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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 $(\theta_i, X_i), i \geq 1$, where $\theta_1$ has an unknown prior distribution $\omega$ and given $\theta_1, X_1$ has distribution $P_{\theta_1}$. This setup arises naturally in the empirical Bayes problems. We put a probability (a hyperprior) on the space of all possible $\omega$ and consider the posterior mean $\hat{\omega}$ of $\omega$. We show that, under reasonable conditions, $P_{\hat{\omega}} = \int P_\theta d\hat{\omega}$ is consistent in $L_1$. Under a identifiability assumption, this result implies that $\hat{\omega}$ is consistent in probability. As another application of the $L_1$ consistency, we consider a general empirical Bayes problem with compact state space. We prove that the Bayes empirical Bayes rules are asymptotically optimal.

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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

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Vol.19 • No. 1 • March, 1991
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