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

E. Regazzini ; V. V. Sazonov

Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 1, pp. 103-124

Résumé

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 $\mathcal P$.

Détail

Keywords: approximation of priors and posteriors, Dirichlet distributions, Dirichlet processes, elicitation of prior beliefs, Lévy metric, Prokhorov metric, random measures.