{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T07:23:11Z","timestamp":1774941791419,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,6]],"date-time":"2021-03-06T00:00:00Z","timestamp":1614988800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1816499; DMS-1812197"],"award-info":[{"award-number":["IIS-1816499; DMS-1812197"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Datasets displaying temporal dependencies abound in science and engineering applications, with Markov models representing a simplified and popular view of the temporal dependence structure. In this paper, we consider Bayesian settings that place prior distributions over the parameters of the transition kernel of a Markov model, and seek to characterize the resulting, typically intractable, posterior distributions. We present a Probably Approximately Correct (PAC)-Bayesian analysis of variational Bayes (VB) approximations to tempered Bayesian posterior distributions, bounding the model risk of the VB approximations. Tempered posteriors are known to be robust to model misspecification, and their variational approximations do not suffer the usual problems of over confident approximations. Our results tie the risk bounds to the mixing and ergodic properties of the Markov data generating model. We illustrate the PAC-Bayes bounds through a number of example Markov models, and also consider the situation where the Markov model is misspecified.<\/jats:p>","DOI":"10.3390\/e23030313","type":"journal-article","created":{"date-parts":[[2021,3,7]],"date-time":"2021-03-07T20:20:43Z","timestamp":1615148443000},"page":"313","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models"],"prefix":"10.3390","volume":"23","author":[{"given":"Imon","family":"Banerjee","sequence":"first","affiliation":[{"name":"Department of Statistics, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vinayak A.","family":"Rao","sequence":"additional","affiliation":[{"name":"Department of Statistics, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Harsha","family":"Honnappa","sequence":"additional","affiliation":[{"name":"School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000001","article-title":"Introduction to Variational Methods for Graphical Models","volume":"1","author":"Wainwright","year":"2008","journal-title":"Found. 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