The Annals of Statistics

The consistency of posterior distributions in nonparametric problems

Andrew Barron, Mark J. Schervish, and Larry Wasserman

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We give conditions that guarantee that the posterior probability of every Hellinger neighborhood of the true distribution tends to 1 almost surely. The conditions are (1) a requirement that the prior not put high mass near distributions with very rough densities and (2) a requirement that the prior put positive mass in Kullback-Leibler neighborhoods of the true distribution. The results are based on the idea of approximating the set of distributions with a finite-dimensional set of distributions with sufficiently small Hellinger bracketing metric entropy. We apply the results to some examples.

Article information

Ann. Statist., Volume 27, Number 2 (1999), 536-561.

First available in Project Euclid: 5 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties

Exponential families Hellinger distance nonparametric Bayesian inference Pólya trees.


Barron, Andrew; Schervish, Mark J.; Wasserman, Larry. The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 (1999), no. 2, 536--561. doi:10.1214/aos/1018031206.

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