On posterior concentration in misspecified models

We investigate the asymptotic behavior of Bayesian posterior distributions under model misspecification in the nonparametric context. Although a few results on this topic are available, we believe that that these are somewhat inaccessible due, in part, to the technicalities and the subtle differences compared to the more familiar well-specified model case. Our goal in this paper is to make some of the available results more accessible and transparent. In particular, we give a simple proof that sets which satisfy the separation condition in Kleijn and van der Vaart (2006) will have vanishing posterior mass. Unlike their approach, we do not require construction of test sequences and we get almost sure convergence as compared to mean convergence. Further, we relate the Kleijn-van der Vaart separation condition to a roughly equivalent condition involving the more familiar and more natural Kullback-Leibler contrast. We show by example that, the standard assumption in the misspecified case that the prior puts positive mass on Kullback-Leibler neighborhoods of the true model, does not necessarily ensure that the true model is in the support of the prior with respect to natural topologies (such as $L_1$). This observation leads to a new condition involving the Kullback-Leibler contrast and, in turn, a posterior concentration result for weak neighborhoods in general and for certain $L_1$ neighborhoods under a compactness condition.