Approximation of Laws of Multinomial Parameters by Mixtures of Dirichlet Distributions with Applications to Bayesian Inference

Abstract

Within the framework of Bayesian inference, when observations are exchangeable and take values in a finite space X, a prior P is approximated (in the Prokhorov metric) with any precision by explicitly constructed mixtures of Dirichlet distributions. Likewise, the posteriors are approximated with some precision by the posteriors of these mixtures of Dirichlet distributions. Approximations in the uniform metric for distribution functions are also given. These results are applied to obtain a method for eliciting prior beliefs and to approximate both the predictive distribution (in the variational metric) and the posterior distribution function of ∫ψd $$\tilde p$$ (in the Levy metric), when $$\tilde p$$ is a random probability having distribution P.