The Consistency of Posterior Distribtions in Nonparametric Problems
Andrew Barron, Mark J. Schervish and Larry Wasserman
We give conditions that guarantee that the posterior probability of
every Hellinger neighborhood of the true density tends to 1 almost
surely. The conditions are (i) a smoothness condition on the prior and
(ii) a requirement that the prior put positive mass in appropriate
neighborhoods of the true density. The results are based on the idea
of approximating the set of densities with a finite dimensional set of
densities and then computing the Hellinger bracketing metric entropy
of the approximating set. We apply the results to some examples.
Keywords: Exponential families, Hellinger distance,
Nonparametric Bayesian inference, Polya trees.
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