On posterior concentration in misspecified models

We investigate the asymptotic behavior of Bayesian posterior distributions under independent and identically distributed (i.i.d.) misspecified models. More specifically, we study the concentration of the posterior distribution on neighborhoods of $f^\star$, the density that is closest in the Kullback-Leibler sense to the true model $f_0$. We note through examples, the need for assumptions beyond the usual Kullback--Leibler support assumption. We then investigate consistency with respect to a general metric under three assumptions, each based on a notion of divergence measure, and then apply these to a weighted $L_1$ metric in convex models and non-convex models. Although a few results on this topic are available, we believe that these are somewhat inaccessible due, in part, to the technicalities and the subtle differences compared to the more familiar well-specified model case. One of our goals is to make some of the available results, especially that of Kleijn and van der Vaart (2006) more accessible and transparent. Unlike their paper, our approach does not require construction of test sequences. We also discuss a preliminary extension of the i.i.d. results to the independent but not identically distributed (i.n.i.d.) case.