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Datasets displaying temporal dependencies abound in science and engineeringapplications, with Markov models representing a simplified and popular view ofthe temporal dependence structure. In this paper, we consider Bayesian settingsthat place prior distributions over the parameters of the transition kernel ofa Markov model, and seeks to characterize the resulting, typically intractable,posterior distributions. We present a PAC-Bayesian analysis of variationalBayes (VB) approximations to tempered Bayesian posterior distributions,bounding the model risk of the VB approximations. Tempered posteriors are knownto be robust to model misspecification, and their variational approximations donot suffer the usual problems of over confident approximations. Our results tiethe risk bounds to the mixing and ergodic properties of the Markov datagenerating model. We illustrate the PAC-Bayes bounds through a number ofexample Markov models, and also consider the situation where the Markov modelis misspecified.

PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models