Model selection and order selection problems frequently arise in statistical practice. A popular approach to addressing these problems in the frequentist setting involves information criteria based on penalized maxima of log-likelihoods for competing models. In the Bayesian context, similar criteria are employed, replacing the maxima of log-likelihoods with their posterior expectations. Despite their popularity in applications, the large-sample behavior of these criteria -- such as the deviance information criterion (DIC), Bayesian predictive information criterion (BPIC), and widely-applicable Bayesian information criterion (WBIC) -- has received relatively little attention. In this work, we investigate the almost sure limits of these criteria and establish novel results on posterior and generalized posterior consistency, which are of independent interest. The utility of our theoretical findings is demonstrated via illustrative technical and numerical examples.

翻译：模型选择与阶数选择问题在统计实践中频繁出现。在频率论框架下，处理这类问题的常用方法涉及基于竞争模型对数似然惩罚最大化的信息准则。在贝叶斯语境中，采用类似准则，将对数似然的最大值替换为其后验期望。尽管这些准则——如偏差信息准则（DIC）、贝叶斯预测信息准则（BPIC）和广泛适用贝叶斯信息准则（WBIC）——在应用中广受欢迎，但其大样本行为却鲜有研究关注。本文中，我们探究了这些准则的几乎必然极限，并在后验一致性及广义后验一致性方面建立了新颖结论，这些结论本身亦具有独立的理论价值。我们通过阐释性的技术与数值算例，展示了本理论成果的实用价值。