PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models

TL;DR

This paper develops PAC-Bayesian bounds for variational approximations of tempered Bayesian posteriors in Markov models, linking risk to model properties and addressing misspecification.

Contribution

It introduces a novel PAC-Bayesian analysis for variational Bayes in Markov models, accounting for tempering and model misspecification.

Findings

Risk bounds depend on mixing and ergodic properties.

Temperate posteriors are robust to model misspecification.

Bounds are illustrated on various Markov models.

Abstract

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 seeks to characterize the resulting, typically intractable, posterior distributions. We present a 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.