SS3.p3.1.m1.1.2"><times id="S2.SS3.p3.1.m1.1.2.1.cmml" xref="S2.SS3.p3.1.m1.1.2.1"></times><apply id="S2.SS3.p3.1.m1.1.2.2.cmml" xref="S2.SS3.p3.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.1.m1.1.2.2.1.cmml" xref="S2.SS3.p3.1.m1.1.2.2">subscript</csymbol><apply id="S2.SS3.p3.1.m1.1.2.2.2.cmml" xref="S2.SS3.p3.1.m1.1.2.2.2"><ci id="S2.SS3.p3.1.m1.1.2.2.2.1.cmml" xref="S2.SS3.p3.1.m1.1.2.2.2.1">^</ci><ci id="S2.SS3.p3.1.m1.1.2.2.2.2.cmml" xref="S2.SS3.p3.1.m1.1.2.2.2.2">𝑓</ci></apply><cn id="S2.SS3.p3.1.m1.1.2.2.3.cmml" type="integer" xref="S2.SS3.p3.1.m1.1.2.2.3">0</cn></apply><ci id="S2.SS3.p3.1.m1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.1c">\hat{f}_{0}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.1d">over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>, where we set <math alttext="\lambda_{i}=1" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.1"><semantics id="S2.SS3.p3.2.m2.1a"><mrow id="S2.SS3.p3.2.m2.1.1" xref="S2.SS3.p3.2.m2.1.1.cmml"><msub id="S2.SS3.p3.2.m2.1.1.2" xref="S2.SS3.p3.2.m2.1.1.2.cmml"><mi id="S2.SS3.p3.2.m2.1.1.2.2" xref="S2.SS3.p3.2.m2.1.1.2.2.cmml">λ</mi><mi id="S2.SS3.p3.2.m2.1.1.2.3" xref="S2.SS3.p3.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS3.p3.2.m2.1.1.1" xref="S2.SS3.p3.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS3.p3.2.m2.1.1.3" xref="S2.SS3.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.2.m2.1b"><apply id="S2.SS3.p3.2.m2.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1"><eq id="S2.SS3.p3.2.m2.1.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1.1"></eq><apply id="S2.SS3.p3.2.m2.1.1.2.cmml" xref="S2.SS3.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p3.2.m2.1.1.2.1.cmml" xref="S2.SS3.p3.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS3.p3.2.m2.1.1.2.2.cmml" xref="S2.SS3.p3.2.m2.1.1.2.2">𝜆</ci><ci id="S2.SS3.p3.2.m2.1.1.2.3.cmml" xref="S2.SS3.p3.2.m2.1.1.2.3">𝑖</ci></apply><cn id="S2.SS3.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS3.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.2.m2.1c">\lambda_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.2.m2.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> for all events. Then choosing a sensitivity parameter <math alttext="0<\alpha\leq 1" class="ltx_Math" display="inline" id="S2.SS3.p3.3.m3.1"><semantics id="S2.SS3.p3.3.m3.1a"><mrow id="S2.SS3.p3.3.m3.1.1" xref="S2.SS3.p3.3.m3.1.1.cmml"><mn id="S2.SS3.p3.3.m3.1.1.2" xref="S2.SS3.p3.3.m3.1.1.2.cmml">0</mn><mo id="S2.SS3.p3.3.m3.1.1.3" xref="S2.SS3.p3.3.m3.1.1.3.cmml"><</mo><mi id="S2.SS3.p3.3.m3.1.1.4" xref="S2.SS3.p3.3.m3.1.1.4.cmml">α</mi><mo id="S2.SS3.p3.3.m3.1.1.5" xref="S2.SS3.p3.3.m3.1.1.5.cmml">≤</mo><mn id="S2.SS3.p3.3.m3.1.1.6" xref="S2.SS3.p3.3.m3.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.3.m3.1b"><apply id="S2.SS3.p3.3.m3.1.1.cmml" xref="S2.SS3.p3.3.m3.1.1"><and id="S2.SS3.p3.3.m3.1.1a.cmml" xref="S2.SS3.p3.3.m3.1.1"></and><apply id="S2.SS3.p3.3.m3.1.1b.cmml" xref="S2.SS3.p3.3.m3.1.1"><lt id="S2.SS3.p3.3.m3.1.1.3.cmml" xref="S2.SS3.p3.3.m3.1.1.3"></lt><cn id="S2.SS3.p3.3.m3.1.1.2.cmml" type="integer" xref="S2.SS3.p3.3.m3.1.1.2">0</cn><ci id="S2.SS3.p3.3.m3.1.1.4.cmml" xref="S2.SS3.p3.3.m3.1.1.4">𝛼</ci></apply><apply id="S2.SS3.p3.3.m3.1.1c.cmml" xref="S2.SS3.p3.3.m3.1.1"><leq id="S2.SS3.p3.3.m3.1.1.5.cmml" xref="S2.SS3.p3.3.m3.1.1.5"></leq><share href="https://arxiv.org/html/2410.17056v2#S2.SS3.p3.3.m3.1.1.4.cmml" id="S2.SS3.p3.3.m3.1.1d.cmml" xref="S2.SS3.p3.3.m3.1.1"></share><cn id="S2.SS3.p3.3.m3.1.1.6.cmml" type="integer" xref="S2.SS3.p3.3.m3.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.3.m3.1c">0<\alpha\leq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.3.m3.1d">0 < italic_α ≤ 1</annotation></semantics></math> we determine the local adaptive bandwidth factor as</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\lambda_{i}=\left(\frac{\hat{f}_{0}(X_{i})}{g}\right)^{-\alpha},\,\,\log g=n^{% -1}\sum_{i=1}^{n}\log\hat{f}_{0}(X_{i})\,," class="ltx_Math" display="block" id="S2.E5.m1.2"><semantics id="S2.E5.m1.2a"><mrow id="S2.E5.m1.2.2.1"><mrow id="S2.E5.m1.2.2.1.1.2" xref="S2.E5.m1.2.2.1.1.3.cmml"><mrow id="S2.E5.m1.2.2.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.cmml"><msub id="S2.E5.m1.2.2.1.1.1.1.2" xref="S2.E5.m1.2.2.1.1.1.1.2.cmml"><mi id="S2.E5.m1.2.2.1.1.1.1.2.2" xref="S2.E5.m1.2.2.1.1.1.1.2.2.cmml">λ</mi><mi id="S2.E5.m1.2.2.1.1.1.1.2.3" xref="S2.E5.m1.2.2.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.E5.m1.2.2.1.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.1.cmml">=</mo><msup id="S2.E5.m1.2.2.1.1.1.1.3" xref="S2.E5.m1.2.2.1.1.1.1.3.cmml"><mrow id="S2.E5.m1.2.2.1.1.1.1.3.2.2" xref="S2.E5.m1.1.1.cmml"><mo id="S2.E5.m1.2.2.1.1.1.1.3.2.2.1" xref="S2.E5.m1.1.1.cmml">(</mo><mfrac id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml"><mrow id="S2.E5.m1.1.1.1" xref="S2.E5.m1.1.1.1.cmml"><msub id="S2.E5.m1.1.1.1.3" xref="S2.E5.m1.1.1.1.3.cmml"><mover accent="true" id="S2.E5.m1.1.1.1.3.2" xref="S2.E5.m1.1.1.1.3.2.cmml"><mi id="S2.E5.m1.1.1.1.3.2.2" xref="S2.E5.m1.1.1.1.3.2.2.cmml">f</mi><mo id="S2.E5.m1.1.1.1.3.2.1" xref="S2.E5.m1.1.1.1.3.2.1.cmml">^</mo></mover><mn id="S2.E5.m1.1.1.1.3.3" xref="S2.E5.m1.1.1.1.3.3.cmml">0</mn></msub><mo id="S2.E5.m1.1.1.1.2" xref="S2.E5.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E5.m1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.1.cmml"><mo id="S2.E5.m1.1.1.1.1.1.2" stretchy="false" xref="S2.E5.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E5.m1.1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.1.cmml"><mi id="S2.E5.m1.1.1.1.1.1.1.2" xref="S2.E5.m1.1.1.1.1.1.1.2.cmml">X</mi><mi id="S2.E5.m1.1.1.1.1.1.1.3" xref="S2.E5.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.E5.m1.1.1.1.1.1.3" stretchy="false" xref="S2.E5.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mi id="S2.E5.m1.1.1.3" xref="S2.E5.m1.1.1.3.cmml">g</mi></mfrac><mo id="S2.E5.m1.2.2.1.1.1.1.3.2.2.2" xref="S2.E5.m1.1.1.cmml">)</mo></mrow><mrow id="S2.E5.m1.2.2.1.1.1.1.3.3" xref="S2.E5.m1.2.2.1.1.1.1.3.3.cmml"><mo id="S2.E5.m1.2.2.1.1.1.1.3.3a" xref="S2.E5.m1.2.2.1.1.1.1.3.3.cmml">−</mo><mi id="S2.E5.m1.2.2.1.1.1.1.3.3.2" xref="S2.E5.m1.2.2.1.1.1.1.3.3.2.cmml">α</mi></mrow></msup></mrow><mo id="S2.E5.m1.2.2.1.1.2.3" rspace="0.497em" xref="S2.E5.m1.2.2.1.1.3a.cmml">,</mo><mrow id="S2.E5.m1.2.2.1.1.2.2" xref="S2.E5.m1.2.2.1.1.2.2.cmml"><mrow id="S2.E5.m1.2.2.1.1.2.2.3" xref="S2.E5.m1.2.2.1.1.2.2.3.cmml"><mi id="S2.E5.m1.2.2.1.1.2.2.3.1" xref="S2.E5.m1.2.2.1.1.2.2.3.1.cmml">log</mi><mo id="S2.E5.m1.2.2.1.1.2.2.3a" lspace="0.167em" xref="S2.E5.m1.2.2.1.1.2.2.3.cmml">⁡</mo><mi id="S2.E5.m1.2.2.1.1.2.2.3.2" xref="S2.E5.m1.2.2.1.1.2.2.3.2.cmml">g</mi></mrow><mo id="S2.E5.m1.2.2.1.1.2.2.2" xref="S2.E5.m1.2.2.1.1.2.2.2.cmml">=</mo><mrow id="S2.E5.m1.2.2.1.1.2.2.1" xref="S2.E5.m1.2.2.1.1.2.2.1.cmml"><msup id="S2.E5.m1.2.2.1.1.2.2.1.3" xref="S2.E5.m1.2.2.1.1.2.2.1.3.cmml"><mi id="S2.E5.m1.2.2.1.1.2.2.1.3.2" 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id="S2.E5.m1.2d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ( divide start_ARG over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG italic_g end_ARG ) start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT , roman_log italic_g = italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_log over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_i 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">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p3.4">and finally evaluate the KDE via (<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.E4" title="In II.3 Population reconstruction via iterative KDE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">4</span></a>) using these <math alttext="\lambda_{i}" class="ltx_Math" display="inline" id="S2.SS3.p3.4.m1.1"><semantics id="S2.SS3.p3.4.m1.1a"><msub id="S2.SS3.p3.4.m1.1.1" xref="S2.SS3.p3.4.m1.1.1.cmml"><mi id="S2.SS3.p3.4.m1.1.1.2" xref="S2.SS3.p3.4.m1.1.1.2.cmml">λ</mi><mi id="S2.SS3.p3.4.m1.1.1.3" xref="S2.SS3.p3.4.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.4.m1.1b"><apply id="S2.SS3.p3.4.m1.1.1.cmml" xref="S2.SS3.p3.4.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p3.4.m1.1.1.1.cmml" xref="S2.SS3.p3.4.m1.1.1">subscript</csymbol><ci id="S2.SS3.p3.4.m1.1.1.2.cmml" xref="S2.SS3.p3.4.m1.1.1.2">𝜆</ci><ci id="S2.SS3.p3.4.m1.1.1.3.cmml" xref="S2.SS3.p3.4.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.4.m1.1c">\lambda_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.4.m1.1d">italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> values.</p> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.3">The global bandwidth <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS3.p4.1.m1.1"><semantics id="S2.SS3.p4.1.m1.1a"><mi id="S2.SS3.p4.1.m1.1.1" xref="S2.SS3.p4.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.1.m1.1b"><ci id="S2.SS3.p4.1.m1.1.1.cmml" xref="S2.SS3.p4.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.1.m1.1d">italic_β</annotation></semantics></math> and sensitivity parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS3.p4.2.m2.1"><semantics id="S2.SS3.p4.2.m2.1a"><mi id="S2.SS3.p4.2.m2.1.1" xref="S2.SS3.p4.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.2.m2.1b"><ci id="S2.SS3.p4.2.m2.1.1.cmml" xref="S2.SS3.p4.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.2.m2.1d">italic_α</annotation></semantics></math> remain to be chosen: we determine their optimal values by grid search, using the total log likelihood as a figure of merit. We evaluate the likelihood via <span class="ltx_text ltx_font_italic" id="S2.SS3.p4.3.1">K-fold cross-validation</span>, setting <math alttext="K=5" class="ltx_Math" display="inline" id="S2.SS3.p4.3.m3.1"><semantics id="S2.SS3.p4.3.m3.1a"><mrow id="S2.SS3.p4.3.m3.1.1" xref="S2.SS3.p4.3.m3.1.1.cmml"><mi id="S2.SS3.p4.3.m3.1.1.2" xref="S2.SS3.p4.3.m3.1.1.2.cmml">K</mi><mo id="S2.SS3.p4.3.m3.1.1.1" xref="S2.SS3.p4.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS3.p4.3.m3.1.1.3" xref="S2.SS3.p4.3.m3.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.3.m3.1b"><apply id="S2.SS3.p4.3.m3.1.1.cmml" xref="S2.SS3.p4.3.m3.1.1"><eq id="S2.SS3.p4.3.m3.1.1.1.cmml" xref="S2.SS3.p4.3.m3.1.1.1"></eq><ci id="S2.SS3.p4.3.m3.1.1.2.cmml" xref="S2.SS3.p4.3.m3.1.1.2">𝐾</ci><cn id="S2.SS3.p4.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.p4.3.m3.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.3.m3.1c">K=5</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.3.m3.1d">italic_K = 5</annotation></semantics></math> unless otherwise specified (see Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib99" title="">99</a>]</cite> section 3.4.4 and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib100" title="">100</a>]</cite>). The cross-validated (log) likelihood is</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\log\mathcal{L}_{\rm Kfold}=\sum_{k=1}^{K}\sum_{i\in\text{Fold}_{k}}\log\hat{f% }_{{\rm Kfold},i}(X_{i})," class="ltx_Math" display="block" id="S2.E6.m1.3"><semantics id="S2.E6.m1.3a"><mrow id="S2.E6.m1.3.3.1" xref="S2.E6.m1.3.3.1.1.cmml"><mrow id="S2.E6.m1.3.3.1.1" xref="S2.E6.m1.3.3.1.1.cmml"><mrow id="S2.E6.m1.3.3.1.1.3" xref="S2.E6.m1.3.3.1.1.3.cmml"><mi id="S2.E6.m1.3.3.1.1.3.1" xref="S2.E6.m1.3.3.1.1.3.1.cmml">log</mi><mo id="S2.E6.m1.3.3.1.1.3a" lspace="0.167em" xref="S2.E6.m1.3.3.1.1.3.cmml">⁡</mo><msub id="S2.E6.m1.3.3.1.1.3.2" xref="S2.E6.m1.3.3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E6.m1.3.3.1.1.3.2.2" xref="S2.E6.m1.3.3.1.1.3.2.2.cmml">ℒ</mi><mi id="S2.E6.m1.3.3.1.1.3.2.3" 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italic_i 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="S2.SS3.p4.9">where the data points are split into <math alttext="K" class="ltx_Math" display="inline" id="S2.SS3.p4.4.m1.1"><semantics id="S2.SS3.p4.4.m1.1a"><mi id="S2.SS3.p4.4.m1.1.1" xref="S2.SS3.p4.4.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.4.m1.1b"><ci id="S2.SS3.p4.4.m1.1.1.cmml" xref="S2.SS3.p4.4.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.4.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.4.m1.1d">italic_K</annotation></semantics></math> subsets of equal size: <math alttext="\text{Fold}_{k}" class="ltx_Math" display="inline" id="S2.SS3.p4.5.m2.1"><semantics id="S2.SS3.p4.5.m2.1a"><msub id="S2.SS3.p4.5.m2.1.1" xref="S2.SS3.p4.5.m2.1.1.cmml"><mtext id="S2.SS3.p4.5.m2.1.1.2" xref="S2.SS3.p4.5.m2.1.1.2a.cmml">Fold</mtext><mi id="S2.SS3.p4.5.m2.1.1.3" xref="S2.SS3.p4.5.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.5.m2.1b"><apply id="S2.SS3.p4.5.m2.1.1.cmml" xref="S2.SS3.p4.5.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p4.5.m2.1.1.1.cmml" xref="S2.SS3.p4.5.m2.1.1">subscript</csymbol><ci id="S2.SS3.p4.5.m2.1.1.2a.cmml" xref="S2.SS3.p4.5.m2.1.1.2"><mtext id="S2.SS3.p4.5.m2.1.1.2.cmml" xref="S2.SS3.p4.5.m2.1.1.2">Fold</mtext></ci><ci id="S2.SS3.p4.5.m2.1.1.3.cmml" xref="S2.SS3.p4.5.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.5.m2.1c">\text{Fold}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.5.m2.1d">Fold start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> represents the <math alttext="k" class="ltx_Math" display="inline" id="S2.SS3.p4.6.m3.1"><semantics id="S2.SS3.p4.6.m3.1a"><mi id="S2.SS3.p4.6.m3.1.1" xref="S2.SS3.p4.6.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.6.m3.1b"><ci id="S2.SS3.p4.6.m3.1.1.cmml" xref="S2.SS3.p4.6.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.6.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.6.m3.1d">italic_k</annotation></semantics></math>-th subset, and <math alttext="\hat{f}_{{\rm Kfold},i}" class="ltx_Math" display="inline" id="S2.SS3.p4.7.m4.2"><semantics id="S2.SS3.p4.7.m4.2a"><msub id="S2.SS3.p4.7.m4.2.3" xref="S2.SS3.p4.7.m4.2.3.cmml"><mover accent="true" id="S2.SS3.p4.7.m4.2.3.2" xref="S2.SS3.p4.7.m4.2.3.2.cmml"><mi id="S2.SS3.p4.7.m4.2.3.2.2" xref="S2.SS3.p4.7.m4.2.3.2.2.cmml">f</mi><mo id="S2.SS3.p4.7.m4.2.3.2.1" xref="S2.SS3.p4.7.m4.2.3.2.1.cmml">^</mo></mover><mrow id="S2.SS3.p4.7.m4.2.2.2.4" xref="S2.SS3.p4.7.m4.2.2.2.3.cmml"><mi id="S2.SS3.p4.7.m4.1.1.1.1" xref="S2.SS3.p4.7.m4.1.1.1.1.cmml">Kfold</mi><mo id="S2.SS3.p4.7.m4.2.2.2.4.1" xref="S2.SS3.p4.7.m4.2.2.2.3.cmml">,</mo><mi id="S2.SS3.p4.7.m4.2.2.2.2" xref="S2.SS3.p4.7.m4.2.2.2.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.7.m4.2b"><apply id="S2.SS3.p4.7.m4.2.3.cmml" xref="S2.SS3.p4.7.m4.2.3"><csymbol cd="ambiguous" id="S2.SS3.p4.7.m4.2.3.1.cmml" xref="S2.SS3.p4.7.m4.2.3">subscript</csymbol><apply id="S2.SS3.p4.7.m4.2.3.2.cmml" xref="S2.SS3.p4.7.m4.2.3.2"><ci id="S2.SS3.p4.7.m4.2.3.2.1.cmml" xref="S2.SS3.p4.7.m4.2.3.2.1">^</ci><ci id="S2.SS3.p4.7.m4.2.3.2.2.cmml" xref="S2.SS3.p4.7.m4.2.3.2.2">𝑓</ci></apply><list id="S2.SS3.p4.7.m4.2.2.2.3.cmml" xref="S2.SS3.p4.7.m4.2.2.2.4"><ci id="S2.SS3.p4.7.m4.1.1.1.1.cmml" xref="S2.SS3.p4.7.m4.1.1.1.1">Kfold</ci><ci id="S2.SS3.p4.7.m4.2.2.2.2.cmml" xref="S2.SS3.p4.7.m4.2.2.2.2">𝑖</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.7.m4.2c">\hat{f}_{{\rm Kfold},i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.7.m4.2d">over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT roman_Kfold , italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the KDE constructed using the samples from the remaining <math alttext="K-1" class="ltx_Math" display="inline" id="S2.SS3.p4.8.m5.1"><semantics id="S2.SS3.p4.8.m5.1a"><mrow id="S2.SS3.p4.8.m5.1.1" xref="S2.SS3.p4.8.m5.1.1.cmml"><mi id="S2.SS3.p4.8.m5.1.1.2" xref="S2.SS3.p4.8.m5.1.1.2.cmml">K</mi><mo id="S2.SS3.p4.8.m5.1.1.1" xref="S2.SS3.p4.8.m5.1.1.1.cmml">−</mo><mn id="S2.SS3.p4.8.m5.1.1.3" xref="S2.SS3.p4.8.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.8.m5.1b"><apply id="S2.SS3.p4.8.m5.1.1.cmml" xref="S2.SS3.p4.8.m5.1.1"><minus id="S2.SS3.p4.8.m5.1.1.1.cmml" xref="S2.SS3.p4.8.m5.1.1.1"></minus><ci id="S2.SS3.p4.8.m5.1.1.2.cmml" xref="S2.SS3.p4.8.m5.1.1.2">𝐾</ci><cn id="S2.SS3.p4.8.m5.1.1.3.cmml" type="integer" xref="S2.SS3.p4.8.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.8.m5.1c">K-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.8.m5.1d">italic_K - 1</annotation></semantics></math> subsets (excluding the points in <math alttext="\text{Fold}_{k}" class="ltx_Math" display="inline" id="S2.SS3.p4.9.m6.1"><semantics id="S2.SS3.p4.9.m6.1a"><msub id="S2.SS3.p4.9.m6.1.1" xref="S2.SS3.p4.9.m6.1.1.cmml"><mtext id="S2.SS3.p4.9.m6.1.1.2" xref="S2.SS3.p4.9.m6.1.1.2a.cmml">Fold</mtext><mi id="S2.SS3.p4.9.m6.1.1.3" xref="S2.SS3.p4.9.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.9.m6.1b"><apply id="S2.SS3.p4.9.m6.1.1.cmml" xref="S2.SS3.p4.9.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.p4.9.m6.1.1.1.cmml" xref="S2.SS3.p4.9.m6.1.1">subscript</csymbol><ci id="S2.SS3.p4.9.m6.1.1.2a.cmml" xref="S2.SS3.p4.9.m6.1.1.2"><mtext id="S2.SS3.p4.9.m6.1.1.2.cmml" xref="S2.SS3.p4.9.m6.1.1.2">Fold</mtext></ci><ci id="S2.SS3.p4.9.m6.1.1.3.cmml" xref="S2.SS3.p4.9.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.9.m6.1c">\text{Fold}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.9.m6.1d">Fold start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S2.SS3.p5"> <p class="ltx_p" id="S2.SS3.p5.1">We further apply an iterative reweighting method, introduced in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib97" title="">97</a>]</cite>, in order to self-consistently treat measurement uncertainties in the detected events and avoid an over-dispersed estimate of the true distribution. The motivation of this reweighting is as follows: posterior PE samples are produced using uniform or uninformative parameter priors, but we would obtain more accurate and precise measurements of astrophysically interesting parameters by instead using a prior close(r) to the true astrophysical distribution. Conversely, with more accurate individual event parameters we would obtain a more accurate KDE for the population. Our method addresses parameter errors by using the current estimate of KDE, obtained from some draw of samples from each event, to re-weight the samples of each event with probability proportional to the estimated density at each sample, in order to draw a new set of samples for the next bootstrap iteration. The new samples are then used to obtain a new KDE, and the process is repeated for many iterations, in a process similar in spirit to the expectation-maximization algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib101" title="">101</a>]</cite>. We have shown that iterative reweighting can reduce the excess KDE dispersion due to measurement uncertainty <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib97" title="">97</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.SS3.p6"> <p class="ltx_p" id="S2.SS3.p6.1">In our iterative process, the first sample reweighting uses a KDE based on the medians of 100 samples from each event. In subsequent iterations the KDE from the previous step is used for reweighting, producing a Markov chain of density estimates. We discard the first 100 iterations to exclude the initial transient behavior; after 100 additional iterations, at each step we then use the median of KDEs from a buffer containing the previous 100 iterations to derive sample weights for the next step’s KDE, as explained in detail in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib97" title="">97</a>]</cite>. We then collect 1000 bootstrap iterations using this median buffer reweighting, from which we calculate summary estimates via the median, 5th and 95th percentile of the KDEs.</p> </div> <div class="ltx_para" id="S2.SS3.p7"> <p class="ltx_p" id="S2.SS3.p7.1">Our KDE derived from detected events is susceptible to selection bias, due to the detector’s limitations in observing merging binaries over a finite range of masses and redshifts. It is crucial to account for selection effects to avoid biased estimates of the population properties. In the following section, we describe our treatment of selection effects via estimating the probability of detection.</p> </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.4 </span>Selection effects and validation of PE</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.8">The probability of detection <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.1"><semantics id="S2.SS4.p1.1.m1.1a"><msub id="S2.SS4.p1.1.m1.1.1" xref="S2.SS4.p1.1.m1.1.1.cmml"><mi id="S2.SS4.p1.1.m1.1.1.2" xref="S2.SS4.p1.1.m1.1.1.2.cmml">p</mi><mi id="S2.SS4.p1.1.m1.1.1.3" xref="S2.SS4.p1.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.1b"><apply id="S2.SS4.p1.1.m1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.1.m1.1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.p1.1.m1.1.1.2.cmml" xref="S2.SS4.p1.1.m1.1.1.2">𝑝</ci><ci id="S2.SS4.p1.1.m1.1.1.3.cmml" xref="S2.SS4.p1.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> for a given coalescing binary source is, in general, a function of all the parameters that we estimate (see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS2" title="II.2 Parameter Estimation ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.2</span></a>). As we do not attempt detailed modelling of non-ideal data and the complexity of search pipelines, we use a relatively simple detectability criterion based on the SNR of simulated signals. Specifically, we suppose that a search would detect an event if the matched filter SNR is above a given threshold <math alttext="\rho_{\rm th}" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m2.1"><semantics id="S2.SS4.p1.2.m2.1a"><msub id="S2.SS4.p1.2.m2.1.1" xref="S2.SS4.p1.2.m2.1.1.cmml"><mi id="S2.SS4.p1.2.m2.1.1.2" xref="S2.SS4.p1.2.m2.1.1.2.cmml">ρ</mi><mi id="S2.SS4.p1.2.m2.1.1.3" xref="S2.SS4.p1.2.m2.1.1.3.cmml">th</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.2.m2.1b"><apply id="S2.SS4.p1.2.m2.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.2.m2.1.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS4.p1.2.m2.1.1.2.cmml" xref="S2.SS4.p1.2.m2.1.1.2">𝜌</ci><ci id="S2.SS4.p1.2.m2.1.1.3.cmml" xref="S2.SS4.p1.2.m2.1.1.3">th</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m2.1c">\rho_{\rm th}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m2.1d">italic_ρ start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT</annotation></semantics></math> which we set to <math alttext="8" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m3.1"><semantics id="S2.SS4.p1.3.m3.1a"><mn id="S2.SS4.p1.3.m3.1.1" xref="S2.SS4.p1.3.m3.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m3.1b"><cn id="S2.SS4.p1.3.m3.1.1.cmml" type="integer" xref="S2.SS4.p1.3.m3.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m3.1c">8</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m3.1d">8</annotation></semantics></math>. Consistent with the treatment of PE to estimate measurement uncertainties for noiseless injections, we take the matched filter SNR <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m4.1"><semantics id="S2.SS4.p1.4.m4.1a"><mi id="S2.SS4.p1.4.m4.1.1" xref="S2.SS4.p1.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m4.1b"><ci id="S2.SS4.p1.4.m4.