  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.
