## The Annals of Statistics

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

### On the Consistency of Posterior Mixtures and Its Applications

#### 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