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m4.1d">italic_ρ</annotation></semantics></math> to be a Gaussian with mean equal to the signal optimal SNR <math alttext="\bar{\rho}" class="ltx_Math" display="inline" id="S2.SS4.p1.5.m5.1"><semantics id="S2.SS4.p1.5.m5.1a"><mover accent="true" id="S2.SS4.p1.5.m5.1.1" xref="S2.SS4.p1.5.m5.1.1.cmml"><mi id="S2.SS4.p1.5.m5.1.1.2" xref="S2.SS4.p1.5.m5.1.1.2.cmml">ρ</mi><mo id="S2.SS4.p1.5.m5.1.1.1" xref="S2.SS4.p1.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.5.m5.1b"><apply id="S2.SS4.p1.5.m5.1.1.cmml" xref="S2.SS4.p1.5.m5.1.1"><ci id="S2.SS4.p1.5.m5.1.1.1.cmml" xref="S2.SS4.p1.5.m5.1.1.1">¯</ci><ci id="S2.SS4.p1.5.m5.1.1.2.cmml" xref="S2.SS4.p1.5.m5.1.1.2">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m5.1c">\bar{\rho}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m5.1d">over¯ start_ARG italic_ρ end_ARG</annotation></semantics></math> and standard deviation of unity. (Strictly, for detection of a signal of unknown phase we should consider a non-central chi-squared distribution with 2 degrees of freedom <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib102" title="">102</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib18" title="">18</a>]</cite>; however, in practice the difference in <math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S2.SS4.p1.6.m6.1"><semantics id="S2.SS4.p1.6.m6.1a"><msub id="S2.SS4.p1.6.m6.1.1" xref="S2.SS4.p1.6.m6.1.1.cmml"><mi id="S2.SS4.p1.6.m6.1.1.2" xref="S2.SS4.p1.6.m6.1.1.2.cmml">p</mi><mi id="S2.SS4.p1.6.m6.1.1.3" xref="S2.SS4.p1.6.m6.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.6.m6.1b"><apply id="S2.SS4.p1.6.m6.1.1.cmml" xref="S2.SS4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.1.1.1.cmml" xref="S2.SS4.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS4.p1.6.m6.1.1.2.cmml" xref="S2.SS4.p1.6.m6.1.1.2">𝑝</ci><ci id="S2.SS4.p1.6.m6.1.1.3.cmml" xref="S2.SS4.p1.6.m6.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.6.m6.1c">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.6.m6.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> relative to the simple Gaussian is small.) Thus, the probability of detection is the tail distribution function (complementary cumulative distribution function) of a unit Gaussian centered on <math alttext="\bar{\rho}" class="ltx_Math" display="inline" id="S2.SS4.p1.7.m7.1"><semantics id="S2.SS4.p1.7.m7.1a"><mover accent="true" id="S2.SS4.p1.7.m7.1.1" xref="S2.SS4.p1.7.m7.1.1.cmml"><mi id="S2.SS4.p1.7.m7.1.1.2" xref="S2.SS4.p1.7.m7.1.1.2.cmml">ρ</mi><mo id="S2.SS4.p1.7.m7.1.1.1" xref="S2.SS4.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.7.m7.1b"><apply id="S2.SS4.p1.7.m7.1.1.cmml" xref="S2.SS4.p1.7.m7.1.1"><ci id="S2.SS4.p1.7.m7.1.1.1.cmml" xref="S2.SS4.p1.7.m7.1.1.1">¯</ci><ci id="S2.SS4.p1.7.m7.1.1.2.cmml" xref="S2.SS4.p1.7.m7.1.1.2">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.7.m7.1c">\bar{\rho}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.7.m7.1d">over¯ start_ARG italic_ρ end_ARG</annotation></semantics></math>, evaluated at <math alttext="\rho_{\rm th}" class="ltx_Math" display="inline" id="S2.SS4.p1.8.m8.1"><semantics id="S2.SS4.p1.8.m8.1a"><msub id="S2.SS4.p1.8.m8.1.1" xref="S2.SS4.p1.8.m8.1.1.cmml"><mi id="S2.SS4.p1.8.m8.1.1.2" xref="S2.SS4.p1.8.m8.1.1.2.cmml">ρ</mi><mi id="S2.SS4.p1.8.m8.1.1.3" xref="S2.SS4.p1.8.m8.1.1.3.cmml">th</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.8.m8.1b"><apply id="S2.SS4.p1.8.m8.1.1.cmml" xref="S2.SS4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.8.m8.1.1.1.cmml" xref="S2.SS4.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS4.p1.8.m8.1.1.2.cmml" xref="S2.SS4.p1.8.m8.1.1.2">𝜌</ci><ci id="S2.SS4.p1.8.m8.1.1.3.cmml" xref="S2.SS4.p1.8.m8.1.1.3">th</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.8.m8.1c">\rho_{\rm th}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.8.m8.1d">italic_ρ start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.6">In a realistic situation, though, the true signal parameters are unknown: we do not have access to <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.1"><semantics id="S2.SS4.p2.1.m1.1a"><msub id="S2.SS4.p2.1.m1.1.1" xref="S2.SS4.p2.1.m1.1.1.cmml"><mi id="S2.SS4.p2.1.m1.1.1.2" xref="S2.SS4.p2.1.m1.1.1.2.cmml">p</mi><mi id="S2.SS4.p2.1.m1.1.1.3" xref="S2.SS4.p2.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.1.m1.1b"><apply id="S2.SS4.p2.1.m1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.1.m1.1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.p2.1.m1.1.1.2.cmml" xref="S2.SS4.p2.1.m1.1.1.2">𝑝</ci><ci id="S2.SS4.p2.1.m1.1.1.3.cmml" xref="S2.SS4.p2.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.1.m1.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> for these parameters, but can only obtain an estimate of it based on the PE results. We thus begin with order(<math alttext="10^{3}" class="ltx_Math" display="inline" id="S2.SS4.p2.2.m2.1"><semantics id="S2.SS4.p2.2.m2.1a"><msup id="S2.SS4.p2.2.m2.1.1" xref="S2.SS4.p2.2.m2.1.1.cmml"><mn id="S2.SS4.p2.2.m2.1.1.2" xref="S2.SS4.p2.2.m2.1.1.2.cmml">10</mn><mn id="S2.SS4.p2.2.m2.1.1.3" xref="S2.SS4.p2.2.m2.1.1.3.cmml">3</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.2.m2.1b"><apply id="S2.SS4.p2.2.m2.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.2.m2.1.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1">superscript</csymbol><cn id="S2.SS4.p2.2.m2.1.1.2.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.2">10</cn><cn id="S2.SS4.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.2.m2.1c">10^{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.2.m2.1d">10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>) posterior parameter samples for each event, obtained as described in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS2" title="II.2 Parameter Estimation ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.2</span></a>, and compute the optimal SNR for all samples. This enables us to validate the output of PE, as for a given event we expect a relatively narrow spread of posterior sample <math alttext="\bar{\rho}" class="ltx_Math" display="inline" id="S2.SS4.p2.3.m3.1"><semantics id="S2.SS4.p2.3.m3.1a"><mover accent="true" id="S2.SS4.p2.3.m3.1.1" xref="S2.SS4.p2.3.m3.1.1.cmml"><mi id="S2.SS4.p2.3.m3.1.1.2" xref="S2.SS4.p2.3.m3.1.1.2.cmml">ρ</mi><mo id="S2.SS4.p2.3.m3.1.1.1" xref="S2.SS4.p2.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.3.m3.1b"><apply id="S2.SS4.p2.3.m3.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1"><ci id="S2.SS4.p2.3.m3.1.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1.1">¯</ci><ci id="S2.SS4.p2.3.m3.1.1.2.cmml" xref="S2.SS4.p2.3.m3.1.1.2">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.3.m3.1c">\bar{\rho}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.3.m3.1d">over¯ start_ARG italic_ρ end_ARG</annotation></semantics></math>. However, for a small number of events we find samples at very high redshift (<math alttext="z>30" class="ltx_Math" display="inline" id="S2.SS4.p2.4.m4.1"><semantics id="S2.SS4.p2.4.m4.1a"><mrow id="S2.SS4.p2.4.m4.1.1" xref="S2.SS4.p2.4.m4.1.1.cmml"><mi id="S2.SS4.p2.4.m4.1.1.2" xref="S2.SS4.p2.4.m4.1.1.2.cmml">z</mi><mo id="S2.SS4.p2.4.m4.1.1.1" xref="S2.SS4.p2.4.m4.1.1.1.cmml">></mo><mn id="S2.SS4.p2.4.m4.1.1.3" xref="S2.SS4.p2.4.m4.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.4.m4.1b"><apply id="S2.SS4.p2.4.m4.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1"><gt id="S2.SS4.p2.4.m4.1.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1.1"></gt><ci id="S2.SS4.p2.4.m4.1.1.2.cmml" xref="S2.SS4.p2.4.m4.1.1.2">𝑧</ci><cn id="S2.SS4.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS4.p2.4.m4.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.4.m4.1c">z>30</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.4.m4.1d">italic_z > 30</annotation></semantics></math>) and/or large mass ratio (<math alttext="q>100" class="ltx_Math" display="inline" id="S2.SS4.p2.5.m5.1"><semantics id="S2.SS4.p2.5.m5.1a"><mrow id="S2.SS4.p2.5.m5.1.1" xref="S2.SS4.p2.5.m5.1.1.cmml"><mi id="S2.SS4.p2.5.m5.1.1.2" xref="S2.SS4.p2.5.m5.1.1.2.cmml">q</mi><mo id="S2.SS4.p2.5.m5.1.1.1" xref="S2.SS4.p2.5.m5.1.1.1.cmml">></mo><mn id="S2.SS4.p2.5.m5.1.1.3" xref="S2.SS4.p2.5.m5.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.5.m5.1b"><apply id="S2.SS4.p2.5.m5.1.1.cmml" xref="S2.SS4.p2.5.m5.1.1"><gt id="S2.SS4.p2.5.m5.1.1.1.cmml" xref="S2.SS4.p2.5.m5.1.1.1"></gt><ci id="S2.SS4.p2.5.m5.1.1.2.cmml" xref="S2.SS4.p2.5.m5.1.1.2">𝑞</ci><cn id="S2.SS4.p2.5.m5.1.1.3.cmml" type="integer" xref="S2.SS4.p2.5.m5.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.5.m5.1c">q>100</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.5.m5.1d">italic_q > 100</annotation></semantics></math>) having very low SNR (<math alttext="<4" class="ltx_Math" display="inline" id="S2.SS4.p2.6.m6.1"><semantics id="S2.SS4.p2.6.m6.1a"><mrow id="S2.SS4.p2.6.m6.1.1" xref="S2.SS4.p2.6.m6.1.1.cmml"><mi id="S2.SS4.p2.6.m6.1.1.2" xref="S2.SS4.p2.6.m6.1.1.2.cmml"></mi><mo id="S2.SS4.p2.6.m6.1.1.1" xref="S2.SS4.p2.6.m6.1.1.1.cmml"><</mo><mn id="S2.SS4.p2.6.m6.1.1.3" xref="S2.SS4.p2.6.m6.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.6.m6.1b"><apply id="S2.SS4.p2.6.m6.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1"><lt id="S2.SS4.p2.6.m6.1.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1"></lt><csymbol cd="latexml" id="S2.SS4.p2.6.m6.1.1.2.cmml" xref="S2.SS4.p2.6.m6.1.1.2">absent</csymbol><cn id="S2.SS4.p2.6.m6.1.1.3.cmml" type="integer" xref="S2.SS4.p2.6.m6.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.6.m6.1c"><4</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.6.m6.1d">< 4</annotation></semantics></math>). We interpret this as a failure of PE to converge to a posterior distribution reflecting the event’s true parameters rather than the prior. To ensure data quality, we calculate the median and standard deviation of sample optimal SNR for each event. Events with a median SNR below 7 or a standard deviation above 2 were discarded, accounting for less than 4% of the total dataset. Additionally, we filtered individual samples for the remaining events to remove those with an optimal SNR below 4.</p> </div> <div class="ltx_para" id="S2.SS4.p3"> <p class="ltx_p" id="S2.SS4.p3.6">To treat selection effects in our population analysis, we consider three distinct groups of source parameters: first, the KDE parameters <math alttext="\{x\}" class="ltx_Math" display="inline" id="S2.SS4.p3.1.m1.1"><semantics id="S2.SS4.p3.1.m1.1a"><mrow id="S2.SS4.p3.1.m1.1.2.2" xref="S2.SS4.p3.1.m1.1.2.1.cmml"><mo id="S2.SS4.p3.1.m1.1.2.2.1" stretchy="false" xref="S2.SS4.p3.1.m1.1.2.1.cmml">{</mo><mi id="S2.SS4.p3.1.m1.1.1" xref="S2.SS4.p3.1.m1.1.1.cmml">x</mi><mo id="S2.SS4.p3.1.m1.1.2.2.2" stretchy="false" xref="S2.SS4.p3.1.m1.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.1.m1.1b"><set id="S2.SS4.p3.1.m1.1.2.1.cmml" xref="S2.SS4.p3.1.m1.1.2.2"><ci id="S2.SS4.p3.1.m1.1.1.cmml" xref="S2.SS4.p3.1.m1.1.1">𝑥</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.1.m1.1c">\{x\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.1.m1.1d">{ italic_x }</annotation></semantics></math>, i.e. those of most physical interest in reconstructing the population distribution, here comprising the source-frame total mass and redshift. Quantifying the source distribution over <math alttext="M" class="ltx_Math" display="inline" id="S2.SS4.p3.2.m2.1"><semantics id="S2.SS4.p3.2.m2.1a"><mi id="S2.SS4.p3.2.m2.1.1" xref="S2.SS4.p3.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.2.m2.1b"><ci id="S2.SS4.p3.2.m2.1.1.cmml" xref="S2.SS4.p3.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.2.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.2.m2.1d">italic_M</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="S2.SS4.p3.3.m3.1"><semantics id="S2.SS4.p3.3.m3.1a"><mi id="S2.SS4.p3.3.m3.1.1" xref="S2.SS4.p3.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.3.m3.1b"><ci id="S2.SS4.p3.3.m3.1.1.cmml" xref="S2.SS4.p3.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.3.m3.1d">italic_z</annotation></semantics></math> is crucial to understand the formation and evolution of MBH binaries and their role in galaxy formation. Second, other intrinsic parameters <math alttext="\{\theta\}" class="ltx_Math" display="inline" id="S2.SS4.p3.4.m4.1"><semantics id="S2.SS4.p3.4.m4.1a"><mrow id="S2.SS4.p3.4.m4.1.2.2" xref="S2.SS4.p3.4.m4.1.2.1.cmml"><mo id="S2.SS4.p3.4.m4.1.2.2.1" stretchy="false" xref="S2.SS4.p3.4.m4.1.2.1.cmml">{</mo><mi id="S2.SS4.p3.4.m4.1.1" xref="S2.SS4.p3.4.m4.1.1.cmml">θ</mi><mo id="S2.SS4.p3.4.m4.1.2.2.2" stretchy="false" xref="S2.SS4.p3.4.m4.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.4.m4.1b"><set id="S2.SS4.p3.4.m4.1.2.1.cmml" xref="S2.SS4.p3.4.m4.1.2.2"><ci id="S2.SS4.p3.4.m4.1.1.cmml" xref="S2.SS4.p3.4.m4.1.1">𝜃</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.4.m4.1c">\{\theta\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.4.m4.1d">{ italic_θ }</annotation></semantics></math>, which may contain some astrophysically relevant information, as they may depend on the binary formation and evolution mechanisms, and may influence <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p3.5.m5.1"><semantics id="S2.SS4.p3.5.m5.1a"><msub id="S2.SS4.p3.5.m5.1.1" xref="S2.SS4.p3.5.m5.1.1.cmml"><mi id="S2.SS4.p3.5.m5.1.1.2" xref="S2.SS4.p3.5.m5.1.1.2.cmml">p</mi><mi id="S2.SS4.p3.5.m5.1.1.3" xref="S2.SS4.p3.5.m5.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.5.m5.1b"><apply id="S2.SS4.p3.5.m5.1.1.cmml" xref="S2.SS4.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS4.p3.5.m5.1.1.1.cmml" xref="S2.SS4.p3.5.m5.1.1">subscript</csymbol><ci id="S2.SS4.p3.5.m5.1.1.2.cmml" xref="S2.SS4.p3.5.m5.1.1.2">𝑝</ci><ci id="S2.SS4.p3.5.m5.1.1.3.cmml" xref="S2.SS4.p3.5.m5.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.5.m5.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.5.m5.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>: these include the mass ratio and component spins. Lastly, extrinsic parameters <math alttext="\{\psi\}" class="ltx_Math" display="inline" id="S2.SS4.p3.6.m6.1"><semantics id="S2.SS4.p3.6.m6.1a"><mrow id="S2.SS4.p3.6.m6.1.2.2" xref="S2.SS4.p3.6.m6.1.2.1.cmml"><mo id="S2.SS4.p3.6.m6.1.2.2.1" stretchy="false" xref="S2.SS4.p3.6.m6.1.2.1.cmml">{</mo><mi id="S2.SS4.p3.6.m6.1.1" xref="S2.SS4.p3.6.m6.1.1.cmml">ψ</mi><mo id="S2.SS4.p3.6.m6.1.2.2.2" stretchy="false" xref="S2.SS4.p3.6.m6.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.6.m6.1b"><set id="S2.SS4.p3.6.m6.1.2.1.cmml" xref="S2.SS4.p3.6.m6.1.2.2"><ci id="S2.SS4.p3.6.m6.1.1.cmml" xref="S2.SS4.p3.6.m6.1.1">𝜓</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.6.m6.1c">\{\psi\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.6.m6.1d">{ italic_ψ }</annotation></semantics></math>, which are expected to have uniform/isotropic or uninformative distributions from symmetry arguments: these comprise the source direction and orientation relative to the Solar System and the time of coalescence.</p> </div> <div class="ltx_para" id="S2.SS4.p4"> <p class="ltx_p" id="S2.SS4.p4.1">Taking a uniform/isotropic distribution of sources over the time and angular parameters, <math alttext="p(\psi)" class="ltx_Math" display="inline" id="S2.SS4.p4.1.m1.1"><semantics id="S2.SS4.p4.1.m1.1a"><mrow id="S2.SS4.p4.1.m1.1.2" xref="S2.SS4.p4.1.m1.1.2.cmml"><mi id="S2.SS4.p4.1.m1.1.2.2" xref="S2.SS4.p4.1.m1.1.2.2.cmml">p</mi><mo id="S2.SS4.p4.1.m1.1.2.1" xref="S2.SS4.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p4.1.m1.1.2.3.2" xref="S2.SS4.p4.1.m1.1.2.cmml"><mo id="S2.SS4.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS4.p4.1.m1.1.2.cmml">(</mo><mi id="S2.SS4.p4.1.m1.1.1" xref="S2.SS4.p4.1.m1.1.1.cmml">ψ</mi><mo id="S2.SS4.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS4.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p4.1.m1.1b"><apply id="S2.SS4.p4.1.m1.1.2.cmml" 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id="S2.E7.m1.7.7.1.1.1.1.7.cmml" xref="S2.E7.m1.7.7.1.1.1.1.7">𝜓</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.7c">p_{\rm det}(x,\theta)=\int p(\rho>\rho_{\rm th}|\bar{\rho}(x,\theta,\psi))p(% \psi)\,d^{n}\psi\,,</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.7d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ ) = ∫ italic_p ( italic_ρ > italic_ρ start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT | over¯ start_ARG italic_ρ end_ARG ( italic_x , italic_θ , italic_ψ ) ) italic_p ( italic_ψ ) italic_d start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_ψ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p4.4">i.e. the probability of matched filter SNR to be above threshold given the optimal SNR for a binary with fixed <math alttext="x,\theta" class="ltx_Math" display="inline" id="S2.SS4.p4.2.m1.2"><semantics id="S2.SS4.p4.2.m1.2a"><mrow id="S2.SS4.p4.2.m1.2.3.2" xref="S2.SS4.p4.2.m1.2.3.1.cmml"><mi id="S2.SS4.p4.2.m1.1.1" xref="S2.SS4.p4.2.m1.1.1.cmml">x</mi><mo id="S2.SS4.p4.2.m1.2.3.2.1" xref="S2.SS4.p4.2.m1.2.3.1.cmml">,</mo><mi id="S2.SS4.p4.2.m1.2.2" xref="S2.SS4.p4.2.m1.2.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p4.2.m1.2b"><list id="S2.SS4.p4.2.m1.2.3.1.cmml" xref="S2.SS4.p4.2.m1.2.3.2"><ci id="S2.SS4.p4.2.m1.1.1.cmml" xref="S2.SS4.p4.2.m1.1.1">𝑥</ci><ci id="S2.SS4.p4.2.m1.2.2.cmml" xref="S2.SS4.p4.2.m1.2.2">𝜃</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p4.2.m1.2c">x,\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p4.2.m1.2d">italic_x , italic_θ</annotation></semantics></math> but unknown <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS4.p4.3.m2.1"><semantics id="S2.SS4.p4.3.m2.1a"><mi id="S2.SS4.p4.3.m2.1.1" xref="S2.SS4.p4.3.m2.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p4.3.m2.1b"><ci id="S2.SS4.p4.3.m2.1.1.cmml" xref="S2.SS4.p4.3.m2.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p4.3.m2.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p4.3.m2.1d">italic_ψ</annotation></semantics></math>. We compute this probability for the redshift, masses and spins of any given posterior sample via Monte Carlo integration over <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS4.p4.4.m3.1"><semantics id="S2.SS4.p4.4.m3.1a"><mi id="S2.SS4.p4.4.m3.1.1" xref="S2.SS4.p4.4.m3.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p4.4.m3.1b"><ci id="S2.SS4.p4.4.m3.1.1.cmml" xref="S2.SS4.p4.4.m3.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p4.4.m3.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p4.4.m3.1d">italic_ψ</annotation></semantics></math>, where for each draw of angular extrinsic parameters we also rescale the luminosity distance by a random weak lensing error given by Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.E3" title="In II.2 Parameter Estimation ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">3</span></a>).</p> </div> <div class="ltx_para" id="S2.SS4.p5"> <p class="ltx_p" id="S2.SS4.p5.7">The astrophysical rate density of mergers over intrinsic parameters and redshift, <math alttext="R(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p5.1.m1.2"><semantics id="S2.SS4.p5.1.m1.2a"><mrow id="S2.SS4.p5.1.m1.2.3" xref="S2.SS4.p5.1.m1.2.3.cmml"><mi id="S2.SS4.p5.1.m1.2.3.2" xref="S2.SS4.p5.1.m1.2.3.2.cmml">R</mi><mo id="S2.SS4.p5.1.m1.2.3.1" xref="S2.SS4.p5.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.p5.1.m1.2.3.3.2" xref="S2.SS4.p5.1.m1.2.3.3.1.cmml"><mo id="S2.SS4.p5.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS4.p5.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS4.p5.1.m1.1.1" xref="S2.SS4.p5.1.m1.1.1.cmml">x</mi><mo 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to the rate density of detected events <math alttext="R_{\rm det}(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p5.2.m2.2"><semantics id="S2.SS4.p5.2.m2.2a"><mrow id="S2.SS4.p5.2.m2.2.3" xref="S2.SS4.p5.2.m2.2.3.cmml"><msub id="S2.SS4.p5.2.m2.2.3.2" xref="S2.SS4.p5.2.m2.2.3.2.cmml"><mi id="S2.SS4.p5.2.m2.2.3.2.2" xref="S2.SS4.p5.2.m2.2.3.2.2.cmml">R</mi><mi id="S2.SS4.p5.2.m2.2.3.2.3" xref="S2.SS4.p5.2.m2.2.3.2.3.cmml">det</mi></msub><mo id="S2.SS4.p5.2.m2.2.3.1" xref="S2.SS4.p5.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.p5.2.m2.2.3.3.2" xref="S2.SS4.p5.2.m2.2.3.3.1.cmml"><mo id="S2.SS4.p5.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS4.p5.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.SS4.p5.2.m2.1.1" xref="S2.SS4.p5.2.m2.1.1.cmml">x</mi><mo id="S2.SS4.p5.2.m2.2.3.3.2.2" xref="S2.SS4.p5.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p5.2.m2.2.2" xref="S2.SS4.p5.2.m2.2.2.cmml">θ</mi><mo id="S2.SS4.p5.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS4.p5.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.2.m2.2b"><apply id="S2.SS4.p5.2.m2.2.3.cmml" xref="S2.SS4.p5.2.m2.2.3"><times id="S2.SS4.p5.2.m2.2.3.1.cmml" xref="S2.SS4.p5.2.m2.2.3.1"></times><apply id="S2.SS4.p5.2.m2.2.3.2.cmml" xref="S2.SS4.p5.2.m2.2.3.2"><csymbol cd="ambiguous" id="S2.SS4.p5.2.m2.2.3.2.1.cmml" xref="S2.SS4.p5.2.m2.2.3.2">subscript</csymbol><ci id="S2.SS4.p5.2.m2.2.3.2.2.cmml" xref="S2.SS4.p5.2.m2.2.3.2.2">𝑅</ci><ci id="S2.SS4.p5.2.m2.2.3.2.3.cmml" xref="S2.SS4.p5.2.m2.2.3.2.3">det</ci></apply><interval closure="open" id="S2.SS4.p5.2.m2.2.3.3.1.cmml" xref="S2.SS4.p5.2.m2.2.3.3.2"><ci id="S2.SS4.p5.2.m2.1.1.cmml" xref="S2.SS4.p5.2.m2.1.1">𝑥</ci><ci id="S2.SS4.p5.2.m2.2.2.cmml" xref="S2.SS4.p5.2.m2.2.2">𝜃</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.2.m2.2c">R_{\rm det}(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.2.m2.2d">italic_R start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ )</annotation></semantics></math> via <math alttext="\mathbf{R_{\rm det}(x,\theta)=p_{\rm det}(x,\theta)R(x,\theta)}" class="ltx_Math" display="inline" id="S2.SS4.p5.3.m3.6"><semantics id="S2.SS4.p5.3.m3.6a"><mrow id="S2.SS4.p5.3.m3.6.7" xref="S2.SS4.p5.3.m3.6.7.cmml"><mrow id="S2.SS4.p5.3.m3.6.7.2" xref="S2.SS4.p5.3.m3.6.7.2.cmml"><msub id="S2.SS4.p5.3.m3.6.7.2.2" xref="S2.SS4.p5.3.m3.6.7.2.2.cmml"><mi id="S2.SS4.p5.3.m3.6.7.2.2.2" xref="S2.SS4.p5.3.m3.6.7.2.2.2.cmml">𝐑</mi><mi id="S2.SS4.p5.3.m3.6.7.2.2.3" xref="S2.SS4.p5.3.m3.6.7.2.2.3.cmml">det</mi></msub><mo id="S2.SS4.p5.3.m3.6.7.2.1" xref="S2.SS4.p5.3.m3.6.7.2.1.cmml">⁢</mo><mrow id="S2.SS4.p5.3.m3.6.7.2.3.2" xref="S2.SS4.p5.3.m3.6.7.2.3.1.cmml"><mo id="S2.SS4.p5.3.m3.6.7.2.3.2.1" stretchy="false" xref="S2.SS4.p5.3.m3.6.7.2.3.1.cmml">(</mo><mi id="S2.SS4.p5.3.m3.1.1" xref="S2.SS4.p5.3.m3.1.1.cmml">𝐱</mi><mo id="S2.SS4.p5.3.m3.6.7.2.3.2.2" xref="S2.SS4.p5.3.m3.6.7.2.3.1.cmml">,</mo><mi id="S2.SS4.p5.3.m3.2.2" xref="S2.SS4.p5.3.m3.2.2.cmml">θ</mi><mo id="S2.SS4.p5.3.m3.6.7.2.3.2.3" stretchy="false" xref="S2.SS4.p5.3.m3.