Generalized Variational Inference: Three arguments for deriving new Posteriors

TL;DR

This paper introduces a generalized Bayesian inference framework called the Rule of Three (RoT) and its special case, Generalized Variational Inference (GVI), which address limitations of standard Bayesian methods and improve predictive performance in machine learning models.

Contribution

It provides an axiomatic derivation of a new generalized inference method, recovering existing posteriors as special cases and demonstrating improved robustness and accuracy in neural network applications.

Findings

GVI offers a flexible family of belief distributions with desirable properties.

Standard VI's ELBO maximization is shown to be optimal under the proposed view.

GVI improves predictive performance in Bayesian Neural Networks and Deep Gaussian Processes.

Abstract

We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes' rule as infinite-dimensional optimization problem (Csiszar, 1975; Donsker and Varadhan; 1975, Zellner; 1988). First, we use it to prove an optimality result of standard Variational Inference (VI): Under the proposed view, the standard Evidence Lower Bound (ELBO) maximizing VI posterior is preferable to alternative approximations of the Bayesian posterior. Next, we argue for generalizing standard Bayesian inference. The need for this arises in situations of severe misalignment between reality and three assumptions underlying standard Bayesian inference: (1) Well-specified priors, (2) well-specified likelihoods, (3) the availability of infinite computing power. Our generalization addresses these shortcomings with three arguments and is called the Rule of Three (RoT). We derive it axiomatically and recover existing posteriors as special cases, including the Bayesian posterior and its approximation by standard VI. In contrast, approximations based on alternative ELBO-like objectives violate the axioms. Finally, we study a special case of the RoT that we call Generalized Variational Inference (GVI). GVI posteriors are a large and tractable family of belief distributions specified by three arguments: A loss, a divergence and a variational family. GVI posteriors have appealing properties, including consistency and an interpretation as approximate ELBO. The last part of the paper explores some attractive applications of GVI in popular machine learning models, including robustness and more appropriate marginals. After deriving black box inference schemes for GVI posteriors, their predictive performance is investigated on Bayesian Neural Networks and Deep Gaussian Processes, where GVI can comprehensively improve upon existing methods.