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