Bayesian Estimation of Mixing Distributions

Let = \(t): 0 \ t \ 1\\ be a family of probability distributions on the positive integers parameterized on [0, 1], that is \{equation*}\{1} P_t(X = x) = q_x(t);\ x = 1,2,\⋯.\{equation*} If G is a distribution on \ 0, 1\ the distribution of X is a G-mixture over if \{equation*}\{2} P_G(X = x) = \∫^1_0 q_x(t) dG(t) = q_x(G).\{equation*} G is called the mixing distribution. It is assumed at the outset that the family is known to be identifiable, that is if qₓ(G₁) = qₓ(G₂) for x = 1,2,\⋯, then G₁ = G₂. See [12] and [13] for conditions insuring identifiability. Thus it makes sense to attempt to estimate G when one has independent observations on X. Some work on estimating G has been done when the mixture is finite [4], [2], [9] and for special 's [6], [14]. The problem is of interest not only in an estimation context, but also in the construction of empirical Bayes decision procedures [9]. Our approach is to define a prior distribution on possible values of G and then construct consistent Bayes estimates of G from the posterior distribution. Section 2 gives the needed background on moment spaces, sets up the prior distribution and derives the posterior distribution. In Section 3, the Bayes estimates are defined while Section 4 proves the consistency of the posterior distribution and thus of the estimates. Here, Theorem 1 is not directly applicable to our problem, but is included because of its possible independent interest. Sections 5 and 6 generalize the earlier results.