The consistency of posterior distributions in nonparametric problems
Andrew Barron, Mark J. Schervish, Larry Wasserman
The Annals of Statistics · 1999 · Volume 27 · Issue 2
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.
Keywords
62G20, exponential families, Hellinger distance, nonparametric Bayesian inference, Pólya trees.
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