On the large-sample limits of some Bayesian model evaluation statistics

TL;DR

This paper investigates the large-sample behavior of Bayesian model evaluation criteria like DIC, BPIC, and WBIC, establishing new theoretical results on their limits and consistency, supported by examples.

Contribution

It provides the first detailed analysis of the almost-sure limits of Bayesian information criteria, advancing understanding of their asymptotic properties.

Findings

Established almost-sure limits for Bayesian criteria

Proved posterior and generalized posterior consistency

Demonstrated theoretical results with numerical examples

Abstract

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 penalised maxima of log-likelihoods for competing models. In the Bayesian context, similar criteria are employed, replacing the maximised log-likelihoods with posterior expectations of the log-likelihood. Despite their popularity in applications, the large-sample behaviour 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 generalised posterior consistency, which are of independent interest. The utility of our theoretical findings is demonstrated via illustrative technical and numerical examples.