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December 2015 On Posterior Concentration in Misspecified Models
R. V. Ramamoorthi, Karthik Sriram, Ryan Martin
Bayesian Anal. 10(4): 759-789 (December 2015). DOI: 10.1214/15-BA941

Abstract

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.

Citation

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R. V. Ramamoorthi. Karthik Sriram. Ryan Martin. "On Posterior Concentration in Misspecified Models." Bayesian Anal. 10 (4) 759 - 789, December 2015. https://doi.org/10.1214/15-BA941

Information

Published: December 2015
First available in Project Euclid: 4 February 2015

zbMATH: 1335.62022
MathSciNet: MR3432239
Digital Object Identifier: 10.1214/15-BA941

Subjects:
Primary: 62C10
Secondary: 62C10

Rights: Copyright © 2015 International Society for Bayesian Analysis

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Vol.10 • No. 4 • December 2015
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