Bayes Empirical Bayes: Finite Parameter Case

Since Robbins (1956), a large literature has evolved treating the empirical Bayes formulation of a sequence of decision problems. In this paper we look at the formulation and the asymptotic optimality criterion as a classical iid problem and a consistency property. With this point of view and the finite state component problem, we evoke classical results on Bayes procedures to prove a complete class theorem and to establish the asymptotic optimality of Bayes empirical Bayes procedures.