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.

翻译：在科学和工程应用中,数据集显示了大量时间依赖性,Markov模型代表了对时间依赖结构的简化和流行观点。在本文中,我们考虑了先前将分布置于Markov模型过渡核心参数之上的Bayesian设置,并试图对由此产生的典型难以解决的后部分布进行定性。我们展示了PAC-Bayesian(VB)对温和的Bayesian后方分布的变异近似(VB)分析,对VB近似(VB)的模型风险进行了约束。已知Tenderered后方非常强大,可以模拟误差,而其变近似不会遇到过于自信近似的常见问题。我们的结果将风险界限与Markov数据生成模型的混合和随机特性挂钩。我们通过一些Markov模型来说明PAC-Bayes(VB)的界限,还考虑了Markov模型被错误描述的情况。