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p5.3.m3.6.7.1" xref="S2.SS4.p5.3.m3.6.7.1.cmml">=</mo><mrow id="S2.SS4.p5.3.m3.6.7.3" xref="S2.SS4.p5.3.m3.6.7.3.cmml"><msub id="S2.SS4.p5.3.m3.6.7.3.2" xref="S2.SS4.p5.3.m3.6.7.3.2.cmml"><mi id="S2.SS4.p5.3.m3.6.7.3.2.2" xref="S2.SS4.p5.3.m3.6.7.3.2.2.cmml">𝐩</mi><mi id="S2.SS4.p5.3.m3.6.7.3.2.3" xref="S2.SS4.p5.3.m3.6.7.3.2.3.cmml">det</mi></msub><mo id="S2.SS4.p5.3.m3.6.7.3.1" xref="S2.SS4.p5.3.m3.6.7.3.1.cmml">⁢</mo><mrow id="S2.SS4.p5.3.m3.6.7.3.3.2" xref="S2.SS4.p5.3.m3.6.7.3.3.1.cmml"><mo id="S2.SS4.p5.3.m3.6.7.3.3.2.1" stretchy="false" xref="S2.SS4.p5.3.m3.6.7.3.3.1.cmml">(</mo><mi id="S2.SS4.p5.3.m3.3.3" xref="S2.SS4.p5.3.m3.3.3.cmml">𝐱</mi><mo id="S2.SS4.p5.3.m3.6.7.3.3.2.2" xref="S2.SS4.p5.3.m3.6.7.3.3.1.cmml">,</mo><mi id="S2.SS4.p5.3.m3.4.4" xref="S2.SS4.p5.3.m3.4.4.cmml">θ</mi><mo id="S2.SS4.p5.3.m3.6.7.3.3.2.3" stretchy="false" xref="S2.SS4.p5.3.m3.6.7.3.3.1.cmml">)</mo></mrow><mo id="S2.SS4.p5.3.m3.6.7.3.1a" xref="S2.SS4.p5.3.m3.6.7.3.1.cmml">⁢</mo><mi id="S2.SS4.p5.3.m3.6.7.3.4" xref="S2.SS4.p5.3.m3.6.7.3.4.cmml">𝐑</mi><mo id="S2.SS4.p5.3.m3.6.7.3.1b" xref="S2.SS4.p5.3.m3.6.7.3.1.cmml">⁢</mo><mrow id="S2.SS4.p5.3.m3.6.7.3.5.2" xref="S2.SS4.p5.3.m3.6.7.3.5.1.cmml"><mo id="S2.SS4.p5.3.m3.6.7.3.5.2.1" stretchy="false" xref="S2.SS4.p5.3.m3.6.7.3.5.1.cmml">(</mo><mi id="S2.SS4.p5.3.m3.5.5" xref="S2.SS4.p5.3.m3.5.5.cmml">𝐱</mi><mo id="S2.SS4.p5.3.m3.6.7.3.5.2.2" xref="S2.SS4.p5.3.m3.6.7.3.5.1.cmml">,</mo><mi id="S2.SS4.p5.3.m3.6.6" xref="S2.SS4.p5.3.m3.6.6.cmml">θ</mi><mo id="S2.SS4.p5.3.m3.6.7.3.5.2.3" stretchy="false" xref="S2.SS4.p5.3.m3.6.7.3.5.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.3.m3.6b"><apply id="S2.SS4.p5.3.m3.6.7.cmml" xref="S2.SS4.p5.3.m3.6.7"><eq id="S2.SS4.p5.3.m3.6.7.1.cmml" xref="S2.SS4.p5.3.m3.6.7.1"></eq><apply id="S2.SS4.p5.3.m3.6.7.2.cmml" xref="S2.SS4.p5.3.m3.6.7.2"><times id="S2.SS4.p5.3.m3.6.7.2.1.cmml" xref="S2.SS4.p5.3.m3.6.7.2.1"></times><apply id="S2.SS4.p5.3.m3.6.7.2.2.cmml" xref="S2.SS4.p5.3.m3.6.7.2.2"><csymbol cd="ambiguous" id="S2.SS4.p5.3.m3.6.7.2.2.1.cmml" xref="S2.SS4.p5.3.m3.6.7.2.2">subscript</csymbol><ci id="S2.SS4.p5.3.m3.6.7.2.2.2.cmml" xref="S2.SS4.p5.3.m3.6.7.2.2.2">𝐑</ci><ci id="S2.SS4.p5.3.m3.6.7.2.2.3.cmml" xref="S2.SS4.p5.3.m3.6.7.2.2.3">det</ci></apply><interval closure="open" id="S2.SS4.p5.3.m3.6.7.2.3.1.cmml" xref="S2.SS4.p5.3.m3.6.7.2.3.2"><ci id="S2.SS4.p5.3.m3.1.1.cmml" xref="S2.SS4.p5.3.m3.1.1">𝐱</ci><ci id="S2.SS4.p5.3.m3.2.2.cmml" xref="S2.SS4.p5.3.m3.2.2">𝜃</ci></interval></apply><apply id="S2.SS4.p5.3.m3.6.7.3.cmml" xref="S2.SS4.p5.3.m3.6.7.3"><times id="S2.SS4.p5.3.m3.6.7.3.1.cmml" xref="S2.SS4.p5.3.m3.6.7.3.1"></times><apply id="S2.SS4.p5.3.m3.6.7.3.2.cmml" xref="S2.SS4.p5.3.m3.6.7.3.2"><csymbol cd="ambiguous" id="S2.SS4.p5.3.m3.6.7.3.2.1.cmml" xref="S2.SS4.p5.3.m3.6.7.3.2">subscript</csymbol><ci id="S2.SS4.p5.3.m3.6.7.3.2.2.cmml" xref="S2.SS4.p5.3.m3.6.7.3.2.2">𝐩</ci><ci id="S2.SS4.p5.3.m3.6.7.3.2.3.cmml" xref="S2.SS4.p5.3.m3.6.7.3.2.3">det</ci></apply><interval closure="open" id="S2.SS4.p5.3.m3.6.7.3.3.1.cmml" xref="S2.SS4.p5.3.m3.6.7.3.3.2"><ci id="S2.SS4.p5.3.m3.3.3.cmml" xref="S2.SS4.p5.3.m3.3.3">𝐱</ci><ci id="S2.SS4.p5.3.m3.4.4.cmml" xref="S2.SS4.p5.3.m3.4.4">𝜃</ci></interval><ci id="S2.SS4.p5.3.m3.6.7.3.4.cmml" xref="S2.SS4.p5.3.m3.6.7.3.4">𝐑</ci><interval closure="open" id="S2.SS4.p5.3.m3.6.7.3.5.1.cmml" xref="S2.SS4.p5.3.m3.6.7.3.5.2"><ci id="S2.SS4.p5.3.m3.5.5.cmml" xref="S2.SS4.p5.3.m3.5.5">𝐱</ci><ci id="S2.SS4.p5.3.m3.6.6.cmml" xref="S2.SS4.p5.3.m3.6.6">𝜃</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.3.m3.6c">\mathbf{R_{\rm det}(x,\theta)=p_{\rm det}(x,\theta)R(x,\theta)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.3.m3.6d">bold_R start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( bold_x , italic_θ ) = bold_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( bold_x , italic_θ ) bold_R ( bold_x , italic_θ )</annotation></semantics></math>. The astrophysical density over <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.p5.4.m4.1"><semantics id="S2.SS4.p5.4.m4.1a"><mi id="S2.SS4.p5.4.m4.1.1" xref="S2.SS4.p5.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.4.m4.1b"><ci id="S2.SS4.p5.4.m4.1.1.cmml" xref="S2.SS4.p5.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.4.m4.1d">italic_x</annotation></semantics></math> requires marginalization over <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS4.p5.5.m5.1"><semantics id="S2.SS4.p5.5.m5.1a"><mi id="S2.SS4.p5.5.m5.1.1" xref="S2.SS4.p5.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.5.m5.1b"><ci id="S2.SS4.p5.5.m5.1.1.cmml" xref="S2.SS4.p5.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.5.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.5.m5.1d">italic_θ</annotation></semantics></math> via <math alttext="R(x)=\int R(x,\theta)\,d^{n}\theta" class="ltx_Math" display="inline" id="S2.SS4.p5.6.m6.3"><semantics id="S2.SS4.p5.6.m6.3a"><mrow id="S2.SS4.p5.6.m6.3.4" xref="S2.SS4.p5.6.m6.3.4.cmml"><mrow id="S2.SS4.p5.6.m6.3.4.2" xref="S2.SS4.p5.6.m6.3.4.2.cmml"><mi id="S2.SS4.p5.6.m6.3.4.2.2" xref="S2.SS4.p5.6.m6.3.4.2.2.cmml">R</mi><mo id="S2.SS4.p5.6.m6.3.4.2.1" xref="S2.SS4.p5.6.m6.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS4.p5.6.m6.3.4.2.3.2" xref="S2.SS4.p5.6.m6.3.4.2.cmml"><mo id="S2.SS4.p5.6.m6.3.4.2.3.2.1" stretchy="false" xref="S2.SS4.p5.6.m6.3.4.2.cmml">(</mo><mi id="S2.SS4.p5.6.m6.1.1" xref="S2.SS4.p5.6.m6.1.1.cmml">x</mi><mo id="S2.SS4.p5.6.m6.3.4.2.3.2.2" stretchy="false" xref="S2.SS4.p5.6.m6.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p5.6.m6.3.4.1" rspace="0.111em" xref="S2.SS4.p5.6.m6.3.4.1.cmml">=</mo><mrow id="S2.SS4.p5.6.m6.3.4.3" xref="S2.SS4.p5.6.m6.3.4.3.cmml"><mo id="S2.SS4.p5.6.m6.3.4.3.1" xref="S2.SS4.p5.6.m6.3.4.3.1.cmml">∫</mo><mrow id="S2.SS4.p5.6.m6.3.4.3.2" xref="S2.SS4.p5.6.m6.3.4.3.2.cmml"><mi id="S2.SS4.p5.6.m6.3.4.3.2.2" xref="S2.SS4.p5.6.m6.3.4.3.2.2.cmml">R</mi><mo id="S2.SS4.p5.6.m6.3.4.3.2.1" xref="S2.SS4.p5.6.m6.3.4.3.2.1.cmml">⁢</mo><mrow id="S2.SS4.p5.6.m6.3.4.3.2.3.2" xref="S2.SS4.p5.6.m6.3.4.3.2.3.1.cmml"><mo id="S2.SS4.p5.6.m6.3.4.3.2.3.2.1" stretchy="false" xref="S2.SS4.p5.6.m6.3.4.3.2.3.1.cmml">(</mo><mi id="S2.SS4.p5.6.m6.2.2" xref="S2.SS4.p5.6.m6.2.2.cmml">x</mi><mo id="S2.SS4.p5.6.m6.3.4.3.2.3.2.2" xref="S2.SS4.p5.6.m6.3.4.3.2.3.1.cmml">,</mo><mi id="S2.SS4.p5.6.m6.3.3" xref="S2.SS4.p5.6.m6.3.3.cmml">θ</mi><mo id="S2.SS4.p5.6.m6.3.4.3.2.3.2.3" stretchy="false" xref="S2.SS4.p5.6.m6.3.4.3.2.3.1.cmml">)</mo></mrow><mo id="S2.SS4.p5.6.m6.3.4.3.2.1a" lspace="0.170em" xref="S2.SS4.p5.6.m6.3.4.3.2.1.cmml">⁢</mo><msup id="S2.SS4.p5.6.m6.3.4.3.2.4" xref="S2.SS4.p5.6.m6.3.4.3.2.4.cmml"><mi id="S2.SS4.p5.6.m6.3.4.3.2.4.2" xref="S2.SS4.p5.6.m6.3.4.3.2.4.2.cmml">d</mi><mi id="S2.SS4.p5.6.m6.3.4.3.2.4.3" xref="S2.SS4.p5.6.m6.3.4.3.2.4.3.cmml">n</mi></msup><mo id="S2.SS4.p5.6.m6.3.4.3.2.1b" xref="S2.SS4.p5.6.m6.3.4.3.2.1.cmml">⁢</mo><mi id="S2.SS4.p5.6.m6.3.4.3.2.5" xref="S2.SS4.p5.6.m6.3.4.3.2.5.cmml">θ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.6.m6.3b"><apply id="S2.SS4.p5.6.m6.3.4.cmml" xref="S2.SS4.p5.6.m6.3.4"><eq id="S2.SS4.p5.6.m6.3.4.1.cmml" xref="S2.SS4.p5.6.m6.3.4.1"></eq><apply id="S2.SS4.p5.6.m6.3.4.2.cmml" xref="S2.SS4.p5.6.m6.3.4.2"><times id="S2.SS4.p5.6.m6.3.4.2.1.cmml" xref="S2.SS4.p5.6.m6.3.4.2.1"></times><ci id="S2.SS4.p5.6.m6.3.4.2.2.cmml" xref="S2.SS4.p5.6.m6.3.4.2.2">𝑅</ci><ci id="S2.SS4.p5.6.m6.1.1.cmml" xref="S2.SS4.p5.6.m6.1.1">𝑥</ci></apply><apply id="S2.SS4.p5.6.m6.3.4.3.cmml" xref="S2.SS4.p5.6.m6.3.4.3"><int id="S2.SS4.p5.6.m6.3.4.3.1.cmml" xref="S2.SS4.p5.6.m6.3.4.3.1"></int><apply id="S2.SS4.p5.6.m6.3.4.3.2.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2"><times id="S2.SS4.p5.6.m6.3.4.3.2.1.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.1"></times><ci id="S2.SS4.p5.6.m6.3.4.3.2.2.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.2">𝑅</ci><interval closure="open" id="S2.SS4.p5.6.m6.3.4.3.2.3.1.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.3.2"><ci id="S2.SS4.p5.6.m6.2.2.cmml" xref="S2.SS4.p5.6.m6.2.2">𝑥</ci><ci id="S2.SS4.p5.6.m6.3.3.cmml" xref="S2.SS4.p5.6.m6.3.3">𝜃</ci></interval><apply id="S2.SS4.p5.6.m6.3.4.3.2.4.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.4"><csymbol cd="ambiguous" id="S2.SS4.p5.6.m6.3.4.3.2.4.1.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.4">superscript</csymbol><ci id="S2.SS4.p5.6.m6.3.4.3.2.4.2.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.4.2">𝑑</ci><ci id="S2.SS4.p5.6.m6.3.4.3.2.4.3.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.4.3">𝑛</ci></apply><ci id="S2.SS4.p5.6.m6.3.4.3.2.5.cmml" xref="S2.SS4.p5.6.m6.3.4.3.2.5">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.6.m6.3c">R(x)=\int R(x,\theta)\,d^{n}\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.6.m6.3d">italic_R ( italic_x ) = ∫ italic_R ( italic_x , italic_θ ) italic_d start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_θ</annotation></semantics></math>. However, the density of <em class="ltx_emph ltx_font_italic" id="S2.SS4.p5.7.1">detected</em> events over <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.p5.7.m7.1"><semantics id="S2.SS4.p5.7.m7.1a"><mi id="S2.SS4.p5.7.m7.1.1" xref="S2.SS4.p5.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.7.m7.1b"><ci id="S2.SS4.p5.7.m7.1.1.cmml" xref="S2.SS4.p5.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.7.m7.1d">italic_x</annotation></semantics></math>, which we have direct access to via the KDE, is given by</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="R_{\rm det}(x)=\int R(x,\theta)p_{\rm det}(x,\theta)\,d^{n}\theta\,." class="ltx_Math" display="block" id="S2.E8.m1.6"><semantics id="S2.E8.m1.6a"><mrow id="S2.E8.m1.6.6.1" xref="S2.E8.m1.6.6.1.1.cmml"><mrow id="S2.E8.m1.6.6.1.1" xref="S2.E8.m1.6.6.1.1.cmml"><mrow id="S2.E8.m1.6.6.1.1.2" xref="S2.E8.m1.6.6.1.1.2.cmml"><msub id="S2.E8.m1.6.6.1.1.2.2" xref="S2.E8.m1.6.6.1.1.2.2.cmml"><mi id="S2.E8.m1.6.6.1.1.2.2.2" xref="S2.E8.m1.6.6.1.1.2.2.2.cmml">R</mi><mi id="S2.E8.m1.6.6.1.1.2.2.3" xref="S2.E8.m1.6.6.1.1.2.2.3.cmml">det</mi></msub><mo id="S2.E8.m1.6.6.1.1.2.1" xref="S2.E8.m1.6.6.1.1.2.1.cmml">⁢</mo><mrow id="S2.E8.m1.6.6.1.1.2.3.2" xref="S2.E8.m1.6.6.1.1.2.cmml"><mo id="S2.E8.m1.6.6.1.1.2.3.2.1" stretchy="false" xref="S2.E8.m1.6.6.1.1.2.cmml">(</mo><mi id="S2.E8.m1.1.1" xref="S2.E8.m1.1.1.cmml">x</mi><mo id="S2.E8.m1.6.6.1.1.2.3.2.2" stretchy="false" xref="S2.E8.m1.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E8.m1.6.6.1.1.1" rspace="0.111em" xref="S2.E8.m1.6.6.1.1.1.cmml">=</mo><mrow id="S2.E8.m1.6.6.1.1.3" xref="S2.E8.m1.6.6.1.1.3.cmml"><mo id="S2.E8.m1.6.6.1.1.3.1" xref="S2.E8.m1.6.6.1.1.3.1.cmml">∫</mo><mrow id="S2.E8.m1.6.6.1.1.3.2" xref="S2.E8.m1.6.6.1.1.3.2.cmml"><mi id="S2.E8.m1.6.6.1.1.3.2.2" xref="S2.E8.m1.6.6.1.1.3.2.2.cmml">R</mi><mo id="S2.E8.m1.6.6.1.1.3.2.1" xref="S2.E8.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.E8.m1.6.6.1.1.3.2.3.2" xref="S2.E8.m1.6.6.1.1.3.2.3.1.cmml"><mo id="S2.E8.m1.6.6.1.1.3.2.3.2.1" stretchy="false" xref="S2.E8.m1.6.6.1.1.3.2.3.1.cmml">(</mo><mi id="S2.E8.m1.2.2" xref="S2.E8.m1.2.2.cmml">x</mi><mo id="S2.E8.m1.6.6.1.1.3.2.3.2.2" xref="S2.E8.m1.6.6.1.1.3.2.3.1.cmml">,</mo><mi id="S2.E8.m1.3.3" xref="S2.E8.m1.3.3.cmml">θ</mi><mo id="S2.E8.m1.6.6.1.1.3.2.3.2.3" stretchy="false" xref="S2.E8.m1.6.6.1.1.3.2.3.1.cmml">)</mo></mrow><mo id="S2.E8.m1.6.6.1.1.3.2.1a" xref="S2.E8.m1.6.6.1.1.3.2.1.cmml">⁢</mo><msub id="S2.E8.m1.6.6.1.1.3.2.4" xref="S2.E8.m1.6.6.1.1.3.2.4.cmml"><mi id="S2.E8.m1.6.6.1.1.3.2.4.2" xref="S2.E8.m1.6.6.1.1.3.2.4.2.cmml">p</mi><mi id="S2.E8.m1.6.6.1.1.3.2.4.3" xref="S2.E8.m1.6.6.1.1.3.2.4.3.cmml">det</mi></msub><mo id="S2.E8.m1.6.6.1.1.3.2.1b" xref="S2.E8.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.E8.m1.6.6.1.1.3.2.5.2" xref="S2.E8.m1.6.6.1.1.3.2.5.1.cmml"><mo id="S2.E8.m1.6.6.1.1.3.2.5.2.1" stretchy="false" xref="S2.E8.m1.6.6.1.1.3.2.5.1.cmml">(</mo><mi id="S2.E8.m1.4.4" xref="S2.E8.m1.4.4.cmml">x</mi><mo id="S2.E8.m1.6.6.1.1.3.2.5.2.2" xref="S2.E8.m1.6.6.1.1.3.2.5.1.cmml">,</mo><mi id="S2.E8.m1.5.5" xref="S2.E8.m1.5.5.cmml">θ</mi><mo id="S2.E8.m1.6.6.1.1.3.2.5.2.3" stretchy="false" xref="S2.E8.m1.6.6.1.1.3.2.5.1.cmml">)</mo></mrow><mo id="S2.E8.m1.6.6.1.1.3.2.1c" lspace="0.170em" xref="S2.E8.m1.6.6.1.1.3.2.1.cmml">⁢</mo><msup id="S2.E8.m1.6.6.1.1.3.2.6" xref="S2.E8.m1.6.6.1.1.3.2.6.cmml"><mi id="S2.E8.m1.6.6.1.1.3.2.6.2" xref="S2.E8.m1.6.6.1.1.3.2.6.2.cmml">d</mi><mi id="S2.E8.m1.6.6.1.1.3.2.6.3" xref="S2.E8.m1.6.6.1.1.3.2.6.3.cmml">n</mi></msup><mo id="S2.E8.m1.6.6.1.1.3.2.1d" xref="S2.E8.m1.6.6.1.1.3.2.1.cmml">⁢</mo><mi id="S2.E8.m1.6.6.1.1.3.2.7" xref="S2.E8.m1.6.6.1.1.3.2.7.cmml">θ</mi></mrow></mrow></mrow><mo id="S2.E8.m1.6.6.1.2" lspace="0.170em" xref="S2.E8.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.m1.6b"><apply id="S2.E8.m1.6.6.1.1.cmml" xref="S2.E8.m1.6.6.1"><eq id="S2.E8.m1.6.6.1.1.1.cmml" xref="S2.E8.m1.6.6.1.1.1"></eq><apply id="S2.E8.m1.6.6.1.1.2.cmml" xref="S2.E8.m1.6.6.1.1.2"><times id="S2.E8.m1.6.6.1.1.2.1.cmml" xref="S2.E8.m1.6.6.1.1.2.1"></times><apply id="S2.E8.m1.6.6.1.1.2.2.cmml" xref="S2.E8.m1.6.6.1.1.2.2"><csymbol cd="ambiguous" id="S2.E8.m1.6.6.1.1.2.2.1.cmml" xref="S2.E8.m1.6.6.1.1.2.2">subscript</csymbol><ci id="S2.E8.m1.6.6.1.1.2.2.2.cmml" xref="S2.E8.m1.6.6.1.1.2.2.2">𝑅</ci><ci id="S2.E8.m1.6.6.1.1.2.2.3.cmml" xref="S2.E8.m1.6.6.1.1.2.2.3">det</ci></apply><ci id="S2.E8.m1.1.1.cmml" xref="S2.E8.m1.1.1">𝑥</ci></apply><apply id="S2.E8.m1.6.6.1.1.3.cmml" xref="S2.E8.m1.6.6.1.1.3"><int id="S2.E8.m1.6.6.1.1.3.1.cmml" xref="S2.E8.m1.6.6.1.1.3.1"></int><apply id="S2.E8.m1.6.6.1.1.3.2.cmml" xref="S2.E8.m1.6.6.1.1.3.2"><times id="S2.E8.m1.6.6.1.1.3.2.1.cmml" xref="S2.E8.m1.6.6.1.1.3.2.1"></times><ci id="S2.E8.m1.6.6.1.1.3.2.2.cmml" xref="S2.E8.m1.6.6.1.1.3.2.2">𝑅</ci><interval closure="open" id="S2.E8.m1.6.6.1.1.3.2.3.1.cmml" xref="S2.E8.m1.6.6.1.1.3.2.3.2"><ci id="S2.E8.m1.2.2.cmml" xref="S2.E8.m1.2.2">𝑥</ci><ci id="S2.E8.m1.3.3.cmml" xref="S2.E8.m1.3.3">𝜃</ci></interval><apply id="S2.E8.m1.6.6.1.1.3.2.4.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4"><csymbol cd="ambiguous" id="S2.E8.m1.6.6.1.1.3.2.4.1.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4">subscript</csymbol><ci id="S2.E8.m1.6.6.1.1.3.2.4.2.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4.2">𝑝</ci><ci id="S2.E8.m1.6.6.1.1.3.2.4.3.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4.3">det</ci></apply><interval closure="open" id="S2.E8.m1.6.6.1.1.3.2.5.1.cmml" xref="S2.E8.m1.6.6.1.1.3.2.5.2"><ci id="S2.E8.m1.4.4.cmml" xref="S2.E8.m1.4.4">𝑥</ci><ci id="S2.E8.m1.5.5.cmml" xref="S2.E8.m1.5.5">𝜃</ci></interval><apply id="S2.E8.m1.6.6.1.1.3.2.6.cmml" xref="S2.E8.m1.6.6.1.1.3.2.6"><csymbol cd="ambiguous" id="S2.E8.m1.6.6.1.1.3.2.6.1.cmml" xref="S2.E8.m1.6.6.1.1.3.2.6">superscript</csymbol><ci id="S2.E8.m1.6.6.1.1.3.2.6.2.cmml" xref="S2.E8.m1.6.6.1.1.3.2.6.2">𝑑</ci><ci id="S2.E8.m1.6.6.1.1.3.2.6.3.cmml" xref="S2.E8.m1.6.6.1.1.3.2.6.3">𝑛</ci></apply><ci id="S2.E8.m1.6.6.1.1.3.2.7.cmml" xref="S2.E8.m1.6.6.1.1.3.2.7">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.6c">R_{\rm det}(x)=\int R(x,\theta)p_{\rm det}(x,\theta)\,d^{n}\theta\,.</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.6d">italic_R start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x ) = ∫ italic_R ( italic_x , italic_θ ) italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ ) italic_d start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT 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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p5.10">Here, the joint density <math alttext="R(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p5.8.m1.2"><semantics id="S2.SS4.p5.8.m1.2a"><mrow id="S2.SS4.p5.8.m1.2.3" xref="S2.SS4.p5.8.m1.2.3.cmml"><mi id="S2.SS4.p5.8.m1.2.3.2" xref="S2.SS4.p5.8.m1.2.3.2.cmml">R</mi><mo id="S2.SS4.p5.8.m1.2.3.1" xref="S2.SS4.p5.8.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.p5.8.m1.2.3.3.2" xref="S2.SS4.p5.8.m1.2.3.3.1.cmml"><mo id="S2.SS4.p5.8.m1.2.3.3.2.1" stretchy="false" xref="S2.SS4.p5.8.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS4.p5.8.m1.1.1" xref="S2.SS4.p5.8.m1.1.1.cmml">x</mi><mo id="S2.SS4.p5.8.m1.2.3.3.2.2" xref="S2.SS4.p5.8.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p5.8.m1.2.2" xref="S2.SS4.p5.8.m1.2.2.cmml">θ</mi><mo id="S2.SS4.p5.8.m1.2.3.3.2.3" stretchy="false" xref="S2.SS4.p5.8.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.8.m1.2b"><apply id="S2.SS4.p5.8.m1.2.3.cmml" xref="S2.SS4.p5.8.m1.2.3"><times id="S2.SS4.p5.8.m1.2.3.1.cmml" xref="S2.SS4.p5.8.m1.2.3.1"></times><ci id="S2.SS4.p5.8.m1.2.3.2.cmml" xref="S2.SS4.p5.8.m1.2.3.2">𝑅</ci><interval closure="open" id="S2.SS4.p5.8.m1.2.3.3.1.cmml" xref="S2.SS4.p5.8.m1.2.3.3.2"><ci id="S2.SS4.p5.8.m1.1.1.cmml" xref="S2.SS4.p5.8.m1.1.1">𝑥</ci><ci id="S2.SS4.p5.8.m1.2.2.cmml" xref="S2.SS4.p5.8.m1.2.2">𝜃</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.8.m1.2c">R(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.8.m1.2d">italic_R ( italic_x , italic_θ )</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS4.p5.10.1">a priori</em> unknown, and we are currently unable to reconstruct the full <em class="ltx_emph ltx_font_italic" id="S2.SS4.p5.10.2">detected</em> distribution over <math alttext="x,\theta" class="ltx_Math" display="inline" id="S2.SS4.p5.9.m2.2"><semantics id="S2.SS4.p5.9.m2.2a"><mrow id="S2.SS4.p5.9.m2.2.3.2" xref="S2.SS4.p5.9.m2.2.3.1.cmml"><mi id="S2.SS4.p5.9.m2.1.1" xref="S2.SS4.p5.9.m2.1.1.cmml">x</mi><mo id="S2.SS4.p5.9.m2.2.3.2.1" xref="S2.SS4.p5.9.m2.2.3.1.cmml">,</mo><mi id="S2.SS4.p5.9.m2.2.2" xref="S2.SS4.p5.9.m2.2.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.9.m2.2b"><list id="S2.SS4.p5.9.m2.2.3.1.cmml" xref="S2.SS4.p5.9.m2.2.3.2"><ci id="S2.SS4.p5.9.m2.1.1.cmml" xref="S2.SS4.p5.9.m2.1.1">𝑥</ci><ci id="S2.SS4.p5.9.m2.2.2.cmml" xref="S2.SS4.p5.9.m2.2.2">𝜃</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.9.m2.2c">x,\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.9.m2.2d">italic_x , italic_θ</annotation></semantics></math> due to its high dimensionality. To proceed, either some simplifying assumptions are needed (in the absence of more sophisticated methods for multi-dimensional KDE), or a strategy which allows us to estimate <math alttext="R(x)" class="ltx_Math" display="inline" id="S2.SS4.p5.10.m3.1"><semantics id="S2.SS4.p5.10.m3.1a"><mrow id="S2.SS4.p5.10.m3.1.2" xref="S2.SS4.p5.10.m3.1.2.cmml"><mi id="S2.SS4.p5.10.m3.1.2.2" xref="S2.SS4.p5.10.m3.1.2.2.cmml">R</mi><mo id="S2.SS4.p5.10.m3.1.2.1" xref="S2.SS4.p5.10.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p5.10.m3.1.2.3.2" xref="S2.SS4.p5.10.m3.1.2.cmml"><mo id="S2.SS4.p5.10.m3.1.2.3.2.1" stretchy="false" xref="S2.SS4.p5.10.m3.1.2.cmml">(</mo><mi id="S2.SS4.p5.10.m3.1.1" xref="S2.SS4.p5.10.m3.1.1.cmml">x</mi><mo id="S2.SS4.p5.10.m3.1.2.3.2.2" stretchy="false" xref="S2.SS4.p5.10.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.10.m3.1b"><apply id="S2.SS4.p5.10.m3.1.2.cmml" xref="S2.SS4.p5.10.m3.1.2"><times id="S2.SS4.p5.10.m3.1.2.1.cmml" xref="S2.SS4.p5.10.m3.1.2.1"></times><ci id="S2.SS4.p5.10.m3.1.2.2.cmml" xref="S2.SS4.p5.10.m3.1.2.2">𝑅</ci><ci id="S2.SS4.p5.10.m3.1.1.cmml" xref="S2.SS4.p5.10.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.10.m3.1c">R(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.10.m3.1d">italic_R ( italic_x )</annotation></semantics></math> directly without considering the joint density.</p> </div> <div class="ltx_para" id="S2.SS4.p6"> <p class="ltx_p" id="S2.SS4.p6.2">In previous works, it was assumed that the joint distribution factorizes as <math alttext="R(x,\theta)=R(x)p_{\rm pop}(\theta)" class="ltx_Math" display="inline" id="S2.SS4.p6.1.m1.4"><semantics id="S2.SS4.p6.1.m1.4a"><mrow id="S2.SS4.p6.1.m1.4.5" xref="S2.SS4.p6.1.m1.4.5.cmml"><mrow id="S2.SS4.p6.1.m1.4.5.2" xref="S2.SS4.p6.1.m1.4.5.2.cmml"><mi id="S2.SS4.p6.1.m1.4.5.2.2" xref="S2.SS4.p6.1.m1.4.5.2.2.cmml">R</mi><mo id="S2.SS4.p6.1.m1.4.5.2.1" xref="S2.SS4.p6.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S2.SS4.p6.1.m1.4.5.2.3.2" xref="S2.SS4.p6.1.m1.4.5.2.3.1.cmml"><mo id="S2.SS4.p6.1.m1.4.5.2.3.2.1" stretchy="false" xref="S2.SS4.p6.1.m1.4.5.2.3.1.cmml">(</mo><mi id="S2.SS4.p6.1.m1.1.1" xref="S2.SS4.p6.1.m1.1.1.cmml">x</mi><mo id="S2.SS4.p6.1.m1.4.5.2.3.2.2" xref="S2.SS4.p6.1.m1.4.5.2.3.1.cmml">,</mo><mi id="S2.SS4.p6.1.m1.2.2" xref="S2.SS4.p6.1.m1.2.2.cmml">θ</mi><mo id="S2.SS4.p6.1.m1.4.5.2.3.2.3" stretchy="false" xref="S2.SS4.p6.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p6.1.m1.4.5.1" xref="S2.SS4.p6.1.m1.4.5.1.cmml">=</mo><mrow id="S2.SS4.p6.1.m1.4.5.3" xref="S2.SS4.p6.1.m1.4.5.3.cmml"><mi id="S2.SS4.p6.1.m1.4.5.3.2" xref="S2.SS4.p6.1.m1.4.5.3.2.cmml">R</mi><mo id="S2.SS4.p6.1.m1.4.5.3.1" xref="S2.SS4.p6.1.m1.4.5.3.1.cmml">⁢</mo><mrow id="S2.SS4.p6.1.m1.4.5.3.3.2" xref="S2.SS4.p6.1.m1.4.5.3.cmml"><mo id="S2.SS4.p6.1.m1.4.5.3.3.2.1" stretchy="false" xref="S2.SS4.p6.1.m1.4.5.3.cmml">(</mo><mi id="S2.SS4.p6.1.m1.3.3" 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id="S2.SS4.p6.1.m1.4.5.3.4.2.cmml" xref="S2.SS4.p6.1.m1.4.5.3.4.2">𝑝</ci><ci id="S2.SS4.p6.1.m1.4.5.3.4.3.cmml" xref="S2.SS4.p6.1.m1.4.5.3.4.3">pop</ci></apply><ci id="S2.SS4.p6.1.m1.4.4.cmml" xref="S2.SS4.p6.1.m1.4.4">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.1.m1.4c">R(x,\theta)=R(x)p_{\rm pop}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.1.m1.4d">italic_R ( italic_x , italic_θ ) = italic_R ( italic_x ) italic_p start_POSTSUBSCRIPT roman_pop end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math>: the “additional” intrinsic parameters <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS4.p6.2.m2.1"><semantics id="S2.SS4.p6.2.m2.1a"><mi id="S2.SS4.p6.2.m2.1.1" xref="S2.SS4.p6.2.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p6.2.m2.1b"><ci id="S2.SS4.p6.2.m2.1.1.cmml" xref="S2.SS4.p6.2.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.2.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.2.m2.1d">italic_θ</annotation></semantics></math> were taken to follow a simple, fixed distribution <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib95" title="">95</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib97" title="">97</a>]</cite> based on independent population studies (e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib103" title="">103</a>]</cite>). Then Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.E8" title="In II.4 Selection effects and validation of PE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">8</span></a>) factorizes, and we may define</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="R_{\rm det}(x)=R(x)\int p_{\rm pop}(\theta)p_{\rm det}(x,\theta)\,d^{n}\theta% \,\\ \equiv R(x)p_{\rm det}(x;p_{\rm pop})\,," class="ltx_Math" display="block" id="S2.E9.m1.42"><semantics id="S2.E9.m1.42a"><mtable displaystyle="true" id="S2.E9.m1.42.42.2" rowspacing="0pt"><mtr id="S2.E9.m1.42.42.2a"><mtd class="ltx_align_left" columnalign="left" id="S2.E9.m1.42.42.2b"><mrow id="S2.E9.m1.26.26.26.26.26"><mrow 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italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x ; italic_p start_POSTSUBSCRIPT roman_pop 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">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p6.7">enabling us to pass between the detected and astrophysical distributions via a function of <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.p6.3.m1.1"><semantics id="S2.SS4.p6.3.m1.1a"><mi id="S2.SS4.p6.3.m1.1.1" xref="S2.SS4.p6.3.