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On Posterior Concentration in Misspecified Models

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journal contribution
posted on 21.11.2015, 00:00 authored by RV Ramamoorthi, K. Sriram, R. Martin
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 , 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.

Funding

The work of Karthik Sriram was supported by a research grant provided by Indian Institute of Management Ahmedabad, India.

History

Publisher Statement

This is the copy of an article published in Bayesian Analysis © 2015 International Society for Bayesian Analysis Publications.

Publisher

International Society for Bayesian Analysis

issn

1936-0975

Issue date

01/01/2015

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