Inference as risk identification

Abstract

The Bayes posterior solves the variational problem of maximizing the expected likelihood of the data minus the Kullback–Leibler divergence from the prior distribution. We generalize Bayesian inference from this variational point of view. In particular, we introduce a variational inference (VI) problem whose solutions we call posteriors and show that these posteriors are risk identifiers, i.e., subgradients of risk measures. This insight leads to: (i) rigorous results concerning the existence and asymptotics of posteriors, (ii) the ability to constrain the divergence from the prior—which motivates a measure-valued relaxation of M-estimation that we call coherent inference—and (iii) insights into VI problems derived from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}-divergences. These insights include semi-analytical expressions for posteriors and connections with utility theory. We use the semi-analytical expressions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}-divergence posteriors to numerically solve a multi-armed bandit problem.