{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"institution":[{"name":"bioRxiv"}],"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T17:12:59Z","timestamp":1786122779317,"version":"build-2736575974"},"posted":{"date-parts":[[2026,5,19]]},"group-title":"Bioinformatics","reference-count":34,"publisher":"openRxiv","license":[{"start":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T00:00:00Z","timestamp":1779148800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NIH","award":["HG004483"],"award-info":[{"award-number":["HG004483"]}]},{"name":"NIH","award":["GM080203"],"award-info":[{"award-number":["GM080203"]}]},{"name":"NIH","award":["HG013117"],"award-info":[{"award-number":["HG013117"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2026,8,4]]},"abstract":"<jats:p>\n                  We investigate the use of Expectation\u2013Maximization (\n                  <jats:sc>em<\/jats:sc>\n                  ) and variational Bayes for inferring rates and interactions under models of molecular coevolution. We first review\n                  <jats:sc>em<\/jats:sc>\n                  theory for continuous-time Markov chains (\n                  <jats:sc>ctmc<\/jats:sc>\n                  s) and develop it for coevolutionary models, exploiting exchangeability and reversibility symmetries to constrain the parameter dimension. We fit several paired amino-acid coevolutionary models to structural alignments and compare the results to previous work. Our richest model trained on pooled coevolutionary data has explanatory power comparable to CherryML\u2019s Q2 matrix (also trained on pooled data), with one quarter the parameters. However, we observe that aggregation of training data can lead to a form of Simpson\u2019s Paradox: a mixture model, whose components are parameter-efficient continuous-time Bayes networks (\n                  <jats:sc>ctbn<\/jats:sc>\n                  s), resolves signals that wash out when a single model tries to capture everything. These signals include both correlated and anticorrelated hydropathy and volume-packing in coevolving amino-acid pairs, as well as the anticorrelated acid\/base compensation that was detected by CherryML\u2019s Q2. We next present a closed-form evidence lower bound (ELBO) for\n                  <jats:sc>ctbn<\/jats:sc>\n                  s using\n                  <jats:sc>em<\/jats:sc>\n                  statistics. Compared to the state of the art in variational modeling of\n                  <jats:sc>ctbn<\/jats:sc>\n                  s, the Euler-Lagrange equations derived by Cohn et al (JMLR, 2010), our closed-form ELBO is competitive in accuracy, considerably more efficient, simpler, and more stable. We conclude by describing a covariant indel model: a Dirichlet process selecting coevolving sites within TKF92, yielding an Infinite Pair HMM over alignments and structures. This model slightly outperforms TKF92 on a structural-alignment benchmark. Code and data are at\n                  <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"https:\/\/tkfdp.net\/\">https:\/\/tkfdp.net\/<\/jats:ext-link>\n                  .\n                <\/jats:p>","DOI":"10.64898\/2026.05.16.725674","type":"posted-content","created":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T01:10:26Z","timestamp":1779239426000},"source":"Crossref","is-referenced-by-count":0,"title":["Variational inference in coupled models of amino acid substitution"],"prefix":"10.64898","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-4122-2259","authenticated-orcid":false,"given":"Annabel","family":"Large","sequence":"first","affiliation":[{"name":"Department of Bioengineering, University of California, Berkeley, Berkeley, CA 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7639-5369","authenticated-orcid":false,"given":"Ian","family":"Holmes","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, University of California, Berkeley, Berkeley, CA 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"54368","reference":[{"key":"2026080704450659000_2026.05.16.725674v2.1","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1239\/aap\/1013540176","article-title":"Probabilistic models of DNA sequence evolution with context dependent rates of substitution","volume":"32","year":"2000","journal-title":"Advances in Applied Probability"},{"key":"2026080704450659000_2026.05.16.725674v2.2","doi-asserted-by":"publisher","DOI":"10.1089\/10665270360688039"},{"key":"2026080704450659000_2026.05.16.725674v2.3","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/bth901"},{"key":"2026080704450659000_2026.05.16.725674v2.4","doi-asserted-by":"publisher","DOI":"10.1093\/molbev\/msh039"},{"key":"2026080704450659000_2026.05.16.725674v2.5","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1198\/106186008X289010","article-title":"A markov chain Monte Carlo expectation maximization algorithm for statistical analysis of DNA sequence evolution with neighbor-dependent substitution rates","volume":"17","year":"2008","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"2026080704450659000_2026.05.16.725674v2.6","doi-asserted-by":"publisher","DOI":"10.1089\/cmb.2005.12.1166"},{"key":"2026080704450659000_2026.05.16.725674v2.7","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/bth917"},{"key":"2026080704450659000_2026.05.16.725674v2.8","unstructured":"Y. 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