Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference

In the general setting of nonparametric Bayesian inference, when observations are exchangeable and take values in a Polish space $X$, priors are approximated (in the Prokhorov metric) with any degree of precision by explicitly constructed mixtures of the distributions of Dirichlet processes. It is shown that if these mixtures ${\mathcal P}_{n}$ converge weakly to a given prior $\mathcal P$, the posteriors derived from ${\mathcal P}_{n}$'s converge weakly to the posterior deduced from $\mathcal P$. The error of approximation is estimated under some further assumptions. 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 $\int \psi d\widetilde{p}$ (in the Lévy metric), where $\widetilde p$ is a random probability having distribution $\mathcal P$.