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p6.3.m1.1b"><ci id="S2.SS4.p6.3.m1.1.1.cmml" xref="S2.SS4.p6.3.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.3.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.3.m1.1d">italic_x</annotation></semantics></math> alone. (We would also obtain a simple conversion <math alttext="p_{\rm det}(x)" class="ltx_Math" display="inline" id="S2.SS4.p6.4.m2.1"><semantics id="S2.SS4.p6.4.m2.1a"><mrow id="S2.SS4.p6.4.m2.1.2" xref="S2.SS4.p6.4.m2.1.2.cmml"><msub id="S2.SS4.p6.4.m2.1.2.2" xref="S2.SS4.p6.4.m2.1.2.2.cmml"><mi id="S2.SS4.p6.4.m2.1.2.2.2" xref="S2.SS4.p6.4.m2.1.2.2.2.cmml">p</mi><mi id="S2.SS4.p6.4.m2.1.2.2.3" xref="S2.SS4.p6.4.m2.1.2.2.3.cmml">det</mi></msub><mo id="S2.SS4.p6.4.m2.1.2.1" xref="S2.SS4.p6.4.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p6.4.m2.1.2.3.2" xref="S2.SS4.p6.4.m2.1.2.cmml"><mo id="S2.SS4.p6.4.m2.1.2.3.2.1" stretchy="false" xref="S2.SS4.p6.4.m2.1.2.cmml">(</mo><mi id="S2.SS4.p6.4.m2.1.1" xref="S2.SS4.p6.4.m2.1.1.cmml">x</mi><mo id="S2.SS4.p6.4.m2.1.2.3.2.2" stretchy="false" xref="S2.SS4.p6.4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p6.4.m2.1b"><apply id="S2.SS4.p6.4.m2.1.2.cmml" xref="S2.SS4.p6.4.m2.1.2"><times id="S2.SS4.p6.4.m2.1.2.1.cmml" xref="S2.SS4.p6.4.m2.1.2.1"></times><apply id="S2.SS4.p6.4.m2.1.2.2.cmml" xref="S2.SS4.p6.4.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p6.4.m2.1.2.2.1.cmml" xref="S2.SS4.p6.4.m2.1.2.2">subscript</csymbol><ci id="S2.SS4.p6.4.m2.1.2.2.2.cmml" xref="S2.SS4.p6.4.m2.1.2.2.2">𝑝</ci><ci id="S2.SS4.p6.4.m2.1.2.2.3.cmml" xref="S2.SS4.p6.4.m2.1.2.2.3">det</ci></apply><ci id="S2.SS4.p6.4.m2.1.1.cmml" xref="S2.SS4.p6.4.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.4.m2.1c">p_{\rm det}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.4.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> by neglecting the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS4.p6.5.m3.1"><semantics id="S2.SS4.p6.5.m3.1a"><mi id="S2.SS4.p6.5.m3.1.1" xref="S2.SS4.p6.5.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p6.5.m3.1b"><ci id="S2.SS4.p6.5.m3.1.1.cmml" xref="S2.SS4.p6.5.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.5.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.5.m3.1d">italic_θ</annotation></semantics></math>-dependence of <math alttext="p_{\rm det}(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p6.6.m4.2"><semantics id="S2.SS4.p6.6.m4.2a"><mrow id="S2.SS4.p6.6.m4.2.3" xref="S2.SS4.p6.6.m4.2.3.cmml"><msub id="S2.SS4.p6.6.m4.2.3.2" xref="S2.SS4.p6.6.m4.2.3.2.cmml"><mi id="S2.SS4.p6.6.m4.2.3.2.2" xref="S2.SS4.p6.6.m4.2.3.2.2.cmml">p</mi><mi id="S2.SS4.p6.6.m4.2.3.2.3" xref="S2.SS4.p6.6.m4.2.3.2.3.cmml">det</mi></msub><mo id="S2.SS4.p6.6.m4.2.3.1" xref="S2.SS4.p6.6.m4.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.p6.6.m4.2.3.3.2" xref="S2.SS4.p6.6.m4.2.3.3.1.cmml"><mo id="S2.SS4.p6.6.m4.2.3.3.2.1" stretchy="false" xref="S2.SS4.p6.6.m4.2.3.3.1.cmml">(</mo><mi id="S2.SS4.p6.6.m4.1.1" xref="S2.SS4.p6.6.m4.1.1.cmml">x</mi><mo id="S2.SS4.p6.6.m4.2.3.3.2.2" xref="S2.SS4.p6.6.m4.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p6.6.m4.2.2" xref="S2.SS4.p6.6.m4.2.2.cmml">θ</mi><mo id="S2.SS4.p6.6.m4.2.3.3.2.3" stretchy="false" xref="S2.SS4.p6.6.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p6.6.m4.2b"><apply id="S2.SS4.p6.6.m4.2.3.cmml" xref="S2.SS4.p6.6.m4.2.3"><times id="S2.SS4.p6.6.m4.2.3.1.cmml" xref="S2.SS4.p6.6.m4.2.3.1"></times><apply id="S2.SS4.p6.6.m4.2.3.2.cmml" xref="S2.SS4.p6.6.m4.2.3.2"><csymbol cd="ambiguous" id="S2.SS4.p6.6.m4.2.3.2.1.cmml" xref="S2.SS4.p6.6.m4.2.3.2">subscript</csymbol><ci id="S2.SS4.p6.6.m4.2.3.2.2.cmml" xref="S2.SS4.p6.6.m4.2.3.2.2">𝑝</ci><ci id="S2.SS4.p6.6.m4.2.3.2.3.cmml" xref="S2.SS4.p6.6.m4.2.3.2.3">det</ci></apply><interval closure="open" id="S2.SS4.p6.6.m4.2.3.3.1.cmml" xref="S2.SS4.p6.6.m4.2.3.3.2"><ci id="S2.SS4.p6.6.m4.1.1.cmml" xref="S2.SS4.p6.6.m4.1.1">𝑥</ci><ci id="S2.SS4.p6.6.m4.2.2.cmml" xref="S2.SS4.p6.6.m4.2.2">𝜃</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.6.m4.2c">p_{\rm det}(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.6.m4.2d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ )</annotation></semantics></math>, but this is unlikely to be a good approximation.) This approach is only as good as the accuracy of its assumptions, in particular that the joint distribution <math alttext="R(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p6.7.m5.2"><semantics id="S2.SS4.p6.7.m5.2a"><mrow id="S2.SS4.p6.7.m5.2.3" xref="S2.SS4.p6.7.m5.2.3.cmml"><mi id="S2.SS4.p6.7.m5.2.3.2" xref="S2.SS4.p6.7.m5.2.3.2.cmml">R</mi><mo id="S2.SS4.p6.7.m5.2.3.1" xref="S2.SS4.p6.7.m5.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.p6.7.m5.2.3.3.2" xref="S2.SS4.p6.7.m5.2.3.3.1.cmml"><mo id="S2.SS4.p6.7.m5.2.3.3.2.1" stretchy="false" xref="S2.SS4.p6.7.m5.2.3.3.1.cmml">(</mo><mi id="S2.SS4.p6.7.m5.1.1" xref="S2.SS4.p6.7.m5.1.1.cmml">x</mi><mo id="S2.SS4.p6.7.m5.2.3.3.2.2" xref="S2.SS4.p6.7.m5.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p6.7.m5.2.2" xref="S2.SS4.p6.7.m5.2.2.cmml">θ</mi><mo id="S2.SS4.p6.7.m5.2.3.3.2.3" stretchy="false" xref="S2.SS4.p6.7.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p6.7.m5.2b"><apply id="S2.SS4.p6.7.m5.2.3.cmml" xref="S2.SS4.p6.7.m5.2.3"><times id="S2.SS4.p6.7.m5.2.3.1.cmml" xref="S2.SS4.p6.7.m5.2.3.1"></times><ci id="S2.SS4.p6.7.m5.2.3.2.cmml" xref="S2.SS4.p6.7.m5.2.3.2">𝑅</ci><interval closure="open" id="S2.SS4.p6.7.m5.2.3.3.1.cmml" xref="S2.SS4.p6.7.m5.2.3.3.2"><ci id="S2.SS4.p6.7.m5.1.1.cmml" xref="S2.SS4.p6.7.m5.1.1">𝑥</ci><ci id="S2.SS4.p6.7.m5.2.2.cmml" xref="S2.SS4.p6.7.m5.2.2">𝜃</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p6.7.m5.2c">R(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p6.7.m5.2d">italic_R ( italic_x , italic_θ )</annotation></semantics></math> is of product form.</p> </div> <div class="ltx_para" id="S2.SS4.p7"> <p class="ltx_p" id="S2.SS4.p7.6">Thus here, we consider different strategies for reconstructing either the observed distribution, proportional to <math alttext="R_{\rm det}(x)" class="ltx_Math" display="inline" id="S2.SS4.p7.1.m1.1"><semantics id="S2.SS4.p7.1.m1.1a"><mrow id="S2.SS4.p7.1.m1.1.2" xref="S2.SS4.p7.1.m1.1.2.cmml"><msub id="S2.SS4.p7.1.m1.1.2.2" xref="S2.SS4.p7.1.m1.1.2.2.cmml"><mi id="S2.SS4.p7.1.m1.1.2.2.2" xref="S2.SS4.p7.1.m1.1.2.2.2.cmml">R</mi><mi id="S2.SS4.p7.1.m1.1.2.2.3" xref="S2.SS4.p7.1.m1.1.2.2.3.cmml">det</mi></msub><mo id="S2.SS4.p7.1.m1.1.2.1" xref="S2.SS4.p7.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p7.1.m1.1.2.3.2" xref="S2.SS4.p7.1.m1.1.2.cmml"><mo id="S2.SS4.p7.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS4.p7.1.m1.1.2.cmml">(</mo><mi id="S2.SS4.p7.1.m1.1.1" xref="S2.SS4.p7.1.m1.1.1.cmml">x</mi><mo id="S2.SS4.p7.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS4.p7.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p7.1.m1.1b"><apply id="S2.SS4.p7.1.m1.1.2.cmml" xref="S2.SS4.p7.1.m1.1.2"><times id="S2.SS4.p7.1.m1.1.2.1.cmml" xref="S2.SS4.p7.1.m1.1.2.1"></times><apply id="S2.SS4.p7.1.m1.1.2.2.cmml" xref="S2.SS4.p7.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p7.1.m1.1.2.2.1.cmml" xref="S2.SS4.p7.1.m1.1.2.2">subscript</csymbol><ci id="S2.SS4.p7.1.m1.1.2.2.2.cmml" xref="S2.SS4.p7.1.m1.1.2.2.2">𝑅</ci><ci id="S2.SS4.p7.1.m1.1.2.2.3.cmml" xref="S2.SS4.p7.1.m1.1.2.2.3">det</ci></apply><ci id="S2.SS4.p7.1.m1.1.1.cmml" xref="S2.SS4.p7.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p7.1.m1.1c">R_{\rm det}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p7.1.m1.1d">italic_R start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>, or the astrophysical distribution <math alttext="R(x)" class="ltx_Math" display="inline" id="S2.SS4.p7.2.m2.1"><semantics id="S2.SS4.p7.2.m2.1a"><mrow id="S2.SS4.p7.2.m2.1.2" xref="S2.SS4.p7.2.m2.1.2.cmml"><mi id="S2.SS4.p7.2.m2.1.2.2" xref="S2.SS4.p7.2.m2.1.2.2.cmml">R</mi><mo id="S2.SS4.p7.2.m2.1.2.1" xref="S2.SS4.p7.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p7.2.m2.1.2.3.2" xref="S2.SS4.p7.2.m2.1.2.cmml"><mo id="S2.SS4.p7.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS4.p7.2.m2.1.2.cmml">(</mo><mi id="S2.SS4.p7.2.m2.1.1" xref="S2.SS4.p7.2.m2.1.1.cmml">x</mi><mo id="S2.SS4.p7.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS4.p7.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p7.2.m2.1b"><apply id="S2.SS4.p7.2.m2.1.2.cmml" xref="S2.SS4.p7.2.m2.1.2"><times id="S2.SS4.p7.2.m2.1.2.1.cmml" xref="S2.SS4.p7.2.m2.1.2.1"></times><ci id="S2.SS4.p7.2.m2.1.2.2.cmml" xref="S2.SS4.p7.2.m2.1.2.2">𝑅</ci><ci id="S2.SS4.p7.2.m2.1.1.cmml" xref="S2.SS4.p7.2.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p7.2.m2.1c">R(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p7.2.m2.1d">italic_R ( italic_x )</annotation></semantics></math>. We expect to a first approximation that the distribution of posterior samples for detected events over <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.p7.3.m3.1"><semantics id="S2.SS4.p7.3.m3.1a"><mi id="S2.SS4.p7.3.m3.1.1" xref="S2.SS4.p7.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p7.3.m3.1b"><ci id="S2.SS4.p7.3.m3.1.1.cmml" xref="S2.SS4.p7.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p7.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p7.3.m3.1d">italic_x</annotation></semantics></math> and <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS4.p7.4.m4.1"><semantics id="S2.SS4.p7.4.m4.1a"><mi id="S2.SS4.p7.4.m4.1.1" xref="S2.SS4.p7.4.m4.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p7.4.m4.1b"><ci id="S2.SS4.p7.4.m4.1.1.cmml" xref="S2.SS4.p7.4.m4.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p7.4.m4.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p7.4.m4.1d">italic_θ</annotation></semantics></math> will mirror the detected distribution <math alttext="R(x,\theta)p_{\rm det}(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p7.5.m5.4"><semantics id="S2.SS4.p7.5.m5.4a"><mrow id="S2.SS4.p7.5.m5.4.5" xref="S2.SS4.p7.5.m5.4.5.cmml"><mi id="S2.SS4.p7.5.m5.4.5.2" xref="S2.SS4.p7.5.m5.4.5.2.cmml">R</mi><mo id="S2.SS4.p7.5.m5.4.5.1" xref="S2.SS4.p7.5.m5.4.5.1.cmml">⁢</mo><mrow id="S2.SS4.p7.5.m5.4.5.3.2" xref="S2.SS4.p7.5.m5.4.5.3.1.cmml"><mo id="S2.SS4.p7.5.m5.4.5.3.2.1" stretchy="false" xref="S2.SS4.p7.5.m5.4.5.3.1.cmml">(</mo><mi id="S2.SS4.p7.5.m5.1.1" xref="S2.SS4.p7.5.m5.1.1.cmml">x</mi><mo id="S2.SS4.p7.5.m5.4.5.3.2.2" xref="S2.SS4.p7.5.m5.4.5.3.1.cmml">,</mo><mi id="S2.SS4.p7.5.m5.2.2" xref="S2.SS4.p7.5.m5.2.2.cmml">θ</mi><mo id="S2.SS4.p7.5.m5.4.5.3.2.3" stretchy="false" xref="S2.SS4.p7.5.m5.4.5.3.1.cmml">)</mo></mrow><mo id="S2.SS4.p7.5.m5.4.5.1a" xref="S2.SS4.p7.5.m5.4.5.1.cmml">⁢</mo><msub id="S2.SS4.p7.5.m5.4.5.4" xref="S2.SS4.p7.5.m5.4.5.4.cmml"><mi id="S2.SS4.p7.5.m5.4.5.4.2" xref="S2.SS4.p7.5.m5.4.5.4.2.cmml">p</mi><mi id="S2.SS4.p7.5.m5.4.5.4.3" xref="S2.SS4.p7.5.m5.4.5.4.3.cmml">det</mi></msub><mo id="S2.SS4.p7.5.m5.4.5.1b" xref="S2.SS4.p7.5.m5.4.5.1.cmml">⁢</mo><mrow id="S2.SS4.p7.5.m5.4.5.5.2" xref="S2.SS4.p7.5.m5.4.5.5.1.cmml"><mo id="S2.SS4.p7.5.m5.4.5.5.2.1" stretchy="false" xref="S2.SS4.p7.5.m5.4.5.5.1.cmml">(</mo><mi id="S2.SS4.p7.5.m5.3.3" xref="S2.SS4.p7.5.m5.3.3.cmml">x</mi><mo id="S2.SS4.p7.5.m5.4.5.5.2.2" xref="S2.SS4.p7.5.m5.4.5.5.1.cmml">,</mo><mi id="S2.SS4.p7.5.m5.4.4" xref="S2.SS4.p7.5.m5.4.4.cmml">θ</mi><mo id="S2.SS4.p7.5.m5.4.5.5.2.3" stretchy="false" xref="S2.SS4.p7.5.m5.4.5.5.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p7.5.m5.4b"><apply id="S2.SS4.p7.5.m5.4.5.cmml" xref="S2.SS4.p7.5.m5.4.5"><times id="S2.SS4.p7.5.m5.4.5.1.cmml" xref="S2.SS4.p7.5.m5.4.5.1"></times><ci id="S2.SS4.p7.5.m5.4.5.2.cmml" xref="S2.SS4.p7.5.m5.4.5.2">𝑅</ci><interval closure="open" id="S2.SS4.p7.5.m5.4.5.3.1.cmml" xref="S2.SS4.p7.5.m5.4.5.3.2"><ci id="S2.SS4.p7.5.m5.1.1.cmml" xref="S2.SS4.p7.5.m5.1.1">𝑥</ci><ci id="S2.SS4.p7.5.m5.2.2.cmml" xref="S2.SS4.p7.5.m5.2.2">𝜃</ci></interval><apply id="S2.SS4.p7.5.m5.4.5.4.cmml" xref="S2.SS4.p7.5.m5.4.5.4"><csymbol cd="ambiguous" id="S2.SS4.p7.5.m5.4.5.4.1.cmml" xref="S2.SS4.p7.5.m5.4.5.4">subscript</csymbol><ci id="S2.SS4.p7.5.m5.4.5.4.2.cmml" xref="S2.SS4.p7.5.m5.4.5.4.2">𝑝</ci><ci id="S2.SS4.p7.5.m5.4.5.4.3.cmml" xref="S2.SS4.p7.5.m5.4.5.4.3">det</ci></apply><interval closure="open" id="S2.SS4.p7.5.m5.4.5.5.1.cmml" xref="S2.SS4.p7.5.m5.4.5.5.2"><ci id="S2.SS4.p7.5.m5.3.3.cmml" xref="S2.SS4.p7.5.m5.3.3">𝑥</ci><ci id="S2.SS4.p7.5.m5.4.4.cmml" xref="S2.SS4.p7.5.m5.4.4">𝜃</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p7.5.m5.4c">R(x,\theta)p_{\rm det}(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p7.5.m5.4d">italic_R ( italic_x , italic_θ ) italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ )</annotation></semantics></math>. Thus, if we apply weights <math alttext="\propto 1/p_{\rm det}(x,\theta)" class="ltx_Math" display="inline" id="S2.SS4.p7.6.m6.2"><semantics id="S2.SS4.p7.6.m6.2a"><mrow id="S2.SS4.p7.6.m6.2.3" xref="S2.SS4.p7.6.m6.2.3.cmml"><mi id="S2.SS4.p7.6.m6.2.3.2" xref="S2.SS4.p7.6.m6.2.3.2.cmml"></mi><mo id="S2.SS4.p7.6.m6.2.3.1" xref="S2.SS4.p7.6.m6.2.3.1.cmml">∝</mo><mrow id="S2.SS4.p7.6.m6.2.3.3" xref="S2.SS4.p7.6.m6.2.3.3.cmml"><mrow id="S2.SS4.p7.6.m6.2.3.3.2" xref="S2.SS4.p7.6.m6.2.3.3.2.cmml"><mn id="S2.SS4.p7.6.m6.2.3.3.2.2" xref="S2.SS4.p7.6.m6.2.3.3.2.2.cmml">1</mn><mo id="S2.SS4.p7.6.m6.2.3.3.2.1" xref="S2.SS4.p7.6.m6.2.3.3.2.1.cmml">/</mo><msub id="S2.SS4.p7.6.m6.2.3.3.2.3" xref="S2.SS4.p7.6.m6.2.3.3.2.3.cmml"><mi id="S2.SS4.p7.6.m6.2.3.3.2.3.2" xref="S2.SS4.p7.6.m6.2.3.3.2.3.2.cmml">p</mi><mi id="S2.SS4.p7.6.m6.2.3.3.2.3.3" xref="S2.SS4.p7.6.m6.2.3.3.2.3.3.cmml">det</mi></msub></mrow><mo id="S2.SS4.p7.6.m6.2.3.3.1" xref="S2.SS4.p7.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS4.p7.6.m6.2.3.3.3.2" xref="S2.SS4.p7.6.m6.2.3.3.3.1.cmml"><mo id="S2.SS4.p7.6.m6.2.3.3.3.2.1" stretchy="false" xref="S2.SS4.p7.6.m6.2.3.3.3.1.cmml">(</mo><mi id="S2.SS4.p7.6.m6.1.1" xref="S2.SS4.p7.6.m6.1.1.cmml">x</mi><mo id="S2.SS4.p7.6.m6.2.3.3.3.2.2" xref="S2.SS4.p7.6.m6.2.3.3.3.1.cmml">,</mo><mi id="S2.SS4.p7.6.m6.2.2" xref="S2.SS4.p7.6.m6.2.2.cmml">θ</mi><mo id="S2.SS4.p7.6.m6.2.3.3.3.2.3" stretchy="false" xref="S2.SS4.p7.6.m6.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p7.6.m6.2b"><apply id="S2.SS4.p7.6.m6.2.3.cmml" xref="S2.SS4.p7.6.m6.2.3"><csymbol cd="latexml" id="S2.SS4.p7.6.m6.2.3.1.cmml" xref="S2.SS4.p7.6.m6.2.3.1">proportional-to</csymbol><csymbol cd="latexml" id="S2.SS4.p7.6.m6.2.3.2.cmml" xref="S2.SS4.p7.6.m6.2.3.2">absent</csymbol><apply id="S2.SS4.p7.6.m6.2.3.3.cmml" xref="S2.SS4.p7.6.m6.2.3.3"><times id="S2.SS4.p7.6.m6.2.3.3.1.cmml" xref="S2.SS4.p7.6.m6.2.3.3.1"></times><apply id="S2.SS4.p7.6.m6.2.3.3.2.cmml" xref="S2.SS4.p7.6.m6.2.3.3.2"><divide id="S2.SS4.p7.6.m6.2.3.3.2.1.cmml" xref="S2.SS4.p7.6.m6.2.3.3.2.1"></divide><cn id="S2.SS4.p7.6.m6.2.3.3.2.2.cmml" type="integer" xref="S2.SS4.p7.6.m6.2.3.3.2.2">1</cn><apply id="S2.SS4.p7.6.m6.2.3.3.2.3.cmml" xref="S2.SS4.p7.6.m6.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.SS4.p7.6.m6.2.3.3.2.3.1.cmml" xref="S2.SS4.p7.6.m6.2.3.3.2.3">subscript</csymbol><ci id="S2.SS4.p7.6.m6.2.3.3.2.3.2.cmml" xref="S2.SS4.p7.6.m6.2.3.3.2.3.2">𝑝</ci><ci id="S2.SS4.p7.6.m6.2.3.3.2.3.3.cmml" xref="S2.SS4.p7.6.m6.2.3.3.2.3.3">det</ci></apply></apply><interval closure="open" id="S2.SS4.p7.6.m6.2.3.3.3.1.cmml" xref="S2.SS4.p7.6.m6.2.3.3.3.2"><ci id="S2.SS4.p7.6.m6.1.1.cmml" xref="S2.SS4.p7.6.m6.1.1">𝑥</ci><ci id="S2.SS4.p7.6.m6.2.2.cmml" xref="S2.SS4.p7.6.m6.2.2">𝜃</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p7.6.m6.2c">\propto 1/p_{\rm det}(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p7.6.m6.2d">∝ 1 / italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ )</annotation></semantics></math> to posterior samples, the resulting weighted distribution will mirror the astrophysical one. We may make use of this in two ways.</p> </div> <div class="ltx_para" id="S2.SS4.p8"> <p class="ltx_p" id="S2.SS4.p8.3">Our <em class="ltx_emph ltx_font_italic" id="S2.SS4.p8.3.1">first method</em> is a reconstruction of the <em class="ltx_emph ltx_font_italic" id="S2.SS4.p8.3.2">detected</em> distribution via a KDE of detected event samples, setting <math alttext="W_{i}=1" class="ltx_Math" display="inline" id="S2.SS4.p8.1.m1.1"><semantics id="S2.SS4.p8.1.m1.1a"><mrow id="S2.SS4.p8.1.m1.1.1" xref="S2.SS4.p8.1.m1.1.1.cmml"><msub id="S2.SS4.p8.1.m1.1.1.2" xref="S2.SS4.p8.1.m1.1.1.2.cmml"><mi id="S2.SS4.p8.1.m1.1.1.2.2" xref="S2.SS4.p8.1.m1.1.1.2.2.cmml">W</mi><mi id="S2.SS4.p8.1.m1.1.1.2.3" xref="S2.SS4.p8.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS4.p8.1.m1.1.1.1" xref="S2.SS4.p8.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS4.p8.1.m1.1.1.3" xref="S2.SS4.p8.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p8.1.m1.1b"><apply id="S2.SS4.p8.1.m1.1.1.cmml" xref="S2.SS4.p8.1.m1.1.1"><eq id="S2.SS4.p8.1.m1.1.1.1.cmml" xref="S2.SS4.p8.1.m1.1.1.1"></eq><apply id="S2.SS4.p8.1.m1.1.1.2.cmml" xref="S2.SS4.p8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.p8.1.m1.1.1.2.1.cmml" xref="S2.SS4.p8.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS4.p8.1.m1.1.1.2.2.cmml" xref="S2.SS4.p8.1.m1.1.1.2.2">𝑊</ci><ci id="S2.SS4.p8.1.m1.1.1.2.3.cmml" xref="S2.SS4.p8.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S2.SS4.p8.1.m1.1.1.3.cmml" type="integer" xref="S2.SS4.p8.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p8.1.m1.1c">W_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p8.1.m1.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.E4" title="In II.3 Population reconstruction via iterative KDE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">4</span></a>). When applying iterative reweighting, to improve the accuracy of event parameters we require the weights to be proportional to the astrophysical distribution <math alttext="R(x)" class="ltx_Math" display="inline" id="S2.SS4.p8.2.m2.1"><semantics id="S2.SS4.p8.2.m2.1a"><mrow id="S2.SS4.p8.2.m2.1.2" xref="S2.SS4.p8.2.m2.1.2.cmml"><mi id="S2.SS4.p8.2.m2.1.2.2" xref="S2.SS4.p8.2.m2.1.2.2.cmml">R</mi><mo id="S2.SS4.p8.2.m2.1.2.1" xref="S2.SS4.p8.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p8.2.m2.1.2.3.2" xref="S2.SS4.p8.2.m2.1.2.cmml"><mo id="S2.SS4.p8.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS4.p8.2.m2.1.2.cmml">(</mo><mi id="S2.SS4.p8.2.m2.1.1" xref="S2.SS4.p8.2.m2.1.1.cmml">x</mi><mo id="S2.SS4.p8.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS4.p8.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p8.2.m2.1b"><apply id="S2.SS4.p8.2.m2.1.2.cmml" xref="S2.SS4.p8.2.m2.1.2"><times id="S2.SS4.p8.2.m2.1.2.1.cmml" xref="S2.SS4.p8.2.m2.1.2.1"></times><ci id="S2.SS4.p8.2.m2.1.2.2.cmml" xref="S2.SS4.p8.2.m2.1.2.2">𝑅</ci><ci id="S2.SS4.p8.2.m2.1.1.cmml" xref="S2.SS4.p8.2.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p8.2.m2.1c">R(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p8.2.m2.1d">italic_R ( italic_x )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib97" title="">97</a>]</cite>. We approximate this weighting here by multiplying the previous iteration’s detected event KDE by a factor <math alttext="\sim\!1/p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S2.SS4.p8.3.m3.1"><semantics id="S2.SS4.p8.3.m3.1a"><mrow id="S2.SS4.p8.3.m3.1.1" xref="S2.SS4.p8.3.m3.1.1.cmml"><mi id="S2.SS4.p8.3.m3.1.1.2" xref="S2.SS4.p8.3.m3.1.1.2.cmml"></mi><mo id="S2.SS4.p8.3.m3.1.1.1" rspace="0.108em" xref="S2.SS4.p8.3.m3.1.1.1.cmml">∼</mo><mrow id="S2.SS4.p8.3.m3.1.1.3" xref="S2.SS4.p8.3.m3.1.1.3.cmml"><mn id="S2.SS4.p8.3.m3.1.1.3.2" xref="S2.SS4.p8.3.m3.1.1.3.2.cmml">1</mn><mo id="S2.SS4.p8.3.m3.1.1.3.1" xref="S2.SS4.p8.3.m3.1.1.3.1.cmml">/</mo><msub id="S2.SS4.p8.3.m3.1.1.3.3" xref="S2.SS4.p8.3.m3.1.1.3.3.cmml"><mi id="S2.SS4.p8.3.m3.1.1.3.3.2" xref="S2.SS4.p8.3.m3.1.1.3.3.2.cmml">p</mi><mi id="S2.SS4.p8.3.m3.1.1.3.3.3" xref="S2.SS4.p8.3.m3.1.1.3.3.3.cmml">det</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p8.3.m3.1b"><apply id="S2.SS4.p8.3.m3.1.1.cmml" xref="S2.SS4.p8.3.m3.1.1"><csymbol cd="latexml" id="S2.SS4.p8.3.m3.1.1.1.cmml" xref="S2.SS4.p8.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS4.p8.3.m3.1.1.2.cmml" xref="S2.SS4.p8.3.m3.1.1.2">absent</csymbol><apply id="S2.SS4.p8.3.m3.1.1.3.cmml" xref="S2.SS4.p8.3.m3.1.1.3"><divide id="S2.SS4.p8.3.m3.1.1.3.1.cmml" xref="S2.SS4.p8.3.m3.1.1.3.1"></divide><cn id="S2.SS4.p8.3.m3.1.1.3.2.cmml" type="integer" xref="S2.SS4.p8.3.m3.1.1.3.2">1</cn><apply id="S2.SS4.p8.3.m3.1.1.3.3.cmml" xref="S2.SS4.p8.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS4.p8.3.m3.1.1.3.3.1.cmml" xref="S2.SS4.p8.3.m3.1.1.3.3">subscript</csymbol><ci id="S2.SS4.p8.3.m3.1.1.3.3.2.cmml" xref="S2.SS4.p8.3.m3.1.1.3.3.2">𝑝</ci><ci id="S2.SS4.p8.3.m3.1.1.3.3.3.cmml" xref="S2.SS4.p8.3.m3.1.1.3.3.3">det</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p8.3.m3.1c">\sim\!1/p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p8.3.m3.1d">∼ 1 / italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>, evaluated at each sample.</p> </div> <div class="ltx_para" id="S2.SS4.p9"> <p class="ltx_p" id="S2.SS4.p9.4">By contrast, our <em class="ltx_emph ltx_font_italic" id="S2.SS4.p9.4.1">second method</em> uses a weighted KDE which incorporates selection effects as KDE weights <math alttext="W_{i}\sim 1/p_{\mathrm{det},i}" class="ltx_Math" display="inline" id="S2.SS4.p9.1.m1.2"><semantics id="S2.SS4.p9.1.m1.2a"><mrow id="S2.SS4.p9.1.m1.2.3" xref="S2.SS4.p9.1.m1.2.3.cmml"><msub id="S2.SS4.p9.1.m1.2.3.2" xref="S2.SS4.p9.1.m1.2.3.2.cmml"><mi id="S2.SS4.p9.1.m1.2.3.2.2" xref="S2.SS4.p9.1.m1.2.3.2.2.cmml">W</mi><mi id="S2.SS4.p9.1.m1.2.3.2.3" xref="S2.SS4.p9.1.m1.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS4.p9.1.m1.2.3.1" xref="S2.SS4.p9.1.m1.2.3.1.cmml">∼</mo><mrow id="S2.SS4.p9.1.m1.2.3.3" xref="S2.SS4.p9.1.m1.2.3.3.cmml"><mn id="S2.SS4.p9.1.m1.2.3.3.2" xref="S2.SS4.p9.1.m1.2.3.3.2.cmml">1</mn><mo id="S2.SS4.p9.1.m1.2.3.3.1" xref="S2.SS4.p9.1.m1.2.3.3.1.cmml">/</mo><msub id="S2.SS4.p9.1.m1.2.3.3.3" xref="S2.SS4.p9.1.m1.2.3.3.3.cmml"><mi id="S2.SS4.p9.1.m1.2.3.3.3.2" xref="S2.SS4.p9.1.m1.2.3.3.3.2.cmml">p</mi><mrow id="S2.SS4.p9.1.m1.2.2.2.4" xref="S2.SS4.p9.1.m1.2.2.2.3.cmml"><mi id="S2.SS4.p9.1.m1.1.1.1.1" xref="S2.SS4.p9.1.m1.1.1.1.1.cmml">det</mi><mo id="S2.SS4.p9.1.m1.2.2.2.4.1" xref="S2.SS4.p9.1.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS4.p9.1.m1.2.2.2.2" xref="S2.SS4.p9.1.m1.2.2.2.2.cmml">i</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p9.1.m1.2b"><apply id="S2.SS4.p9.1.m1.2.3.cmml" xref="S2.SS4.p9.1.m1.2.3"><csymbol cd="latexml" id="S2.SS4.p9.1.m1.2.3.1.cmml" xref="S2.SS4.p9.1.m1.2.3.1">similar-to</csymbol><apply id="S2.SS4.p9.1.m1.2.3.2.cmml" xref="S2.SS4.p9.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS4.p9.1.m1.2.3.2.1.cmml" 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id="S2.SS4.p9.1.m1.2c">W_{i}\sim 1/p_{\mathrm{det},i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p9.1.m1.2d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ 1 / italic_p start_POSTSUBSCRIPT roman_det , italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, in order to obtain a estimate of the astrophysical distribution, at least over regions of <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.p9.2.m2.1"><semantics id="S2.SS4.p9.2.m2.1a"><mi id="S2.SS4.p9.2.m2.1.1" xref="S2.SS4.p9.