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Predictive variational inference (PVI) is introduced as a framework that learns and samples from an optimal posterior density so the posterior predictive distribution matches the true data-generating process, scored by multiple rules. Optimization yields results distinct from Bayesian posterior approximation, supports model diagnosis via detected heterogeneity, and applies to likelihood-exact and likelihood-free settings.",{"@graph":63,"@context":112},[64,80,95],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/predictive-variational-inference-learn-the-predictively-optimal-posterior-distribution/573321/",4,{"url":78,"name":59,"@type":81,"author":82,"headline":59,"publisher":85,"fileFormat":88,"inLanguage":57,"description":61,"dateModified":89,"datePublished":89,"encodingFormat":88,"isAccessibleForFree":90,"interactionStatistic":91},"DigitalDocument",{"name":83,"@type":84},"Mafia Boss","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-14",true,{"@type":92,"interactionType":93,"userInteractionCount":4},"InteractionCounter",{"@type":94},"ViewAction",{"@type":96,"mainEntity":97},"FAQPage",[98,104,108],{"name":99,"@type":100,"acceptedAnswer":101},"What is predictive variational inference (PVI) trying to optimize?","Question",{"text":102,"@type":103},"PVI optimizes an objective that makes the posterior predictive distribution close to the true data-generating process, measured by a divergence defined through scoring rules.","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"How does PVI differ from classical variational inference (VI)?",{"text":107,"@type":103},"PVI directly learns the optimal predictive distribution rather than treating the learned posterior as merely an approximation to the exact Bayesian posterior.",{"name":109,"@type":100,"acceptedAnswer":110},"How does the paper use PVI for model diagnosis?",{"text":111,"@type":103},"The discrepancy between PVI and exact Bayes is interpreted as evidence of model misspecification, helping identify which parameter should vary across the population to improve the model.","https://schema.org",{"og:url":78,"og:type":114,"og:title":59,"og:site_name":86,"og:description":61},"article",{"robots":116,"canonical":78},"index,follow",{"doc_id":118,"site_id":56},573321,1791937098,{"code":4,"msg":5,"data":121},{"doc_id":118,"user_id":122,"nickname":83,"user_avatar":123,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":129,"language":130,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":131,"faqs":132,"seo_title":133,"seo_description":61,"update_tm":119,"read_time":134},962090760730,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Predictive variational inference: Learn the predictively optimal posterior distribution  \nJinlin Lai 1 Antonio Ricardo Linero 2 Yuling Yao 2  \narXiv :2410 . 14843v4 [ stat .ML] 22 Jun 2026  \nAbstract  \nVanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as an implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis.  \nThis framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.  \n1. Introduction  \nTraditional Bayesian inference has often been used for oneof two purposes: estimation of parameters θ in a model p (y|θ) or estimation of a predictive distribution p(y˜|y) to give a probabilistic forecast of a future observation y˜. In many settings, such as decision-making, machine learning, or forecasting, the explicit aim is prediction rather than parameter estimation; fortunately, when both the likelihood p (y|θ) and prior pprior (θ) are correctly-specified, traditional Bayesian inference is optimal for both purposes.  \nThere is, however, an increasing awareness of limitations of  \n1 College of Information and Computer Sciences, University of Massachusetts Amherst 2Department of Statistics and Data Sciences, University of Texas, Austin. Correspondence to: Yuling Yao \u003C[yyao@austin.utexas.edu](yyao@austin.utexas.edu) >.  \nProceedings of the 43 rd International Conference on Machine Learning, Seoul, South Korea. PMLR 306, 2026 . Copyright 2026 by the author(s) .  \nBayes under misspecification (Walker, 2013 ; Gelman et al., 2020 ; Masegosa, 2020 ; Huggins & Miller, 2023 ; Fong et al., 2023) . Assume the observations y = (y1 ,..., yn) are independent and identically distributed (IID), and let y˜ to bethe next unseen data. The Bayesian posterior predictive distribution of y˜ is the sampling distribution averaged over the posterior density: pBayes (y˜|y) = RΘ p (y˜|θ)p (θ|y)dθ . But is this exact Bayesian prediction “optimal”? Or is this Bayesian prediction at least more “optimal” than some pointestimate induced prediction p(y˜| ˆθ) where ˆθ is for example the MLE? The answers are in general both negative (Clarke & Yao, 2025), accompanied by real-data evidence that sometimes Bayesian methods instead produce worse predictions (Wenzel et al., 2020) or overconfident inference (Yang & Zhu, 2018) . The only predictive optimality we may assert is that, if the likelihood and prior are both correct, then among all probabilistic forecasts, the Bayesian posterior prediction minimizes the posterior-averaged KL divergence from the sampling distribution (Aitchison, 1975), i.e.,  \narg mQi(n) Z KL(p(y˜|θ)||Q(y˜))p(θ|y)dθ = pBayes (y˜|y) . (1)  \nBut this model-being-correct assumption is unlikely to be relevant to any realistic application. Even if the model is correct, this optimality statement typically does not hold for other divergences practitioners may care about.  \nTo make Bayesian inference under potential model misspecification, this paper develops predictive variational inference (PVI): a general framework that constructs predictively optimal posterior samples q (θ|y) . Unlike classical VI, we optimize a divergence D between the posterior predictive distribution and the data generating process ptrue (y˜) :  \ninF D 􀀒ZΘ p","cbCaitgTUOxaAeCB","https://ap.wps.com/l/cbCaitgTUOxaAeCB","pdf",1982753,25,"English","# Abstract\n# Introduction\n## Bayesian prediction vs parameter estimation\n## Predictive variational inference (PVI) framework\n## Relationship of PVI to variational inference\n## Practical application in cryoEM","[{\"question\":\"What is predictive variational inference (PVI) trying to optimize?\",\"answer\":\"PVI optimizes an objective that makes the posterior predictive distribution close to the true data-generating process, measured by a divergence defined through scoring rules.\"},{\"question\":\"How does PVI differ from classical variational inference (VI)?\",\"answer\":\"PVI directly learns the optimal predictive distribution rather than treating the learned posterior as merely an approximation to the exact Bayesian posterior.\"},{\"question\":\"How does the paper use PVI for model diagnosis?\",\"answer\":\"The discrepancy between PVI and exact Bayes is interpreted as evidence of model misspecification, helping identify which parameter should vary across the population to improve the model.\"}]","Predictive variational inference - Learn the predictively optimal posterior distribution | PDF",9]