Bayes Estimation of the Mixing Distribution, The Discrete Case

Abstract

Let $X_1, X_2, \cdots$ be independent identically distributed random variables taking on values in the positive integers with a family of possible probability distributions indexed by $G \in \mathscr{G}$, the class of all probability distribution functions on $\lbrack 0, + \infty)$. Under the assumption that the family is identifiable we wish to estimate the true but unknown $G_0$. This is done by constructing a prior probability distribution on $\mathscr{G}$ and showing that the Bayes estimate corresponding to the prior is consistent.