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p9.2.m2.1b"><ci id="S2.SS4.p9.2.m2.1.1.cmml" xref="S2.SS4.p9.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p9.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p9.2.m2.1d">italic_x</annotation></semantics></math>, <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS4.p9.3.m3.1"><semantics id="S2.SS4.p9.3.m3.1a"><mi id="S2.SS4.p9.3.m3.1.1" xref="S2.SS4.p9.3.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p9.3.m3.1b"><ci id="S2.SS4.p9.3.m3.1.1.cmml" xref="S2.SS4.p9.3.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p9.3.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p9.3.m3.1d">italic_θ</annotation></semantics></math> where detections exist and <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p9.4.m4.1"><semantics id="S2.SS4.p9.4.m4.1a"><msub id="S2.SS4.p9.4.m4.1.1" xref="S2.SS4.p9.4.m4.1.1.cmml"><mi id="S2.SS4.p9.4.m4.1.1.2" xref="S2.SS4.p9.4.m4.1.1.2.cmml">p</mi><mi id="S2.SS4.p9.4.m4.1.1.3" xref="S2.SS4.p9.4.m4.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p9.4.m4.1b"><apply id="S2.SS4.p9.4.m4.1.1.cmml" xref="S2.SS4.p9.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.p9.4.m4.1.1.1.cmml" xref="S2.SS4.p9.4.m4.1.1">subscript</csymbol><ci id="S2.SS4.p9.4.m4.1.1.2.cmml" xref="S2.SS4.p9.4.m4.1.1.2">𝑝</ci><ci id="S2.SS4.p9.4.m4.1.1.3.cmml" xref="S2.SS4.p9.4.m4.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p9.4.m4.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p9.4.m4.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> is not very small.</p> </div> <div class="ltx_para" id="S2.SS4.p10"> <p class="ltx_p" id="S2.SS4.p10.4">A naïve application of such “inverse <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p10.1.m1.1"><semantics id="S2.SS4.p10.1.m1.1a"><msub id="S2.SS4.p10.1.m1.1.1" xref="S2.SS4.p10.1.m1.1.1.cmml"><mi id="S2.SS4.p10.1.m1.1.1.2" xref="S2.SS4.p10.1.m1.1.1.2.cmml">p</mi><mi id="S2.SS4.p10.1.m1.1.1.3" xref="S2.SS4.p10.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p10.1.m1.1b"><apply id="S2.SS4.p10.1.m1.1.1.cmml" xref="S2.SS4.p10.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p10.1.m1.1.1.1.cmml" xref="S2.SS4.p10.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.p10.1.m1.1.1.2.cmml" xref="S2.SS4.p10.1.m1.1.1.2">𝑝</ci><ci id="S2.SS4.p10.1.m1.1.1.3.cmml" xref="S2.SS4.p10.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p10.1.m1.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p10.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>” weights can, though, run into problems of statistical stability and bias. Consider moving on a trajectory in the KDE space <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.p10.2.m2.1"><semantics id="S2.SS4.p10.2.m2.1a"><mi id="S2.SS4.p10.2.m2.1.1" xref="S2.SS4.p10.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p10.2.m2.1b"><ci id="S2.SS4.p10.2.m2.1.1.cmml" xref="S2.SS4.p10.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p10.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p10.2.m2.1d">italic_x</annotation></semantics></math> such that <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p10.3.m3.1"><semantics id="S2.SS4.p10.3.m3.1a"><msub id="S2.SS4.p10.3.m3.1.1" xref="S2.SS4.p10.3.m3.1.1.cmml"><mi id="S2.SS4.p10.3.m3.1.1.2" xref="S2.SS4.p10.3.m3.1.1.2.cmml">p</mi><mi id="S2.SS4.p10.3.m3.1.1.3" xref="S2.SS4.p10.3.m3.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p10.3.m3.1b"><apply id="S2.SS4.p10.3.m3.1.1.cmml" xref="S2.SS4.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS4.p10.3.m3.1.1.1.cmml" xref="S2.SS4.p10.3.m3.1.1">subscript</csymbol><ci id="S2.SS4.p10.3.m3.1.1.2.cmml" xref="S2.SS4.p10.3.m3.1.1.2">𝑝</ci><ci id="S2.SS4.p10.3.m3.1.1.3.cmml" xref="S2.SS4.p10.3.m3.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p10.3.m3.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p10.3.m3.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> tends to zero, while the astrophysical distribution is approximately constant. Then to first order, the density of samples will also tend to zero, and our estimate of the astrophysical density will become undefined, in practice becoming excessively sensitive to small number fluctuations rather than approaching a constant. Furthermore, we expect the posterior sample distribution to be somewhat broader (i.e. have wider support) than the actual distribution of detected events, due to PE measurement uncertainty. Hence, a <math alttext="1/p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p10.4.m4.1"><semantics id="S2.SS4.p10.4.m4.1a"><mrow id="S2.SS4.p10.4.m4.1.1" xref="S2.SS4.p10.4.m4.1.1.cmml"><mn id="S2.SS4.p10.4.m4.1.1.2" xref="S2.SS4.p10.4.m4.1.1.2.cmml">1</mn><mo id="S2.SS4.p10.4.m4.1.1.1" xref="S2.SS4.p10.4.m4.1.1.1.cmml">/</mo><msub id="S2.SS4.p10.4.m4.1.1.3" xref="S2.SS4.p10.4.m4.1.1.3.cmml"><mi id="S2.SS4.p10.4.m4.1.1.3.2" xref="S2.SS4.p10.4.m4.1.1.3.2.cmml">p</mi><mi id="S2.SS4.p10.4.m4.1.1.3.3" xref="S2.SS4.p10.4.m4.1.1.3.3.cmml">det</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p10.4.m4.1b"><apply id="S2.SS4.p10.4.m4.1.1.cmml" xref="S2.SS4.p10.4.m4.1.1"><divide id="S2.SS4.p10.4.m4.1.1.1.cmml" xref="S2.SS4.p10.4.m4.1.1.1"></divide><cn id="S2.SS4.p10.4.m4.1.1.2.cmml" type="integer" xref="S2.SS4.p10.4.m4.1.1.2">1</cn><apply id="S2.SS4.p10.4.m4.1.1.3.cmml" xref="S2.SS4.p10.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p10.4.m4.1.1.3.1.cmml" xref="S2.SS4.p10.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS4.p10.4.m4.1.1.3.2.cmml" xref="S2.SS4.p10.4.m4.1.1.3.2">𝑝</ci><ci id="S2.SS4.p10.4.m4.1.1.3.3.cmml" xref="S2.SS4.p10.4.m4.1.1.3.3">det</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p10.4.m4.1c">1/p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p10.4.m4.1d">1 / italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> weighted KDE may be liable to uncontrolled overestimates at the edge of the sample distribution.</p> </div> <div class="ltx_para" id="S2.SS4.p11"> <p class="ltx_p" id="S2.SS4.p11.5">To obtain a well-defined KDE for the astrophysical distribution with variance under control, we must therefore accept some bias in regions where <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p11.1.m1.1"><semantics id="S2.SS4.p11.1.m1.1a"><msub id="S2.SS4.p11.1.m1.1.1" xref="S2.SS4.p11.1.m1.1.1.cmml"><mi id="S2.SS4.p11.1.m1.1.1.2" xref="S2.SS4.p11.1.m1.1.1.2.cmml">p</mi><mi id="S2.SS4.p11.1.m1.1.1.3" xref="S2.SS4.p11.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p11.1.m1.1b"><apply id="S2.SS4.p11.1.m1.1.1.cmml" xref="S2.SS4.p11.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p11.1.m1.1.1.1.cmml" xref="S2.SS4.p11.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.p11.1.m1.1.1.2.cmml" xref="S2.SS4.p11.1.m1.1.1.2">𝑝</ci><ci id="S2.SS4.p11.1.m1.1.1.3.cmml" xref="S2.SS4.p11.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p11.1.m1.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p11.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> approaches zero. For instance, we can control the variance at the cost of allowing the weighted KDE to be an underestimate of the astrophysical distribution in regions with few or zero detected events: this can be achieved by removing samples with very small <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p11.2.m2.1"><semantics id="S2.SS4.p11.2.m2.1a"><msub id="S2.SS4.p11.2.m2.1.1" xref="S2.SS4.p11.2.m2.1.1.cmml"><mi id="S2.SS4.p11.2.m2.1.1.2" xref="S2.SS4.p11.2.m2.1.1.2.cmml">p</mi><mi id="S2.SS4.p11.2.m2.1.1.3" xref="S2.SS4.p11.2.m2.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p11.2.m2.1b"><apply id="S2.SS4.p11.2.m2.1.1.cmml" xref="S2.SS4.p11.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p11.2.m2.1.1.1.cmml" xref="S2.SS4.p11.2.m2.1.1">subscript</csymbol><ci id="S2.SS4.p11.2.m2.1.1.2.cmml" xref="S2.SS4.p11.2.m2.1.1.2">𝑝</ci><ci id="S2.SS4.p11.2.m2.1.1.3.cmml" xref="S2.SS4.p11.2.m2.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p11.2.m2.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p11.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>, or by capping the <math alttext="1/p_{\rm det}" class="ltx_Math" display="inline" id="S2.SS4.p11.3.m3.1"><semantics id="S2.SS4.p11.3.m3.1a"><mrow id="S2.SS4.p11.3.m3.1.1" xref="S2.SS4.p11.3.m3.1.1.cmml"><mn id="S2.SS4.p11.3.m3.1.1.2" xref="S2.SS4.p11.3.m3.1.1.2.cmml">1</mn><mo id="S2.SS4.p11.3.m3.1.1.1" xref="S2.SS4.p11.3.m3.1.1.1.cmml">/</mo><msub id="S2.SS4.p11.3.m3.1.1.3" xref="S2.SS4.p11.3.m3.1.1.3.cmml"><mi id="S2.SS4.p11.3.m3.1.1.3.2" xref="S2.SS4.p11.3.m3.1.1.3.2.cmml">p</mi><mi id="S2.SS4.p11.3.m3.1.1.3.3" xref="S2.SS4.p11.3.m3.1.1.3.3.cmml">det</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p11.3.m3.1b"><apply id="S2.SS4.p11.3.m3.1.1.cmml" xref="S2.SS4.p11.3.m3.1.1"><divide id="S2.SS4.p11.3.m3.1.1.1.cmml" xref="S2.SS4.p11.3.m3.1.1.1"></divide><cn id="S2.SS4.p11.3.m3.1.1.2.cmml" type="integer" xref="S2.SS4.p11.3.m3.1.1.2">1</cn><apply id="S2.SS4.p11.3.m3.1.1.3.cmml" xref="S2.SS4.p11.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p11.3.m3.1.1.3.1.cmml" xref="S2.SS4.p11.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS4.p11.3.m3.1.1.3.2.cmml" xref="S2.SS4.p11.3.m3.1.1.3.2">𝑝</ci><ci id="S2.SS4.p11.3.m3.1.1.3.3.cmml" xref="S2.SS4.p11.3.m3.1.1.3.3">det</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p11.3.m3.1c">1/p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p11.3.m3.1d">1 / italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> weights at a maximum value. In our main analyses, we implement weighting by a factor <math alttext="1/(\max(p_{\rm det},0.1))" class="ltx_Math" display="inline" id="S2.SS4.p11.4.m4.3"><semantics id="S2.SS4.p11.4.m4.3a"><mrow id="S2.SS4.p11.4.m4.3.3" xref="S2.SS4.p11.4.m4.3.3.cmml"><mn id="S2.SS4.p11.4.m4.3.3.3" xref="S2.SS4.p11.4.m4.3.3.3.cmml">1</mn><mo id="S2.SS4.p11.4.m4.3.3.2" xref="S2.SS4.p11.4.m4.3.3.2.cmml">/</mo><mrow id="S2.SS4.p11.4.m4.3.3.1.1" xref="S2.SS4.p11.4.m4.3.3.cmml"><mo id="S2.SS4.p11.4.m4.3.3.1.1.2" stretchy="false" xref="S2.SS4.p11.4.m4.3.3.cmml">(</mo><mrow id="S2.SS4.p11.4.m4.3.3.1.1.1.1" xref="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml"><mi id="S2.SS4.p11.4.m4.1.1" xref="S2.SS4.p11.4.m4.1.1.cmml">max</mi><mo id="S2.SS4.p11.4.m4.3.3.1.1.1.1a" xref="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml">⁡</mo><mrow id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1" xref="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml"><mo id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml">(</mo><msub id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.2" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.2.cmml">p</mi><mi id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.3" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.3.cmml">det</mi></msub><mo id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.3" xref="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml">,</mo><mn id="S2.SS4.p11.4.m4.2.2" xref="S2.SS4.p11.4.m4.2.2.cmml">0.1</mn><mo id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.4" stretchy="false" xref="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p11.4.m4.3.3.1.1.3" stretchy="false" xref="S2.SS4.p11.4.m4.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p11.4.m4.3b"><apply id="S2.SS4.p11.4.m4.3.3.cmml" xref="S2.SS4.p11.4.m4.3.3"><divide id="S2.SS4.p11.4.m4.3.3.2.cmml" xref="S2.SS4.p11.4.m4.3.3.2"></divide><cn id="S2.SS4.p11.4.m4.3.3.3.cmml" type="integer" xref="S2.SS4.p11.4.m4.3.3.3">1</cn><apply id="S2.SS4.p11.4.m4.3.3.1.1.1.2.cmml" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1"><max id="S2.SS4.p11.4.m4.1.1.cmml" xref="S2.SS4.p11.4.m4.1.1"></max><apply id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.cmml" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.2.cmml" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.2">𝑝</ci><ci id="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.3.cmml" xref="S2.SS4.p11.4.m4.3.3.1.1.1.1.1.1.3">det</ci></apply><cn id="S2.SS4.p11.4.m4.2.2.cmml" type="float" xref="S2.SS4.p11.4.m4.2.2">0.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p11.4.m4.3c">1/(\max(p_{\rm det},0.1))</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p11.4.m4.3d">1 / ( roman_max ( italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT , 0.1 ) )</annotation></semantics></math>: thus, we do not expect to reconstruct the astrophysical population well in regions with <math alttext="p_{\rm det}\ll 0.1" class="ltx_Math" display="inline" id="S2.SS4.p11.5.m5.1"><semantics id="S2.SS4.p11.5.m5.1a"><mrow id="S2.SS4.p11.5.m5.1.1" xref="S2.SS4.p11.5.m5.1.1.cmml"><msub id="S2.SS4.p11.5.m5.1.1.2" xref="S2.SS4.p11.5.m5.1.1.2.cmml"><mi id="S2.SS4.p11.5.m5.1.1.2.2" xref="S2.SS4.p11.5.m5.1.1.2.2.cmml">p</mi><mi id="S2.SS4.p11.5.m5.1.1.2.3" xref="S2.SS4.p11.5.m5.1.1.2.3.cmml">det</mi></msub><mo id="S2.SS4.p11.5.m5.1.1.1" xref="S2.SS4.p11.5.m5.1.1.1.cmml">≪</mo><mn id="S2.SS4.p11.5.m5.1.1.3" xref="S2.SS4.p11.5.m5.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p11.5.m5.1b"><apply id="S2.SS4.p11.5.m5.1.1.cmml" xref="S2.SS4.p11.5.m5.1.1"><csymbol cd="latexml" id="S2.SS4.p11.5.m5.1.1.1.cmml" xref="S2.SS4.p11.5.m5.1.1.1">much-less-than</csymbol><apply id="S2.SS4.p11.5.m5.1.1.2.cmml" xref="S2.SS4.p11.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.p11.5.m5.1.1.2.1.cmml" xref="S2.SS4.p11.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS4.p11.5.m5.1.1.2.2.cmml" xref="S2.SS4.p11.5.m5.1.1.2.2">𝑝</ci><ci id="S2.SS4.p11.5.m5.1.1.2.3.cmml" xref="S2.SS4.p11.5.m5.1.1.2.3">det</ci></apply><cn id="S2.SS4.p11.5.m5.1.1.3.cmml" type="float" xref="S2.SS4.p11.5.m5.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p11.5.m5.1c">p_{\rm det}\ll 0.1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p11.5.m5.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ≪ 0.1</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.5 </span>Application to simulated data</h3> <div class="ltx_para" id="S2.SS5.p1"> <p class="ltx_p" id="S2.SS5.p1.4">For our analysis, we randomly choose 100 posterior samples from each event. For each selected sample, we find the corresponding <math alttext="p_{\mathrm{det}}(x,\theta)" class="ltx_Math" display="inline" id="S2.SS5.p1.1.m1.2"><semantics id="S2.SS5.p1.1.m1.2a"><mrow id="S2.SS5.p1.1.m1.2.3" xref="S2.SS5.p1.1.m1.2.3.cmml"><msub id="S2.SS5.p1.1.m1.2.3.2" xref="S2.SS5.p1.1.m1.2.3.2.cmml"><mi id="S2.SS5.p1.1.m1.2.3.2.2" xref="S2.SS5.p1.1.m1.2.3.2.2.cmml">p</mi><mi id="S2.SS5.p1.1.m1.2.3.2.3" xref="S2.SS5.p1.1.m1.2.3.2.3.cmml">det</mi></msub><mo id="S2.SS5.p1.1.m1.2.3.1" xref="S2.SS5.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS5.p1.1.m1.2.3.3.2" xref="S2.SS5.p1.1.m1.2.3.3.1.cmml"><mo id="S2.SS5.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS5.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS5.p1.1.m1.1.1" xref="S2.SS5.p1.1.m1.1.1.cmml">x</mi><mo id="S2.SS5.p1.1.m1.2.3.3.2.2" xref="S2.SS5.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS5.p1.1.m1.2.2" xref="S2.SS5.p1.1.m1.2.2.cmml">θ</mi><mo id="S2.SS5.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS5.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.1.m1.2b"><apply id="S2.SS5.p1.1.m1.2.3.cmml" xref="S2.SS5.p1.1.m1.2.3"><times id="S2.SS5.p1.1.m1.2.3.1.cmml" xref="S2.SS5.p1.1.m1.2.3.1"></times><apply id="S2.SS5.p1.1.m1.2.3.2.cmml" xref="S2.SS5.p1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS5.p1.1.m1.2.3.2.1.cmml" xref="S2.SS5.p1.1.m1.2.3.2">subscript</csymbol><ci id="S2.SS5.p1.1.m1.2.3.2.2.cmml" xref="S2.SS5.p1.1.m1.2.3.2.2">𝑝</ci><ci id="S2.SS5.p1.1.m1.2.3.2.3.cmml" xref="S2.SS5.p1.1.m1.2.3.2.3">det</ci></apply><interval closure="open" id="S2.SS5.p1.1.m1.2.3.3.1.cmml" xref="S2.SS5.p1.1.m1.2.3.3.2"><ci id="S2.SS5.p1.1.m1.1.1.cmml" xref="S2.SS5.p1.1.m1.1.1">𝑥</ci><ci id="S2.SS5.p1.1.m1.2.2.cmml" xref="S2.SS5.p1.1.m1.2.2">𝜃</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.1.m1.2c">p_{\mathrm{det}}(x,\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.1.m1.2d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ( italic_x , italic_θ )</annotation></semantics></math> at the given mass, spins and redshift (distance) via a Monte Carlo over 1000 randomized extrinsic angular parameters and coalescence time values using the <span class="ltx_text ltx_font_italic" id="S2.SS5.p1.4.1">LISAbeta</span> package <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib90" title="">90</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib91" title="">91</a>]</cite>, also applying a random weak lensing factor to the signal amplitude via Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.E3" title="In II.2 Parameter Estimation ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">3</span></a>), to compute the optimal SNR, with a matched filter SNR threshold <math alttext="\rho_{\rm th}=8" class="ltx_Math" display="inline" id="S2.SS5.p1.2.m2.1"><semantics id="S2.SS5.p1.2.m2.1a"><mrow id="S2.SS5.p1.2.m2.1.1" xref="S2.SS5.p1.2.m2.1.1.cmml"><msub id="S2.SS5.p1.2.m2.1.1.2" xref="S2.SS5.p1.2.m2.1.1.2.cmml"><mi id="S2.SS5.p1.2.m2.1.1.2.2" xref="S2.SS5.p1.2.m2.1.1.2.2.cmml">ρ</mi><mi id="S2.SS5.p1.2.m2.1.1.2.3" xref="S2.SS5.p1.2.m2.1.1.2.3.cmml">th</mi></msub><mo id="S2.SS5.p1.2.m2.1.1.1" xref="S2.SS5.p1.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS5.p1.2.m2.1.1.3" xref="S2.SS5.p1.2.m2.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.2.m2.1b"><apply id="S2.SS5.p1.2.m2.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1"><eq id="S2.SS5.p1.2.m2.1.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1.1"></eq><apply id="S2.SS5.p1.2.m2.1.1.2.cmml" xref="S2.SS5.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.2.m2.1.1.2.1.cmml" xref="S2.SS5.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS5.p1.2.m2.1.1.2.2.cmml" xref="S2.SS5.p1.2.m2.1.1.2.2">𝜌</ci><ci id="S2.SS5.p1.2.m2.1.1.2.3.cmml" xref="S2.SS5.p1.2.m2.1.1.2.3">th</ci></apply><cn id="S2.SS5.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS5.p1.2.m2.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.2.m2.1c">\rho_{\rm th}=8</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.2.m2.1d">italic_ρ start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT = 8</annotation></semantics></math>. We then use the two methods of Section <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS4" title="II.4 Selection effects and validation of PE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.4</span></a> to obtain a KDE over redshift <math alttext="z" class="ltx_Math" display="inline" id="S2.SS5.p1.3.m3.1"><semantics id="S2.SS5.p1.3.m3.1a"><mi id="S2.SS5.p1.3.m3.1.1" xref="S2.SS5.p1.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.3.m3.1b"><ci id="S2.SS5.p1.3.m3.1.1.cmml" xref="S2.SS5.p1.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.3.m3.1d">italic_z</annotation></semantics></math> and (log) source-frame total mass, <math alttext="\log_{10}(M/M_{\odot})" class="ltx_Math" display="inline" id="S2.SS5.p1.4.m4.2"><semantics id="S2.SS5.p1.4.m4.2a"><mrow id="S2.SS5.p1.4.m4.2.2.2" xref="S2.SS5.p1.4.m4.2.2.3.cmml"><msub id="S2.SS5.p1.4.m4.1.1.1.1" xref="S2.SS5.p1.4.m4.1.1.1.1.cmml"><mi id="S2.SS5.p1.4.m4.1.1.1.1.2" xref="S2.SS5.p1.4.m4.1.1.1.1.2.cmml">log</mi><mn id="S2.SS5.p1.4.m4.1.1.1.1.3" xref="S2.SS5.p1.4.m4.1.1.1.1.3.cmml">10</mn></msub><mo id="S2.SS5.p1.4.m4.2.2.2a" xref="S2.SS5.p1.4.m4.2.2.3.cmml">⁡</mo><mrow id="S2.SS5.p1.4.m4.2.2.2.2" xref="S2.SS5.p1.4.m4.2.2.3.cmml"><mo id="S2.SS5.p1.4.m4.2.2.2.2.2" stretchy="false" xref="S2.SS5.p1.4.m4.2.2.3.cmml">(</mo><mrow id="S2.SS5.p1.4.m4.2.2.2.2.1" xref="S2.SS5.p1.4.m4.2.2.2.2.1.cmml"><mi id="S2.SS5.p1.4.m4.2.2.2.2.1.2" xref="S2.SS5.p1.4.m4.2.2.2.2.1.2.cmml">M</mi><mo id="S2.SS5.p1.4.m4.2.2.2.2.1.1" xref="S2.SS5.p1.4.m4.2.2.2.2.1.1.cmml">/</mo><msub id="S2.SS5.p1.4.m4.2.2.2.2.1.3" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3.cmml"><mi id="S2.SS5.p1.4.m4.2.2.2.2.1.3.2" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3.2.cmml">M</mi><mo id="S2.SS5.p1.4.m4.2.2.2.2.1.3.3" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3.3.cmml">⊙</mo></msub></mrow><mo id="S2.SS5.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S2.SS5.p1.4.m4.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.4.m4.2b"><apply id="S2.SS5.p1.4.m4.2.2.3.cmml" xref="S2.SS5.p1.4.m4.2.2.2"><apply id="S2.SS5.p1.4.m4.1.1.1.1.cmml" xref="S2.SS5.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.4.m4.1.1.1.1.1.cmml" xref="S2.SS5.p1.4.m4.1.1.1.1">subscript</csymbol><log id="S2.SS5.p1.4.m4.1.1.1.1.2.cmml" xref="S2.SS5.p1.4.m4.1.1.1.1.2"></log><cn id="S2.SS5.p1.4.m4.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p1.4.m4.1.1.1.1.3">10</cn></apply><apply id="S2.SS5.p1.4.m4.2.2.2.2.1.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1"><divide id="S2.SS5.p1.4.m4.2.2.2.2.1.1.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1.1"></divide><ci id="S2.SS5.p1.4.m4.2.2.2.2.1.2.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1.2">𝑀</ci><apply id="S2.SS5.p1.4.m4.2.2.2.2.1.3.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S2.SS5.p1.4.m4.2.2.2.2.1.3.1.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3">subscript</csymbol><ci id="S2.SS5.p1.4.m4.2.2.2.2.1.3.2.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3.2">𝑀</ci><csymbol cd="latexml" id="S2.SS5.p1.4.m4.2.2.2.2.1.3.3.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2.1.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.4.m4.2c">\log_{10}(M/M_{\odot})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.4.m4.2d">roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_M / italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS5.p2"> <p class="ltx_p" id="S2.SS5.p2.5">Considering different strategies to improve the accuracy of our KDE and account for selection effects, as discussed in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS4" title="II.4 Selection effects and validation of PE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.4</span></a>, we employ two distinct methods for population reconstruction. In our <em class="ltx_emph ltx_font_italic" id="S2.SS5.p2.5.1">first method</em> we apply adaptive KDE, computed using <span class="ltx_text ltx_font_smallcaps" id="S2.SS5.p2.5.2">awkde</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib96" title="">96</a>]</cite>, within the iterative framework described in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS3" title="II.3 Population reconstruction via iterative KDE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.3</span></a> to reconstruct the detected event distribution: we then only consider selection effects in the reweighting of samples for each event during iterative estimation. The global bandwidth <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS5.p2.1.m1.1"><semantics id="S2.SS5.p2.1.m1.1a"><mi id="S2.SS5.p2.1.m1.1.1" xref="S2.SS5.p2.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.1.m1.1b"><ci id="S2.SS5.p2.1.m1.1.1.cmml" xref="S2.SS5.p2.