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The Consistency of Posterior Distributions in Nonparametric Problems

Andrew Barron, Mark J. Schervish and Larry Wasserman
The Annals of Statistics
Vol. 27, No. 2 (Apr., 1999), pp. 536-561
Stable URL: http://www.jstor.org/stable/120103
Page Count: 26
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The Consistency of Posterior Distributions in Nonparametric Problems
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Abstract

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.

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