Open Access
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


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.


Download Citation

Somnath Datta. "On the Consistency of Posterior Mixtures and Its Applications." Ann. Statist. 19 (1) 338 - 353, March, 1991.


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

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

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
Back to Top