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March, 1991 On the Consistency of Posterior Mixtures and Its Applications
Somnath Datta
Ann. Statist. 19(1): 338-353 (March, 1991). DOI: 10.1214/aos/1176347986

Abstract

Consider i.i.d. pairs (θi,Xi),i1, where θ1 has an unknown prior distribution ω and given θ1,X1 has distribution Pθ1. This setup arises naturally in the empirical Bayes problems. We put a probability (a hyperprior) on the space of all possible ω and consider the posterior mean ω^ of ω. We show that, under reasonable conditions, Pω^=Pθdω^ is consistent in L1. Under a identifiability assumption, this result implies that ω^ is consistent in probability. As another application of the L1 consistency, we consider a general empirical Bayes problem with compact state space. We prove that the Bayes empirical Bayes rules are asymptotically optimal.

Citation

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Somnath Datta. "On the Consistency of Posterior Mixtures and Its Applications." Ann. Statist. 19 (1) 338 - 353, March, 1991. https://doi.org/10.1214/aos/1176347986

Information

Published: March, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0741.62005
MathSciNet: MR1091855
Digital Object Identifier: 10.1214/aos/1176347986

Subjects:
Primary: 62C10
Secondary: 62C12

Keywords: asymptotic optimality , consistency , Empirical Bayes , mixing distribution , Posterior

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 1 • March, 1991
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