Risk of Predictive Distributions and Bayesian Model Comparison of Misspecified Models

TMüller (2013, Econometrica, 81(5), 1805-1949) shows that Bayesian inference of parameters of interest in a misspecified model can reduce the asymptotic frequentist risk when the standard posterior is replaced with the sandwich posterior. In this paper, we extend the results in Müller (2013) to Bayesian model comparison. Bayesian model comparison of potentially misspecified models can be conducted in a predictive framework with three alternative predictive distributions, namely, the plug-in predictive distribution, the standard posterior predictive distribution, and the sandwich posterior predictive distribution of Müller (2013). Via the Kullback-Leibler (KL) loss function, it is shown that the sandwich posterior predictive distribution yields a lower asymptotic risk than the standard posterior predictive distribution. Moreover, we provide sufficient conditions under which the sandwich posterior predictive distribution yields a lower asymptotic risk than the plug-in predictive distribution. We then propose two new Bayesian penalized information criteria based on the last two predictive distributions to compare misspecified models and establish their relationship with some existing information criteria. The proposed new information criteria are illustrated in several empirical studies.