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.1.m1.1d">italic_β</annotation></semantics></math> and sensitivity parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.p2.2.m2.1"><semantics id="S2.SS5.p2.2.m2.1a"><mi id="S2.SS5.p2.2.m2.1.1" xref="S2.SS5.p2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.2.m2.1b"><ci id="S2.SS5.p2.2.m2.1.1.cmml" xref="S2.SS5.p2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.2.m2.1d">italic_α</annotation></semantics></math> are optimized at each iteration from a grid search, with <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS5.p2.3.m3.1"><semantics id="S2.SS5.p2.3.m3.1a"><mi id="S2.SS5.p2.3.m3.1.1" xref="S2.SS5.p2.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.3.m3.1b"><ci id="S2.SS5.p2.3.m3.1.1.cmml" xref="S2.SS5.p2.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.3.m3.1d">italic_β</annotation></semantics></math> ranging from 0.01 to 0.9 and <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.p2.4.m4.1"><semantics id="S2.SS5.p2.4.m4.1a"><mi id="S2.SS5.p2.4.m4.1.1" xref="S2.SS5.p2.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.4.m4.1b"><ci id="S2.SS5.p2.4.m4.1.1.cmml" xref="S2.SS5.p2.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.4.m4.1d">italic_α</annotation></semantics></math> from 0.1 to 0.8. During iterative reweighting, the value of the previous KDE at each sample is multiplied by an inverse selection function, here taken as <math alttext="1/\max(p_{\mathrm{det}},0.1)" class="ltx_Math" display="inline" id="S2.SS5.p2.5.m5.3"><semantics id="S2.SS5.p2.5.m5.3a"><mrow id="S2.SS5.p2.5.m5.3.3" xref="S2.SS5.p2.5.m5.3.3.cmml"><mn id="S2.SS5.p2.5.m5.3.3.3" xref="S2.SS5.p2.5.m5.3.3.3.cmml">1</mn><mo id="S2.SS5.p2.5.m5.3.3.2" xref="S2.SS5.p2.5.m5.3.3.2.cmml">/</mo><mrow id="S2.SS5.p2.5.m5.3.3.1.1" xref="S2.SS5.p2.5.m5.3.3.1.2.cmml"><mi id="S2.SS5.p2.5.m5.1.1" xref="S2.SS5.p2.5.m5.1.1.cmml">max</mi><mo id="S2.SS5.p2.5.m5.3.3.1.1a" xref="S2.SS5.p2.5.m5.3.3.1.2.cmml">⁡</mo><mrow id="S2.SS5.p2.5.m5.3.3.1.1.1" xref="S2.SS5.p2.5.m5.3.3.1.2.cmml"><mo id="S2.SS5.p2.5.m5.3.3.1.1.1.2" stretchy="false" xref="S2.SS5.p2.5.m5.3.3.1.2.cmml">(</mo><msub id="S2.SS5.p2.5.m5.3.3.1.1.1.1" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1.cmml"><mi id="S2.SS5.p2.5.m5.3.3.1.1.1.1.2" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1.2.cmml">p</mi><mi id="S2.SS5.p2.5.m5.3.3.1.1.1.1.3" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1.3.cmml">det</mi></msub><mo id="S2.SS5.p2.5.m5.3.3.1.1.1.3" xref="S2.SS5.p2.5.m5.3.3.1.2.cmml">,</mo><mn id="S2.SS5.p2.5.m5.2.2" xref="S2.SS5.p2.5.m5.2.2.cmml">0.1</mn><mo id="S2.SS5.p2.5.m5.3.3.1.1.1.4" stretchy="false" xref="S2.SS5.p2.5.m5.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.5.m5.3b"><apply id="S2.SS5.p2.5.m5.3.3.cmml" xref="S2.SS5.p2.5.m5.3.3"><divide id="S2.SS5.p2.5.m5.3.3.2.cmml" xref="S2.SS5.p2.5.m5.3.3.2"></divide><cn id="S2.SS5.p2.5.m5.3.3.3.cmml" type="integer" xref="S2.SS5.p2.5.m5.3.3.3">1</cn><apply id="S2.SS5.p2.5.m5.3.3.1.2.cmml" xref="S2.SS5.p2.5.m5.3.3.1.1"><max id="S2.SS5.p2.5.m5.1.1.cmml" xref="S2.SS5.p2.5.m5.1.1"></max><apply id="S2.SS5.p2.5.m5.3.3.1.1.1.1.cmml" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p2.5.m5.3.3.1.1.1.1.1.cmml" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p2.5.m5.3.3.1.1.1.1.2.cmml" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1.2">𝑝</ci><ci id="S2.SS5.p2.5.m5.3.3.1.1.1.1.3.cmml" xref="S2.SS5.p2.5.m5.3.3.1.1.1.1.3">det</ci></apply><cn id="S2.SS5.p2.5.m5.2.2.cmml" type="float" xref="S2.SS5.p2.5.m5.2.2">0.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.5.m5.3c">1/\max(p_{\mathrm{det}},0.1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.5.m5.3d">1 / roman_max ( italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT , 0.1 )</annotation></semantics></math>. We also performed an analysis where the reweighting steps do not contain this selection factor, the results being presented in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A1" title="Appendix A Results from adaptive KDE reweighting without selection factor ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">A</span></a>.</p> </div> <div class="ltx_para" id="S2.SS5.p3"> <p class="ltx_p" id="S2.SS5.p3.4">Our <em class="ltx_emph ltx_font_italic" id="S2.SS5.p3.4.1">second method</em> aims to reconstruct the astrophysical distribution: we thus take into account selection effects using a weight <math alttext="W_{i}" class="ltx_Math" display="inline" id="S2.SS5.p3.1.m1.1"><semantics id="S2.SS5.p3.1.m1.1a"><msub id="S2.SS5.p3.1.m1.1.1" xref="S2.SS5.p3.1.m1.1.1.cmml"><mi id="S2.SS5.p3.1.m1.1.1.2" xref="S2.SS5.p3.1.m1.1.1.2.cmml">W</mi><mi id="S2.SS5.p3.1.m1.1.1.3" xref="S2.SS5.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.1.m1.1b"><apply id="S2.SS5.p3.1.m1.1.1.cmml" xref="S2.SS5.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.1.m1.1.1.1.cmml" xref="S2.SS5.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS5.p3.1.m1.1.1.2.cmml" xref="S2.SS5.p3.1.m1.1.1.2">𝑊</ci><ci id="S2.SS5.p3.1.m1.1.1.3.cmml" xref="S2.SS5.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.1.m1.1c">W_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.1.m1.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> derived from <math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S2.SS5.p3.2.m2.1"><semantics id="S2.SS5.p3.2.m2.1a"><msub id="S2.SS5.p3.2.m2.1.1" xref="S2.SS5.p3.2.m2.1.1.cmml"><mi id="S2.SS5.p3.2.m2.1.1.2" xref="S2.SS5.p3.2.m2.1.1.2.cmml">p</mi><mi id="S2.SS5.p3.2.m2.1.1.3" xref="S2.SS5.p3.2.m2.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.2.m2.1b"><apply id="S2.SS5.p3.2.m2.1.1.cmml" xref="S2.SS5.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.2.m2.1.1.1.cmml" xref="S2.SS5.p3.2.m2.1.1">subscript</csymbol><ci id="S2.SS5.p3.2.m2.1.1.2.cmml" xref="S2.SS5.p3.2.m2.1.1.2">𝑝</ci><ci id="S2.SS5.p3.2.m2.1.1.3.cmml" xref="S2.SS5.p3.2.m2.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.2.m2.1c">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> for each given sample. We implement the weighting via a standard KDE (without adaptive bandwidth) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib104" title="">104</a>]</cite>, using <math alttext="1/\max(p_{\mathrm{det}},0.1)" class="ltx_Math" display="inline" id="S2.SS5.p3.3.m3.3"><semantics id="S2.SS5.p3.3.m3.3a"><mrow id="S2.SS5.p3.3.m3.3.3" xref="S2.SS5.p3.3.m3.3.3.cmml"><mn id="S2.SS5.p3.3.m3.3.3.3" xref="S2.SS5.p3.3.m3.3.3.3.cmml">1</mn><mo id="S2.SS5.p3.3.m3.3.3.2" xref="S2.SS5.p3.3.m3.3.3.2.cmml">/</mo><mrow id="S2.SS5.p3.3.m3.3.3.1.1" xref="S2.SS5.p3.3.m3.3.3.1.2.cmml"><mi id="S2.SS5.p3.3.m3.1.1" xref="S2.SS5.p3.3.m3.1.1.cmml">max</mi><mo id="S2.SS5.p3.3.m3.3.3.1.1a" xref="S2.SS5.p3.3.m3.3.3.1.2.cmml">⁡</mo><mrow id="S2.SS5.p3.3.m3.3.3.1.1.1" xref="S2.SS5.p3.3.m3.3.3.1.2.cmml"><mo id="S2.SS5.p3.3.m3.3.3.1.1.1.2" stretchy="false" xref="S2.SS5.p3.3.m3.3.3.1.2.cmml">(</mo><msub id="S2.SS5.p3.3.m3.3.3.1.1.1.1" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1.cmml"><mi id="S2.SS5.p3.3.m3.3.3.1.1.1.1.2" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1.2.cmml">p</mi><mi id="S2.SS5.p3.3.m3.3.3.1.1.1.1.3" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1.3.cmml">det</mi></msub><mo id="S2.SS5.p3.3.m3.3.3.1.1.1.3" xref="S2.SS5.p3.3.m3.3.3.1.2.cmml">,</mo><mn id="S2.SS5.p3.3.m3.2.2" xref="S2.SS5.p3.3.m3.2.2.cmml">0.1</mn><mo id="S2.SS5.p3.3.m3.3.3.1.1.1.4" stretchy="false" xref="S2.SS5.p3.3.m3.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.3.m3.3b"><apply id="S2.SS5.p3.3.m3.3.3.cmml" xref="S2.SS5.p3.3.m3.3.3"><divide id="S2.SS5.p3.3.m3.3.3.2.cmml" xref="S2.SS5.p3.3.m3.3.3.2"></divide><cn id="S2.SS5.p3.3.m3.3.3.3.cmml" type="integer" xref="S2.SS5.p3.3.m3.3.3.3">1</cn><apply id="S2.SS5.p3.3.m3.3.3.1.2.cmml" xref="S2.SS5.p3.3.m3.3.3.1.1"><max id="S2.SS5.p3.3.m3.1.1.cmml" xref="S2.SS5.p3.3.m3.1.1"></max><apply id="S2.SS5.p3.3.m3.3.3.1.1.1.1.cmml" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.3.m3.3.3.1.1.1.1.1.cmml" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p3.3.m3.3.3.1.1.1.1.2.cmml" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1.2">𝑝</ci><ci id="S2.SS5.p3.3.m3.3.3.1.1.1.1.3.cmml" xref="S2.SS5.p3.3.m3.3.3.1.1.1.1.3">det</ci></apply><cn id="S2.SS5.p3.3.m3.2.2.cmml" type="float" xref="S2.SS5.p3.3.m3.2.2">0.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.3.m3.3c">1/\max(p_{\mathrm{det}},0.1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.3.m3.3d">1 / roman_max ( italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT , 0.1 )</annotation></semantics></math> as weights for our main result. The only hyperparameter to be optimized is then the global bandwidth <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS5.p3.4.m4.1"><semantics id="S2.SS5.p3.4.m4.1a"><mi id="S2.SS5.p3.4.m4.1.1" xref="S2.SS5.p3.4.m4.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.4.m4.1b"><ci id="S2.SS5.p3.4.m4.1.1.cmml" xref="S2.SS5.p3.4.m4.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.4.m4.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.4.m4.1d">italic_β</annotation></semantics></math>, searching in a grid of values ranking from 0.01 to 0.9 as for the first method. We compare the resulting weighted KDE with the true astrophysical distribution in the next section.</p> </div> <div class="ltx_para" id="S2.SS5.p4"> <p class="ltx_p" id="S2.SS5.p4.4">To derive rate density estimates from our weighted KDEs, we count detected events and use sample weights. For regions with <math alttext="p_{\mathrm{det}}\simeq 1" class="ltx_Math" display="inline" id="S2.SS5.p4.1.m1.1"><semantics id="S2.SS5.p4.1.m1.1a"><mrow id="S2.SS5.p4.1.m1.1.1" xref="S2.SS5.p4.1.m1.1.1.cmml"><msub id="S2.SS5.p4.1.m1.1.1.2" xref="S2.SS5.p4.1.m1.1.1.2.cmml"><mi id="S2.SS5.p4.1.m1.1.1.2.2" xref="S2.SS5.p4.1.m1.1.1.2.2.cmml">p</mi><mi id="S2.SS5.p4.1.m1.1.1.2.3" xref="S2.SS5.p4.1.m1.1.1.2.3.cmml">det</mi></msub><mo id="S2.SS5.p4.1.m1.1.1.1" xref="S2.SS5.p4.1.m1.1.1.1.cmml">≃</mo><mn id="S2.SS5.p4.1.m1.1.1.3" xref="S2.SS5.p4.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.1.m1.1b"><apply id="S2.SS5.p4.1.m1.1.1.cmml" xref="S2.SS5.p4.1.m1.1.1"><csymbol cd="latexml" id="S2.SS5.p4.1.m1.1.1.1.cmml" xref="S2.SS5.p4.1.m1.1.1.1">similar-to-or-equals</csymbol><apply id="S2.SS5.p4.1.m1.1.1.2.cmml" xref="S2.SS5.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p4.1.m1.1.1.2.1.cmml" xref="S2.SS5.p4.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS5.p4.1.m1.1.1.2.2.cmml" xref="S2.SS5.p4.1.m1.1.1.2.2">𝑝</ci><ci id="S2.SS5.p4.1.m1.1.1.2.3.cmml" xref="S2.SS5.p4.1.m1.1.1.2.3">det</ci></apply><cn id="S2.SS5.p4.1.m1.1.1.3.cmml" type="integer" xref="S2.SS5.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.1.m1.1c">p_{\mathrm{det}}\simeq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ≃ 1</annotation></semantics></math> we have a one-to-one correspondence between detected and astrophysical events. For <math alttext="p_{\mathrm{det}}<1" class="ltx_Math" display="inline" id="S2.SS5.p4.2.m2.1"><semantics id="S2.SS5.p4.2.m2.1a"><mrow id="S2.SS5.p4.2.m2.1.1" xref="S2.SS5.p4.2.m2.1.1.cmml"><msub id="S2.SS5.p4.2.m2.1.1.2" xref="S2.SS5.p4.2.m2.1.1.2.cmml"><mi id="S2.SS5.p4.2.m2.1.1.2.2" xref="S2.SS5.p4.2.m2.1.1.2.2.cmml">p</mi><mi id="S2.SS5.p4.2.m2.1.1.2.3" xref="S2.SS5.p4.2.m2.1.1.2.3.cmml">det</mi></msub><mo id="S2.SS5.p4.2.m2.1.1.1" xref="S2.SS5.p4.2.m2.1.1.1.cmml"><</mo><mn id="S2.SS5.p4.2.m2.1.1.3" xref="S2.SS5.p4.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.2.m2.1b"><apply id="S2.SS5.p4.2.m2.1.1.cmml" xref="S2.SS5.p4.2.m2.1.1"><lt id="S2.SS5.p4.2.m2.1.1.1.cmml" xref="S2.SS5.p4.2.m2.1.1.1"></lt><apply id="S2.SS5.p4.2.m2.1.1.2.cmml" xref="S2.SS5.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p4.2.m2.1.1.2.1.cmml" xref="S2.SS5.p4.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS5.p4.2.m2.1.1.2.2.cmml" xref="S2.SS5.p4.2.m2.1.1.2.2">𝑝</ci><ci id="S2.SS5.p4.2.m2.1.1.2.3.cmml" xref="S2.SS5.p4.2.m2.1.1.2.3">det</ci></apply><cn id="S2.SS5.p4.2.m2.1.1.3.cmml" type="integer" xref="S2.SS5.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.2.m2.1c">p_{\mathrm{det}}<1</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT < 1</annotation></semantics></math>, we scale the astrophysical density relative to the detected event density by 1/<math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S2.SS5.p4.3.m3.1"><semantics id="S2.SS5.p4.3.m3.1a"><msub id="S2.SS5.p4.3.m3.1.1" xref="S2.SS5.p4.3.m3.1.1.cmml"><mi id="S2.SS5.p4.3.m3.1.1.2" xref="S2.SS5.p4.3.m3.1.1.2.cmml">p</mi><mi id="S2.SS5.p4.3.m3.1.1.3" xref="S2.SS5.p4.3.m3.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.3.m3.1b"><apply id="S2.SS5.p4.3.m3.1.1.cmml" xref="S2.SS5.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.3.m3.1.1.1.cmml" xref="S2.SS5.p4.3.m3.1.1">subscript</csymbol><ci id="S2.SS5.p4.3.m3.1.1.2.cmml" xref="S2.SS5.p4.3.m3.1.1.2">𝑝</ci><ci id="S2.SS5.p4.3.m3.1.1.3.cmml" xref="S2.SS5.p4.3.m3.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.3.m3.1c">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.3.m3.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> to account for reduced detection probability. Thus, each KDE is normalized by the sum of weights <math alttext="\sum_{i}W_{i}" class="ltx_Math" display="inline" id="S2.SS5.p4.4.m4.1"><semantics id="S2.SS5.p4.4.m4.1a"><mrow id="S2.SS5.p4.4.m4.1.1" xref="S2.SS5.p4.4.m4.1.1.cmml"><msub id="S2.SS5.p4.4.m4.1.1.1" xref="S2.SS5.p4.4.m4.1.1.1.cmml"><mo id="S2.SS5.p4.4.m4.1.1.1.2" xref="S2.SS5.p4.4.m4.1.1.1.2.cmml">∑</mo><mi id="S2.SS5.p4.4.m4.1.1.1.3" xref="S2.SS5.p4.4.m4.1.1.1.3.cmml">i</mi></msub><msub id="S2.SS5.p4.4.m4.1.1.2" xref="S2.SS5.p4.4.m4.1.1.2.cmml"><mi id="S2.SS5.p4.4.m4.1.1.2.2" xref="S2.SS5.p4.4.m4.1.1.2.2.cmml">W</mi><mi id="S2.SS5.p4.4.m4.1.1.2.3" xref="S2.SS5.p4.4.m4.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.4.m4.1b"><apply id="S2.SS5.p4.4.m4.1.1.cmml" xref="S2.SS5.p4.4.m4.1.1"><apply id="S2.SS5.p4.4.m4.1.1.1.cmml" xref="S2.SS5.p4.4.m4.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.4.m4.1.1.1.1.cmml" xref="S2.SS5.p4.4.m4.1.1.1">subscript</csymbol><sum id="S2.SS5.p4.4.m4.1.1.1.2.cmml" xref="S2.SS5.p4.4.m4.1.1.1.2"></sum><ci id="S2.SS5.p4.4.m4.1.1.1.3.cmml" xref="S2.SS5.p4.4.m4.1.1.1.3">𝑖</ci></apply><apply id="S2.SS5.p4.4.m4.1.1.2.cmml" xref="S2.SS5.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p4.4.m4.1.1.2.1.cmml" xref="S2.SS5.p4.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS5.p4.4.m4.1.1.2.2.cmml" xref="S2.SS5.p4.4.m4.1.1.2.2">𝑊</ci><ci id="S2.SS5.p4.4.m4.1.1.2.3.cmml" xref="S2.SS5.p4.4.m4.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.4.m4.1c">\sum_{i}W_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.4.m4.1d">∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to ensure proper scaling (note that the the normalization factor varies between iterations).</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span>Results</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.2">Considering all injected events with SNR above the threshold of 8 in <math alttext="4\," class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mn id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><cn id="S3.p1.1.m1.1.1.cmml" type="integer" xref="S3.p1.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">4\,</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">4</annotation></semantics></math>yr of simulated LISA observations, we determine the optimal SNR for each posterior sample, using both the intrinsic and extrinsic sample parameters. As described in Section <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS4" title="II.4 Selection effects and validation of PE ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.4</span></a>, we exclude certain events with poor PE performance. From the remaining 326 events, only samples with an optimal SNR above 4 are retained. From this subset, we randomly select 100 samples per event and compute their detection probability <math alttext="p_{\mathrm{det}}" 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 id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">p</mi><mi id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">det</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">𝑝</ci><ci id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> using a matched filter SNR threshold of 8.</p> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="180" id="S3.F3.g1" src="extracted/6297058/Figure_3scatter_PE100samplesperevents_with_contours.png" width="287"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Detection probability <math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S3.F3.5.m1.1"><semantics id="S3.F3.5.m1.1b"><msub id="S3.F3.5.m1.1.1" xref="S3.F3.5.m1.1.1.cmml"><mi id="S3.F3.5.m1.1.1.2" xref="S3.F3.5.m1.1.1.2.cmml">p</mi><mi id="S3.F3.5.m1.1.1.3" xref="S3.F3.5.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F3.5.m1.1c"><apply id="S3.F3.5.m1.1.1.cmml" xref="S3.F3.5.m1.1.1"><csymbol cd="ambiguous" id="S3.F3.5.m1.1.1.1.cmml" xref="S3.F3.5.m1.1.1">subscript</csymbol><ci id="S3.F3.5.m1.1.1.2.cmml" xref="S3.F3.5.m1.1.1.2">𝑝</ci><ci id="S3.F3.5.m1.1.1.3.cmml" xref="S3.F3.5.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.5.m1.1d">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.5.m1.1e">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> for 100 randomly chosen posterior samples from each detected event. Taking the intrinsic parameters and redshift of each posterior sample, we perform a Monte Carlo integration over the remaining extrinsic parameters and apply a matched filter SNR threshold of <math alttext="8" class="ltx_Math" display="inline" id="S3.F3.6.m2.1"><semantics id="S3.F3.6.m2.1b"><mn id="S3.F3.6.m2.1.1" xref="S3.F3.6.m2.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S3.F3.6.m2.1c"><cn id="S3.F3.6.m2.1.1.cmml" type="integer" xref="S3.F3.6.m2.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.6.m2.1d">8</annotation><annotation encoding="application/x-llamapun" id="S3.F3.6.m2.1e">8</annotation></semantics></math> to obtain <math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S3.F3.7.m3.1"><semantics id="S3.F3.7.m3.1b"><msub id="S3.F3.7.m3.1.1" xref="S3.F3.7.m3.1.1.cmml"><mi id="S3.F3.7.m3.1.1.2" xref="S3.F3.7.m3.1.1.2.cmml">p</mi><mi id="S3.F3.7.m3.1.1.3" xref="S3.F3.7.m3.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F3.7.m3.1c"><apply id="S3.F3.7.m3.1.1.cmml" xref="S3.F3.7.m3.1.1"><csymbol cd="ambiguous" id="S3.F3.7.m3.1.1.1.cmml" xref="S3.F3.7.m3.1.1">subscript</csymbol><ci id="S3.F3.7.m3.1.1.2.cmml" xref="S3.F3.7.m3.1.1.2">𝑝</ci><ci id="S3.F3.7.m3.1.1.3.cmml" xref="S3.F3.7.m3.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.7.m3.1d">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.7.m3.1e">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>. We add contours (black dashed lines) to guide the eye based on these <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S3.F3.8.m4.1"><semantics id="S3.F3.8.m4.1b"><msub id="S3.F3.8.m4.1.1" xref="S3.F3.8.m4.1.1.cmml"><mi id="S3.F3.8.m4.1.1.2" xref="S3.F3.8.m4.1.1.2.cmml">p</mi><mi id="S3.F3.8.m4.1.1.3" xref="S3.F3.8.m4.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F3.8.m4.1c"><apply id="S3.F3.8.m4.1.1.cmml" xref="S3.F3.8.m4.1.1"><csymbol cd="ambiguous" id="S3.F3.8.m4.1.1.1.cmml" xref="S3.F3.8.m4.1.1">subscript</csymbol><ci id="S3.F3.8.m4.1.1.2.cmml" xref="S3.F3.8.m4.1.1.2">𝑝</ci><ci id="S3.F3.8.m4.1.1.3.cmml" xref="S3.F3.8.m4.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.8.m4.1d">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.8.m4.1e">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> values with nearest-neighbor interpolation and Gaussian smoothing. </figcaption> </figure> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.2">Figure <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F3" title="Figure 3 ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">3</span></a> shows the total source-frame mass and redshift of the chosen samples from all events passing our criteria, colored according to <math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">p</mi><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">𝑝</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>. The range of redshifts covered by these samples is much wider than the true event distribution, reflecting the large PE uncertainties, and the range of source total masses correspondingly extends well below <math alttext="10^{3}\,M_{\odot}" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><msup id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml"><mn id="S3.p2.2.m2.1.1.2.2" xref="S3.p2.2.m2.1.1.2.2.cmml">10</mn><mn id="S3.p2.2.m2.1.1.2.3" xref="S3.p2.2.m2.1.1.2.3.cmml">3</mn></msup><mo id="S3.p2.2.m2.1.1.1" lspace="0.170em" xref="S3.p2.2.m2.1.1.1.cmml">⁢</mo><msub id="S3.p2.2.m2.1.1.3" xref="S3.p2.2.m2.1.1.3.cmml"><mi id="S3.p2.2.m2.1.1.3.2" xref="S3.p2.2.m2.1.1.3.2.cmml">M</mi><mo id="S3.p2.2.m2.1.1.3.3" xref="S3.p2.2.m2.1.1.3.3.cmml">⊙</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><times id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1"></times><apply id="S3.p2.2.m2.1.1.2.cmml" xref="S3.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p2.2.m2.1.1.2.1.cmml" xref="S3.p2.2.m2.1.1.2">superscript</csymbol><cn id="S3.p2.2.m2.1.1.2.2.cmml" type="integer" xref="S3.p2.2.m2.1.1.2.2">10</cn><cn id="S3.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S3.p2.2.m2.1.1.2.3">3</cn></apply><apply id="S3.p2.2.m2.1.1.3.cmml" xref="S3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p2.2.m2.1.1.3.1.cmml" xref="S3.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.p2.2.m2.1.1.3.2.cmml" xref="S3.p2.2.m2.1.1.3.2">𝑀</ci><csymbol cd="latexml" id="S3.p2.2.m2.1.1.3.3.cmml" xref="S3.p2.2.m2.1.1.3.3">direct-product</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">10^{3}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> in contrast to the true values shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.F2" title="Figure 2 ‣ II.1 Astrophysical population model and detectability ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.4">To test the accuracy of our methods, we compare to the properties of the underlying population, which is described in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS1" title="II.1 Astrophysical population model and detectability ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.1</span></a>, focusing on the total mass and redshift of the mergers. For this comparison, we construct KDEs over (true) total mass and redshift values for the underlying population and its detectable sub-population, using <span class="ltx_text ltx_font_smallcaps" id="S3.p3.4.1">awkde</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib96" title="">96</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib95" title="">95</a>]</cite>. These reference KDEs represent the <em class="ltx_emph ltx_font_italic" id="S3.p3.4.2">expected</em> distributions, and can be thought as averages over several realizations of actual 4 yr merger catalogs. In detail, the KDE global bandwidth <math alttext="\beta" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mi id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_β</annotation></semantics></math> and the <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><mi id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><ci id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_α</annotation></semantics></math> parameter are optimized with a grid search with <math alttext="\beta" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><mi id="S3.p3.3.m3.1.1" xref="S3.p3.