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journal contribution
posted on 2015-11-21, 00:00 authored by RV Ramamoorthi, K. Sriram, R. MartinWe 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 , the density that is closest in the Kullback–Leibler sense to
the true model f0. 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 L1-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. 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.