ON THE CONSISTENCY OF POSTERIOR MIXTURES AND ITS APPLICATIONS

Consider i.i.d. pairs (theta-i, X-i), i greater-than-or-equal-to l, where theta-1 has an unknown prior distribution omega and given theta-1, X-1 has distribution P theta-1. This setup aries naturally in the empirical Bayes problems. We put a probability (a hyperprior) on the space of all possible omega and consider the posterior mean omega of omega. We show that, under reasonable conditions, P omega = integral-P-theta d omega is consistent in L1. Under a identifiability assumption, this result implies that omega is consistent in probability. As another application of the L1 consistency, we consider a general empirical Bayes problem with compact state space. We prove that the Bayes empirical Bayes rules are asymptotically optimal.