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><ci id="S3.p3.3.m3.1.1.cmml" xref="S3.p3.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">italic_β</annotation></semantics></math> values ranging from 0.01 to 1.0 and <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.p3.4.m4.1"><semantics id="S3.p3.4.m4.1a"><mi id="S3.p3.4.m4.1.1" xref="S3.p3.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.1b"><ci id="S3.p3.4.m4.1.1.cmml" xref="S3.p3.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.1d">italic_α</annotation></semantics></math> values from 0 to 1.</p> </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="410" id="S3.F4.g1" src="extracted/6297058/beta_Figure_4_AwKDE_withpdet_factor2DKDE_min10.png" width="287"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Top: KDE of the expected distribution over total mass and redshift, obtained using true parameter values sampled from the detectable subpopulation; <math alttext="\beta" class="ltx_Math" display="inline" id="S3.F4.3.m1.1"><semantics id="S3.F4.3.m1.1b"><mi id="S3.F4.3.m1.1.1" xref="S3.F4.3.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.F4.3.m1.1c"><ci id="S3.F4.3.m1.1.1.cmml" xref="S3.F4.3.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.3.m1.1d">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.F4.3.m1.1e">italic_β</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.F4.4.m2.1"><semantics id="S3.F4.4.m2.1b"><mi id="S3.F4.4.m2.1.1" xref="S3.F4.4.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.F4.4.m2.1c"><ci id="S3.F4.4.m2.1.1.cmml" xref="S3.F4.4.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.4.m2.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.F4.4.m2.1e">italic_α</annotation></semantics></math> are the optimized bandwidth and adaptive parameter respectively. Bottom: reconstructed distribution from the median of adaptive iterative KDEs from posterior samples of detected events over a 4 year observation time. </figcaption> </figure> <section class="ltx_paragraph" id="S3.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">First method: Unweighted adaptive KDE for the detected rate density</h4> <div class="ltx_para" id="S3.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p1.1">Following Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS5" title="II.5 Application to simulated data ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.5</span></a>, we reconstruct the detected distribution over total mass and redshift from our posterior sample data, using adaptive but unweighted KDE. The results are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F4" title="Figure 4 ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">4</span></a>, where the median KDE (bottom) is compared with the KDE expected distribution of the detectable subpopulation (top): the two estimates closely resemble each other in most of the parameter space. However, the true expected distribution (top) exhibits structure at high masses <math alttext=">10^{7}\,M_{\odot}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS0.SSS0.Px1.p1.1.m1.1a"><mrow id="S3.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2.cmml"></mi><mo id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.1" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.1.cmml">></mo><mrow id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.cmml"><msup id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.cmml"><mn id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.2" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.2.cmml">10</mn><mn id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.3" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.3.cmml">7</mn></msup><mo id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.1" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.cmml"><mi id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.2" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.3" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.1.m1.1b"><apply id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1"><gt id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.1"></gt><csymbol cd="latexml" id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2">absent</csymbol><apply id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3"><times id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.1"></times><apply id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2">superscript</csymbol><cn id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.2.cmml" type="integer" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.2">10</cn><cn id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.2.3">7</cn></apply><apply id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.1.m1.1c">>10^{7}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p1.1.m1.1d">> 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> which is absent from the reconstruction, due to the lack of detected high-mass mergers in our mock data. Conversely, at high redshift the range of mass covered by the reconstruction is slightly broader than the true detected distribution, likely due to the residual effect of PE uncertainties in source total mass. The iterative reweighting method reduces the impact of single-event measurement uncertainties, ensuring a more accurate and robust reconstruction, but may not be able to completely compensate for these effects.</p> </div> <figure class="ltx_figure" id="S3.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="S3.F5.g1" src="extracted/6297058/Figure_4a_AwKDE_with_pdet_fatcor_Msrc_min10.png" width="287"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="S3.F5.g2" src="extracted/6297058/Figure_4b_AwKDE_with_pdet_fatcor_Redshift_min10.png" width="287"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Rate densities over total source mass (top) and redshift (bottom) for 4 years of LISA observation. We use adaptive unweighted iterative KDE (dark red), plotting the median reconstruction (solid) and <math alttext="90\%" class="ltx_Math" display="inline" id="S3.F5.2.m1.1"><semantics id="S3.F5.2.m1.1b"><mrow id="S3.F5.2.m1.1.1" xref="S3.F5.2.m1.1.1.cmml"><mn id="S3.F5.2.m1.1.1.2" xref="S3.F5.2.m1.1.1.2.cmml">90</mn><mo id="S3.F5.2.m1.1.1.1" xref="S3.F5.2.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.2.m1.1c"><apply id="S3.F5.2.m1.1.1.cmml" xref="S3.F5.2.m1.1.1"><csymbol cd="latexml" id="S3.F5.2.m1.1.1.1.cmml" xref="S3.F5.2.m1.1.1.1">percent</csymbol><cn id="S3.F5.2.m1.1.1.2.cmml" type="integer" xref="S3.F5.2.m1.1.1.2">90</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.2.m1.1d">90\%</annotation><annotation encoding="application/x-llamapun" id="S3.F5.2.m1.1e">90 %</annotation></semantics></math> symmetric interval (dashed) compared with a histogram of true detected event values (light blue).</figcaption> </figure> <div class="ltx_para" id="S3.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p2.3">We convert the two-dimensional KDEs into one-dimensional KDEs by marginalizing over each variable in turn; one-dimensional detected event rate densities are then estimated by multiplying by the total number of observed events, 326. We compare the resulting reconstructed rates against the expected distribution of detectable events obtained from the astrophysical model in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F5" title="Figure 5 ‣ First method: Unweighted adaptive KDE for the detected rate density ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">5</span></a>. 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To assess the impact of this factor, we also conducted an analysis entirely without accounting for selection bias; the resulting rate estimates, presented in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A1" title="Appendix A Results from adaptive KDE reweighting without selection factor ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">A</span></a>, confirm that this neglect leads to a significant downward bias in the redshift distribution.</p> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px1.p4"> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p4.3">We also investigate the effect of using a simple <math alttext="1/p_{\rm det}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p4.1.m1.1"><semantics id="S3.SS0.SSS0.Px1.p4.1.m1.1a"><mrow id="S3.SS0.SSS0.Px1.p4.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.cmml"><mn id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.2" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.2.cmml">1</mn><mo id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.1" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.1.cmml">/</mo><msub id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.2" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.2.cmml">p</mi><mi id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.3" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.3.cmml">det</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p4.1.m1.1b"><apply id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1"><divide id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.1"></divide><cn id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.2.cmml" type="integer" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.2">1</cn><apply id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.2">𝑝</ci><ci id="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p4.1.m1.1.1.3.3">det</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p4.1.m1.1c">1/p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p4.1.m1.1d">1 / italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> factor in reweighting (i.e. without capping <math alttext="p_{\rm det}<0.1" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p4.2.m2.1"><semantics id="S3.SS0.SSS0.Px1.p4.2.m2.1a"><mrow id="S3.SS0.SSS0.Px1.p4.2.m2.1.1" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.cmml"><msub id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.2" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.2.cmml">p</mi><mi id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.3" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.3.cmml">det</mi></msub><mo id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.1" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.1.cmml"><</mo><mn id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.3" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p4.2.m2.1b"><apply id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1"><lt id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.1"></lt><apply id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.2">𝑝</ci><ci id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.2.3">det</ci></apply><cn id="S3.SS0.SSS0.Px1.p4.2.m2.1.1.3.cmml" type="float" xref="S3.SS0.SSS0.Px1.p4.2.m2.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p4.2.m2.1c">p_{\rm det}<0.1</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p4.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT < 0.1</annotation></semantics></math> values), with results shown in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A2" title="Appendix B Results from adaptive KDE using simple 1/𝑝_det reweighting ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">B</span></a>. 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<section class="ltx_paragraph" id="S3.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Second method: Weighted KDE for the astrophysical distribution</h4> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p1.1">Here, we follow Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS5" title="II.5 Application to simulated data ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.5</span></a>, incorporating selection effects via weights <math alttext="W_{i}=1/\max(p_{\rm det,i},0.1)" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.1.m1.5"><semantics id="S3.SS0.SSS0.Px2.p1.1.m1.5a"><mrow id="S3.SS0.SSS0.Px2.p1.1.m1.5.5" xref="S3.SS0.SSS0.Px2.p1.1.m1.5.5.cmml"><msub id="S3.SS0.SSS0.Px2.p1.1.m1.5.5.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.5.5.3.cmml"><mi id="S3.SS0.SSS0.Px2.p1.1.m1.5.5.3.2" 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encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.1.m1.5d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 / roman_max ( italic_p start_POSTSUBSCRIPT roman_det , roman_i end_POSTSUBSCRIPT , 0.1 )</annotation></semantics></math> applied to the posterior samples in order to estimate the underlying astrophysical distribution. For comparison, an adaptive KDE of true mass and redshift value from the full astrophysical population is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F6" title="Figure 6 ‣ Second method: Weighted KDE for the astrophysical distribution ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">6</span></a> (top).</p> </div> <figure class="ltx_figure" id="S3.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="410" id="S3.F6.g1" src="extracted/6297058/beta_Figure_5_WeightedKDE_2DKDE_withPdet_min10.png" width="287"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Top: KDE of the expected distribution of mergers over total source mass and redshift; <math alttext="\beta" class="ltx_Math" display="inline" id="S3.F6.3.m1.1"><semantics id="S3.F6.3.m1.1b"><mi id="S3.F6.3.m1.1.1" xref="S3.F6.3.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.F6.3.m1.1c"><ci id="S3.F6.3.m1.1.1.cmml" xref="S3.F6.3.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.3.m1.1d">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.F6.3.m1.1e">italic_β</annotation></semantics></math>, <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.F6.4.m2.1"><semantics id="S3.F6.4.m2.1b"><mi id="S3.F6.4.m2.1.1" xref="S3.F6.4.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.F6.4.m2.1c"><ci id="S3.F6.4.m2.1.1.cmml" xref="S3.F6.4.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.4.m2.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.F6.4.m2.1e">italic_α</annotation></semantics></math> are the global bandwidth and local adaptive smoothing parameter respectively. Bottom: reconstructed distribution from the median of adaptive weighted KDEs from posterior samples of detected events over a 4 year observation time. </figcaption> </figure> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p2"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p2.5">The reconstructed distribution from iterative weighted KDE is shown in the bottom plot of Figure <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F6" title="Figure 6 ‣ Second method: Weighted KDE for the astrophysical distribution ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">6</span></a>, showing similar behavior over most of the <math alttext="M" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p2.1.m1.1"><semantics id="S3.SS0.SSS0.Px2.p2.1.m1.1a"><mi id="S3.SS0.SSS0.Px2.p2.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.1.m1.1b"><ci id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p2.1.m1.1d">italic_M</annotation></semantics></math>–<math alttext="z" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p2.2.m2.1"><semantics id="S3.SS0.SSS0.Px2.p2.2.m2.1a"><mi id="S3.SS0.SSS0.Px2.p2.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.2.m2.1b"><ci id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p2.2.m2.1d">italic_z</annotation></semantics></math> plane. The KDE sample weights effectively account for selection effects at low mass and high redshift, demonstrating their crucial role in population reconstruction. 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We use weighted iterative KDE (dark red), plotting the median reconstruction (solid) and <math alttext="90\%" class="ltx_Math" display="inline" id="S3.F7.2.m1.1"><semantics id="S3.F7.2.m1.1b"><mrow id="S3.F7.2.m1.1.1" xref="S3.F7.2.m1.1.1.cmml"><mn id="S3.F7.2.m1.1.1.2" xref="S3.F7.2.m1.1.1.2.cmml">90</mn><mo id="S3.F7.2.m1.1.1.1" xref="S3.F7.2.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.2.m1.1c"><apply id="S3.F7.2.m1.1.1.cmml" xref="S3.F7.2.m1.1.1"><csymbol cd="latexml" id="S3.F7.2.m1.1.1.1.cmml" xref="S3.F7.2.m1.1.1.1">percent</csymbol><cn id="S3.F7.2.m1.1.1.2.cmml" type="integer" xref="S3.F7.2.m1.1.1.2">90</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.2.m1.1d">90\%</annotation><annotation encoding="application/x-llamapun" id="S3.F7.2.m1.1e">90 %</annotation></semantics></math> symmetric interval (dashed) compared with a histogram of true values for the whole astrophysical population (light blue). </figcaption> </figure> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p3"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p3.1">From the two-dimensional reconstructed KDE, we compute one-dimensional KDEs as before. We then use sample weights to calculate event-based rates as explained in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S2.SS5" title="II.5 Application to simulated data ‣ II Simulated data ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">II.5</span></a>. The reconstructed one-dimensional rate estimates are compared with the histogram of astrophysical rates, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F7" title="Figure 7 ‣ Second method: Weighted KDE for the astrophysical distribution ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">7</span></a>, showing trends consistent with the two-dimensional comparison, including higher uncertainties around <math alttext="M\sim 10^{6}\,M_{\odot}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p3.1.m1.1"><semantics id="S3.SS0.SSS0.Px2.p3.1.m1.1a"><mrow id="S3.SS0.SSS0.Px2.p3.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.cmml">M</mi><mo id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.1" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.1.cmml">∼</mo><mrow id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3" 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id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2">𝑀</ci><apply id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3"><times id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.1"></times><apply id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2.1.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2">superscript</csymbol><cn id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2.2.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2.2">10</cn><cn id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2.3.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.2.3">6</cn></apply><apply id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3.1.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3.2.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3.3.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p3.1.m1.1c">M\sim 10^{6}\,M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p3.1.m1.1d">italic_M ∼ 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p4"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p4.4">To further explore the impact of selection effects in our second method, we performed an alternative analysis with weights simply given by <math alttext="W_{i}=1/p_{\rm det,i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.1.m1.2"><semantics id="S3.SS0.SSS0.Px2.p4.1.m1.2a"><mrow id="S3.SS0.SSS0.Px2.p4.1.m1.2.3" 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encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.1.m1.2d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 / italic_p start_POSTSUBSCRIPT roman_det , roman_i end_POSTSUBSCRIPT</annotation></semantics></math>, without truncation of small <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.2.m2.1"><semantics id="S3.SS0.SSS0.Px2.p4.2.m2.1a"><msub id="S3.SS0.SSS0.Px2.p4.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2.cmml">p</mi><mi id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.2.m2.1b"><apply id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2.cmml" 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The results as presented in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A3" title="Appendix C Results from weighted KDE using simple 1/𝑝_det weights ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">C</span></a> show some noticeable deviation from the astrophysical distribution: we trace this to a preference for much larger global bandwidths <math alttext="\beta" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.3.m3.1"><semantics id="S3.SS0.SSS0.Px2.p4.3.m3.1a"><mi id="S3.SS0.SSS0.Px2.p4.3.m3.1.1" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.3.m3.1b"><ci id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.3.m3.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.3.m3.1d">italic_β</annotation></semantics></math>, resulting from the fact that the weighted KDE is strongly influenced by a small number of samples with <math alttext="W_{i}\gg 1" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.4.m4.1"><semantics id="S3.SS0.SSS0.Px2.p4.4.m4.1a"><mrow id="S3.SS0.SSS0.Px2.p4.4.m4.1.1" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.2" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.2.cmml">W</mi><mi id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.3" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.cmml">≫</mo><mn id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.3" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.4.m4.1b"><apply id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1"><csymbol cd="latexml" id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1">much-greater-than</csymbol><apply id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.2">𝑊</ci><ci id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.2.3">𝑖</ci></apply><cn id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.3.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.4.m4.1c">W_{i}\gg 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.4.m4.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≫ 1</annotation></semantics></math> close to the low-mass edge of the distribution. Effectively, the bulk of the population must then be fitted by contributions from such “edge-case” events, but this cannot be achieved without significant bias elsewhere. This problem suggests that the dynamic range of KDE weights must be limited to avoid domination of the estimate by very high values.</p> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p5"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p5.1">Our results for both the adaptive unweighted KDE and weighted KDE methods, using the choice of (re)weighting factor <math alttext="1/\max(p_{\rm det},0.1)" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p5.1.m1.3"><semantics id="S3.SS0.SSS0.Px2.p5.1.m1.3a"><mrow id="S3.SS0.SSS0.Px2.p5.1.m1.3.3" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.cmml"><mn id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.3" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.3.cmml">1</mn><mo id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.2" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.2.cmml">/</mo><mrow id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p5.