Normality of Posterior Distribution Under Misspecification and Nonsmoothness, and Bayes Factor for Davies' Problem

We examine the large sample properties of Bayes procedures in a general framework, where data may be dependent and models may be misspecified and nonsmooth. The posterior distribution of parameters is shown to be asymptotically normal, centered at the quasi maximum likelihood estimator, under mild conditions. In this framework, the Bayes factor for the test problem of Davies (1997, 1987), where a parameter is unidentified under the null hypothesis, is analyzed. The probability that the Bayes factor leads to a correct conclusion about the hypotheses in Davies’ problem is shown to approach to one.