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The consistency of posterior distributions in nonparametric problems. (English) Zbl 0980.62039
Summary: 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.

MSC:
62G20 Asymptotic properties of nonparametric inference
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[27] NEW HAVEN, CONNECTICUT 06520 CARNEGIE MELLON UNIVERSITY E-MAIL: barron@stat.yale.edu PITTSBURGH, PENNSYLVANIA 15213 E-MAIL: mark@stat.cmu.edu, larry@stat.cmu.edu
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