Misspecification in Infinite-Dimensional Bayesian Statistics

We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P₀, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the pri...

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Bibliographic Details
Published in:The Annals of statistics 2006-04, Vol.34 (2), p.837-877
Main Authors: Kleijn, B. J. K., van der Vaart, A. W.
Format: Article
Language:English
Subjects:
62F05
62F15
62G07
62G08
62G20
Average linear density
Bayesian Analysis
Convexity
Density
Entropy
Exact sciences and technology
infinite-dimensional model
Logarithms
Mathematical functions
Mathematical models
Mathematics
Minimax
Misspecification
Nonparametric inference
Parametric inference
Parametric models
Perceptron convergence procedure
posterior distribution
Probability and statistics
rate of convergence
Sciences and techniques of general use
Statistical methods
Statistics
Studies
Topological theorems
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Description
Summary:We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P₀, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the prior that minimize the Kullback-Leibler divergence with respect to P₀. An entropy condition and a prior-mass condition determine the rate of convergence. The method is applied to several examples, with special interest for infinite-dimensional models. These include Gaussian mixtures, nonparametric regression and parametric models.
ISSN:0090-5364
2168-8966
DOI:10.1214/009053606000000029