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p5.1.m1.1.1.cmml">max</mi><mo id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1a" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml">⁡</mo><mrow id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml"><mo id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml">(</mo><msub id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.2.cmml">p</mi><mi id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.3.cmml">det</mi></msub><mo id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.3" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml">,</mo><mn id="S3.SS0.SSS0.Px2.p5.1.m1.2.2" xref="S3.SS0.SSS0.Px2.p5.1.m1.2.2.cmml">0.1</mn><mo id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.4" stretchy="false" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p5.1.m1.3b"><apply id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3"><divide id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.2.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.2"></divide><cn id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.3.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.3">1</cn><apply id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.2.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1"><max id="S3.SS0.SSS0.Px2.p5.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.1.1"></max><apply id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.2">𝑝</ci><ci id="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p5.1.m1.3.3.1.1.1.1.3">det</ci></apply><cn id="S3.SS0.SSS0.Px2.p5.1.m1.2.2.cmml" type="float" xref="S3.SS0.SSS0.Px2.p5.1.m1.2.2">0.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p5.1.m1.3c">1/\max(p_{\rm det},0.1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p5.1.m1.3d">1 / roman_max ( italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT , 0.1 )</annotation></semantics></math> to account for selection effects, are summarized in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F8" title="Figure 8 ‣ Second method: Weighted KDE for the astrophysical distribution ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">8</span></a>, where for simplicity we show only the median KDE rate estimate.</p> </div> <figure class="ltx_figure" id="S3.F8"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="S3.F8.g1" src="extracted/6297058/compare_twomethods_Msrc_withpdetreweight_min10.png" width="287"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="S3.F8.g2" src="extracted/6297058/compare_twomethods_Redshift_withpdetreweight_min10.png" width="287"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Rate densities over total source mass (top) and redshift (bottom) for 4 years of LISA observation time. The median weighted iterative KDE (dashed black) is compared with the expected astrophysical distribution (light blue), whereas the unweighted iterative KDE (dashed gray) is compared with the expected distribution of detectable events (orange). </figcaption> </figure> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>Conclusion</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">Understanding the properties of the astrophysical population of binary compact objects throughout the universe is a fundamental aim of GW observatories. However, this task is complicated by the inherent limitations in the strain sensitivity, frequency range, and observation duration of both present and future detectors. As a result, we are typically able to observe only a fraction of the total population, with limited event rates and significant uncertainties in the measured parameters of each detected event. In this paper, we concentrate on (simulated) detections of massive black hole binaries by LISA. If these black holes originate from population III star remnants, or “light seeds”, a large portion of the low-mass, high-redshift binary population could remain undetected. As a result, selection biases must be carefully considered and corrected when reconstructing the underlying population properties from the observed data. Similar problems apply to existing and future terrestrial detectors (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#bib.bib24" title="">24</a>]</cite>).</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">In this paper, we have proposed and tested a non-parametric kernel density estimation (KDE) method with iterative reweighting aimed at addressing these challenges. Our approach tackles two key aspects: selection effects that limit the detectability of certain signals, and the inherent statistical uncertainties in the parameters of observed events.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.1">We employed two methods in our analysis. In the first, we use adaptive KDE without correcting for selection effects directly. Instead, these effects are applied in the iterative reweighting process in order to improve the accuracy of event parameters. The resulting reconstructed rate estimates are then compared to the expected distribution of detected events drawn from the underlying astrophysical model: the detected distribution is recovered across most of parameter space, except in regions where no events are detected. Significant biases would arise either from ignoring selection effects, or by naively attempting to correct such effects without considering the stability of estimates in regions with very small detection probability <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><msub id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml"><mi id="S4.p3.1.m1.1.1.2" xref="S4.p3.1.m1.1.1.2.cmml">p</mi><mi id="S4.p3.1.m1.1.1.3" xref="S4.p3.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p3.1.m1.1.1.1.cmml" xref="S4.p3.1.m1.1.1">subscript</csymbol><ci id="S4.p3.1.m1.1.1.2.cmml" xref="S4.p3.1.m1.1.1.2">𝑝</ci><ci id="S4.p3.1.m1.1.1.3.cmml" xref="S4.p3.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.2">In our second method, we use a weighted KDE, with weights designed to compensate for selection effects, and compare our reconstructed rate estimate with the astrophysical model distribution. Our results show that this approach can effectively reconstruct the astrophysical distribution when regularization is applied to the detection probability <math alttext="p_{\rm det}" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><msub id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml"><mi id="S4.p4.1.m1.1.1.2" xref="S4.p4.1.m1.1.1.2.cmml">p</mi><mi id="S4.p4.1.m1.1.1.3" xref="S4.p4.1.m1.1.1.3.cmml">det</mi></msub><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"><csymbol cd="ambiguous" id="S4.p4.1.m1.1.1.1.cmml" xref="S4.p4.1.m1.1.1">subscript</csymbol><ci id="S4.p4.1.m1.1.1.2.cmml" xref="S4.p4.1.m1.1.1.2">𝑝</ci><ci id="S4.p4.1.m1.1.1.3.cmml" xref="S4.p4.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">p_{\rm det}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>. However, without proper capping on small values of <math alttext="p_{\rm det}\ll 0.1" class="ltx_Math" display="inline" id="S4.p4.2.m2.1"><semantics id="S4.p4.2.m2.1a"><mrow id="S4.p4.2.m2.1.1" xref="S4.p4.2.m2.1.1.cmml"><msub id="S4.p4.2.m2.1.1.2" xref="S4.p4.2.m2.1.1.2.cmml"><mi id="S4.p4.2.m2.1.1.2.2" xref="S4.p4.2.m2.1.1.2.2.cmml">p</mi><mi id="S4.p4.2.m2.1.1.2.3" xref="S4.p4.2.m2.1.1.2.3.cmml">det</mi></msub><mo id="S4.p4.2.m2.1.1.1" xref="S4.p4.2.m2.1.1.1.cmml">≪</mo><mn id="S4.p4.2.m2.1.1.3" xref="S4.p4.2.m2.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.1b"><apply id="S4.p4.2.m2.1.1.cmml" xref="S4.p4.2.m2.1.1"><csymbol cd="latexml" id="S4.p4.2.m2.1.1.1.cmml" xref="S4.p4.2.m2.1.1.1">much-less-than</csymbol><apply id="S4.p4.2.m2.1.1.2.cmml" xref="S4.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.p4.2.m2.1.1.2.1.cmml" xref="S4.p4.2.m2.1.1.2">subscript</csymbol><ci id="S4.p4.2.m2.1.1.2.2.cmml" xref="S4.p4.2.m2.1.1.2.2">𝑝</ci><ci id="S4.p4.2.m2.1.1.2.3.cmml" xref="S4.p4.2.m2.1.1.2.3">det</ci></apply><cn id="S4.p4.2.m2.1.1.3.cmml" type="float" xref="S4.p4.2.m2.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.1c">p_{\rm det}\ll 0.1</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT ≪ 0.1</annotation></semantics></math>, the weighted KDE becomes severely biased as the estimate is dominated by a minority of samples with very high weights. The regularization used here, while apparently effective, is somewhat ad-hoc and deserves further investigation: for instance, determining how the capping of weights should scale with the number of events and the size of their parameter uncertainties.</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.1">Our iterative KDE method has the advantage of being fast and flexible, and it can account for PE uncertainties. It can also complement and improve parametric models by providing non-parametric estimates that can validate or refine model assumptions. So far, we have deployed the method only over a one- or two-dimensional parameter space, with the restriction that the kernel is rotationally symmetric in the two-dimensional space (for standardized data). This approach is suitable only for cases where the relative measurement uncertainties and scales of structure in the population density are comparable between parameters; the immediate further development required to generalize the KDE kernel would be allowing independent bandwidths for different parameters, with a suitable optimization method. In general, density estimation over a higher-dimensional parameter space will also require a larger data set for a stable result (the “curse of dimensionality”). Ideally, it would be preferable to reconstruct the full distribution over the principal binary intrinsic parameters affecting detectability (masses and orbit-aligned spins) and redshift; we leave this as a goal for future work. </p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">E.B. and J.S. acknowledge support from the European Union’s H2020 ERC Consolidator Grant “GRavity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673), the PRIN 2022 grant “GUVIRP - Gravity tests in the UltraViolet and InfraRed with Pulsar timing”, and the EU Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 101007855. K.D. acknowledges IISER Thiruvananthapuram for providing high performance computing resources at HPC Padmanabha. T.D. is supported by research grant PID2020-118635GB-I00 from the Spanish Ministerio de Ciencia e Innovación and also received financial support from Xunta de Galicia (CIGUS Network of research centers) and the European Union. </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_role_refnum ltx_tag_bibitem">Abbott <em class="ltx_emph ltx_font_italic" id="bib.bib1.3.3.1">et al.</em> [2016]</span> <span class="ltx_bibblock">B. P. Abbott <em class="ltx_emph ltx_font_italic" id="bib.bib1.4.1">et al.</em> (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, <a class="ltx_ref ltx_href" href="https://doi.org/10.1103/PhysRevLett.116.061102" title="">Phys. Rev. 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Our results are compared to the expected distribution of detectable events in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A1.F9" title="Figure 9 ‣ Appendix A Results from adaptive KDE reweighting without selection factor ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">9</span></a>.</p> </div> <figure class="ltx_figure" id="A1.F9"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="A1.F9.g1" src="extracted/6297058/Appendix_Figure_4a_AwKDEwithout_pdet_Msrc.png" width="287"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="A1.F9.g2" src="extracted/6297058/Appendix_Figure_4b_AwKDEwithout_pdet_Redshift.png" width="287"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 9: </span>Rate densities over total source mass (top) and redshift (bottom) for 4 years of LISA observation time. The reconstruction (solid dark red: median, dashed dark red: 90% symmetric interval) is performed using adaptive but unweighted KDE, neglecting all selection effects. The light blue histogram represents the distribution of the population’s detectable events. </figcaption> </figure> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.1">While the marginal mass distribution is scarcely affected by neglecting selection effects, the redshift distribution is significantly biased towards lower values relative to the true distribution of detectable events. Given the large redshift uncertainties, the method underestimates the redshift of events above <math alttext="z\simeq 10" class="ltx_Math" display="inline" id="A1.p2.1.m1.1"><semantics id="A1.p2.1.m1.1a"><mrow id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml"><mi id="A1.p2.1.m1.1.1.2" xref="A1.p2.1.m1.1.1.2.cmml">z</mi><mo id="A1.p2.1.m1.1.1.1" xref="A1.p2.1.m1.1.1.1.cmml">≃</mo><mn id="A1.p2.1.m1.1.1.3" xref="A1.p2.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.1b"><apply id="A1.p2.1.m1.1.1.cmml" xref="A1.p2.1.m1.1.1"><csymbol cd="latexml" id="A1.p2.1.m1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1">similar-to-or-equals</csymbol><ci id="A1.p2.1.m1.1.1.2.cmml" xref="A1.p2.1.m1.1.1.2">𝑧</ci><cn id="A1.p2.1.m1.1.1.3.cmml" type="integer" xref="A1.p2.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.1c">z\simeq 10</annotation><annotation encoding="application/x-llamapun" id="A1.p2.1.m1.1d">italic_z ≃ 10</annotation></semantics></math>. The bulk of detected events are at lower redshifts, and here this trend is (wrongly) attributed to a property of the underlying population rather than being due to selection. Hence, selection bias is crucial even to estimation of the detected distribution.</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Results from adaptive KDE using simple 1/<math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="A2.1.m1.1"><semantics id="A2.1.m1.1b"><msub id="A2.1.m1.1.1" xref="A2.1.m1.1.1.cmml"><mi id="A2.1.m1.1.1.2" xref="A2.1.m1.1.1.2.cmml">p</mi><mi id="A2.1.m1.1.1.3" xref="A2.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="A2.1.m1.1c"><apply id="A2.1.m1.1.1.cmml" xref="A2.1.m1.1.1"><csymbol cd="ambiguous" id="A2.1.m1.1.1.1.cmml" xref="A2.1.m1.1.1">subscript</csymbol><ci id="A2.1.m1.1.1.2.cmml" xref="A2.1.m1.1.1.2">𝑝</ci><ci id="A2.1.m1.1.1.3.cmml" xref="A2.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.m1.1d">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="A2.1.m1.1e">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> reweighting </h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.2">Here, we perform an iterative adaptive KDE reconstruction including selection effects in the sample reweighting, but without limits on small <math alttext="p_{\mathrm{det}}" class="ltx_Math" display="inline" id="A2.p1.1.m1.1"><semantics id="A2.p1.1.m1.1a"><msub id="A2.p1.1.m1.1.1" xref="A2.p1.1.m1.1.1.cmml"><mi id="A2.p1.1.m1.1.1.2" xref="A2.p1.1.m1.1.1.2.cmml">p</mi><mi id="A2.p1.1.m1.1.1.3" xref="A2.p1.1.m1.1.1.3.cmml">det</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p1.1.m1.1b"><apply id="A2.p1.1.m1.1.1.cmml" xref="A2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.p1.1.m1.1.1.1.cmml" xref="A2.p1.1.m1.1.1">subscript</csymbol><ci id="A2.p1.1.m1.1.1.2.cmml" xref="A2.p1.1.m1.1.1.2">𝑝</ci><ci id="A2.p1.1.m1.1.1.3.cmml" xref="A2.p1.1.m1.1.1.3">det</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.1.m1.1c">p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.1.m1.1d">italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math> values. In more detail, at each step we draw new samples with weights given by the previous detected KDE multiplied by <math alttext="1/p_{\mathrm{det}}" class="ltx_Math" display="inline" id="A2.p1.2.m2.1"><semantics id="A2.p1.2.m2.1a"><mrow id="A2.p1.2.m2.1.1" xref="A2.p1.2.m2.1.1.cmml"><mn id="A2.p1.2.m2.1.1.2" xref="A2.p1.2.m2.1.1.2.cmml">1</mn><mo id="A2.p1.2.m2.1.1.1" xref="A2.p1.2.m2.1.1.1.cmml">/</mo><msub id="A2.p1.2.m2.1.1.3" xref="A2.p1.2.m2.1.1.3.cmml"><mi id="A2.p1.2.m2.1.1.3.2" xref="A2.p1.2.m2.1.1.3.2.cmml">p</mi><mi id="A2.p1.2.m2.1.1.3.3" xref="A2.p1.2.m2.1.1.3.3.cmml">det</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.2.m2.1b"><apply id="A2.p1.2.m2.1.1.cmml" xref="A2.p1.2.m2.1.1"><divide id="A2.p1.2.m2.1.1.1.cmml" xref="A2.p1.2.m2.1.1.1"></divide><cn id="A2.p1.2.m2.1.1.2.cmml" type="integer" xref="A2.p1.2.m2.1.1.2">1</cn><apply id="A2.p1.2.m2.1.1.3.cmml" xref="A2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A2.p1.2.m2.1.1.3.1.cmml" xref="A2.p1.2.m2.1.1.3">subscript</csymbol><ci id="A2.p1.2.m2.1.1.3.2.cmml" xref="A2.p1.2.m2.1.1.3.2">𝑝</ci><ci id="A2.p1.2.m2.1.1.3.3.cmml" xref="A2.p1.2.m2.1.1.3.3">det</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m2.1c">1/p_{\mathrm{det}}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.2.m2.1d">1 / italic_p start_POSTSUBSCRIPT roman_det end_POSTSUBSCRIPT</annotation></semantics></math>: we compare our results to the distribution of detectable events as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A2.F10" title="Figure 10 ‣ Appendix B Results from adaptive KDE using simple 1/𝑝_det reweighting ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">10</span></a>.</p> </div> <figure class="ltx_figure" id="A2.F10"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="A2.F10.g1" src="extracted/6297058/Figure_4a_AwKDE_with_pdet_fatcor_Msrc.png" width="287"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="A2.F10.g2" src="extracted/6297058/Figure_4b_AwKDE_with_pdet_fatcor_Redshift.png" width="287"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 10: </span>Rate densities over total source mass (top) and redshift (bottom) for 4 years of LISA observation time. Solid (dashed) dark red lines show the median (<math alttext="90\%" class="ltx_Math" display="inline" id="A2.F10.4.m1.1"><semantics id="A2.F10.4.m1.1b"><mrow id="A2.F10.4.m1.1.1" xref="A2.F10.4.m1.1.1.cmml"><mn id="A2.F10.4.m1.1.1.2" xref="A2.F10.4.m1.1.1.2.cmml">90</mn><mo id="A2.F10.4.m1.1.1.1" xref="A2.F10.4.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.F10.4.m1.1c"><apply id="A2.F10.4.m1.1.1.cmml" xref="A2.F10.4.m1.1.1"><csymbol cd="latexml" id="A2.F10.4.m1.1.1.1.cmml" xref="A2.F10.4.m1.1.1.1">percent</csymbol><cn id="A2.F10.4.m1.1.1.2.cmml" type="integer" xref="A2.F10.4.m1.1.1.2">90</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F10.4.m1.1d">90\%</annotation><annotation encoding="application/x-llamapun" id="A2.F10.4.m1.1e">90 %</annotation></semantics></math> symmetric interval) iterative KDE reconstruction from posterior samples: here, the iterative reweighting provides an estimated astrophysical population given by the 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det,i}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.3.m3.2d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 / italic_p start_POSTSUBSCRIPT roman_det , roman_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="A3.F11"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="A3.F11.g1" src="extracted/6297058/Figure_5a_WeightedKDE_Msrc_withPdetNocap.png" width="287"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="180" id="A3.F11.g2" src="extracted/6297058/Figure_5b_WeightedKDE_Redshift_withPdetNocap.png" width="287"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 11: </span>Rate densities over total source mass (top) and redshift (bottom) for 4 years of LISA observation time. The reconstruction (solid dark red, with <math alttext="90\%" class="ltx_Math" display="inline" id="A3.F11.2.m1.1"><semantics id="A3.F11.2.m1.1b"><mrow id="A3.F11.2.m1.1.1" xref="A3.F11.2.m1.1.1.cmml"><mn id="A3.F11.2.m1.1.1.2" xref="A3.F11.2.m1.1.1.2.cmml">90</mn><mo id="A3.F11.2.m1.1.1.1" xref="A3.F11.2.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.F11.2.m1.1c"><apply id="A3.F11.2.m1.1.1.cmml" xref="A3.F11.2.m1.1.1"><csymbol cd="latexml" id="A3.F11.2.m1.1.1.1.cmml" xref="A3.F11.2.m1.1.1.1">percent</csymbol><cn id="A3.F11.2.m1.1.1.2.cmml" type="integer" xref="A3.F11.2.m1.1.1.2">90</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.F11.2.m1.1d">90\%</annotation><annotation encoding="application/x-llamapun" id="A3.F11.2.m1.1e">90 %</annotation></semantics></math> symmetric interval in dashed dark red) is performed using weighted iterative KDE correcting for selection effects. The light blue histogram represents the distribution of the whole astrophysical event population. </figcaption> </figure> <div class="ltx_para" id="A3.p2"> <p class="ltx_p" id="A3.p2.2">Comparing with the true astrophysical distribution in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#A3.F11" title="Figure 11 ‣ Appendix C Results from weighted KDE using simple 1/𝑝_det weights ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">11</span></a>, we find some biases, particularly at low mass and both low and high redshifts. The method now prefers much higher KDE bandwidths, an effect which can be attributed to the estimate being dominated by (the kernels of) a small number of samples with very high <math alttext="W_{i}" class="ltx_Math" display="inline" id="A3.p2.1.m1.1"><semantics id="A3.p2.1.m1.1a"><msub id="A3.p2.1.m1.1.1" xref="A3.p2.1.m1.1.1.cmml"><mi id="A3.p2.1.m1.1.1.2" xref="A3.p2.1.m1.1.1.2.cmml">W</mi><mi id="A3.p2.1.m1.1.1.3" xref="A3.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A3.p2.1.m1.1b"><apply id="A3.p2.1.m1.1.1.cmml" xref="A3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A3.p2.1.m1.1.1.1.cmml" xref="A3.p2.1.m1.1.1">subscript</csymbol><ci id="A3.p2.1.m1.1.1.2.cmml" xref="A3.p2.1.m1.1.1.2">𝑊</ci><ci id="A3.p2.1.m1.1.1.3.cmml" xref="A3.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.1.m1.1c">W_{i}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.1.m1.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> around the low-mass edge seen in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2410.17056v2#S3.F3" title="Figure 3 ‣ III Results ‣ Reconstructing the LISA massive black hole binary population via iterative kernel density estimation"><span class="ltx_text ltx_ref_tag">3</span></a>: these can only fit the rest of the population if their bandwidth is inflated. Hence, the KDE is forced to take a broad Gaussian form with few additional features. This form can, coincidentally, reproduce the true <math alttext="z" class="ltx_Math" display="inline" id="A3.p2.2.m2.1"><semantics id="A3.p2.2.m2.1a"><mi id="A3.p2.2.m2.1.1" xref="A3.p2.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A3.p2.2.m2.1b"><ci id="A3.p2.2.m2.1.1.cmml" xref="A3.p2.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="A3.p2.2.m2.1d">italic_z</annotation></semantics></math> distribution relatively well, as that happens to also be close to Gaussian, but would be unable to reconstruct any more complicated astrophysical population.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Mar 20 15:28:37 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>