{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-ordering-of-divergences-for-variational","title":"Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs","arxiv_id":"2403.13748","date":"2024-03-20","proceeding":null,"authors":["Charles C. Margossian","Loucas Pillaud-Vivien","Lawrence K. Saul"],"abstract":"Given an intractable distribution $p$, the problem of variational inference (VI) is to find the best approximation from some more tractable family $Q$. Commonly, one chooses $Q$ to be a family of factorized distributions (i.e., the mean-field assumption), even though~$p$ itself does not factorize. We show that this mismatch leads to an impossibility theorem: if $p$ does not factorize, then any factorized approximation $q\\in Q$ can correctly estimate at most one of the following three measures of uncertainty: (i) the marginal variances, (ii) the marginal precisions, or (iii) the generalized variance (which can be related to the entropy). In practice, the best variational approximation in $Q$ is found by minimizing some divergence $D(q,p)$ between distributions, and so we ask: how does the choice of divergence determine which measure of uncertainty, if any, is correctly estimated by VI? We consider the classic Kullback-Leibler divergences, the more general $\\alpha$-divergences, and a score-based divergence which compares $\\nabla \\log p$ and $\\nabla \\log q$. We provide a thorough theoretical analysis in the setting where $p$ is a Gaussian and $q$ is a (factorized) Gaussian. We show that all the considered divergences can be \\textit{ordered} based on the estimates of uncertainty they yield as objective functions for~VI. Finally, we empirically evaluate the validity of this ordering when the target distribution $p$ is not Gaussian.","url_abs":"https://arxiv.org/abs/2403.13748v3","url_pdf":"https://arxiv.org/pdf/2403.13748v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"an-ordering-of-divergences-for-variational","repo_url":"https://github.com/charlesm93/vi-ordering","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"uncertainty-quantification","task_name":"Uncertainty Quantification"},{"task_slug":"variational-inference","task_name":"Variational Inference"},{"task_slug":null,"task_name":"valid"}],"methods":[{"method_slug":"variational-inference","method_name":"Variational Inference"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2403.13748","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}