马尔可夫模型的变分温度化后验的PAC-Bayes界
PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models
AI总结：
本文针对马尔可夫模型，提出变分贝叶斯近似温度化后验的PAC-Bayes风险界，该界与数据生成模型的混合和遍历性质相关，并验证了对模型误设的鲁棒性。
摘要：
在科学和工程应用中，展现时间依赖性的数据集比比皆是，而马尔可夫模型代表了对时间依赖结构的一种简化且流行的视角。在本文中，我们考虑贝叶斯设置，即对马尔可夫模型的转移核参数赋予先验分布，并旨在刻画由此产生的、通常难以处理的 posterior 分布。我们对变分贝叶斯（VB）近似温度化贝叶斯后验分布进行了PAC-Bayesian分析，限定了VB近似的模型风险。已知温度化后验对模型误设具有鲁棒性，且其变分近似不会遭受过度自信近似的常见问题。我们的结果将风险界与马尔可夫数据生成模型的混合和遍历性质联系起来。我们通过若干马尔可夫模型示例说明了PAC-Bayes界，并考虑了马尔可夫模型被误设的情况。
英文摘要：
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.
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