The Annals of Statistics

On the Consistency of Posterior Mixtures and Its Applications

Somnath Datta

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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.

Article information

Source
Ann. Statist., Volume 19, Number 1 (1991), 338-353.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347986

Digital Object Identifier
doi:10.1214/aos/1176347986

Mathematical Reviews number (MathSciNet)
MR1091855

Zentralblatt MATH identifier
0741.62005

JSTOR
links.jstor.org

Subjects
Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62C12: Empirical decision procedures; empirical Bayes procedures

Keywords
Posterior consistency mixing distribution empirical Bayes asymptotic optimality

Citation

Datta, Somnath. On the Consistency of Posterior Mixtures and Its Applications. Ann. Statist. 19 (1991), no. 1, 338--353. doi:10.1214/aos/1176347986. https://projecteuclid.org/euclid.aos/1176347986


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