On the posterior distribution of classes of random means

TL;DR

This paper derives exact posterior distributions for mean functionals of normalized random measures with independent increments, extending Bayesian nonparametric results beyond Dirichlet processes to broader classes like NRMI and Poisson-Dirichlet.

Contribution

It provides the first explicit posterior distribution formulas for classes of priors beyond the Dirichlet process, including NRMI and mixtures, advancing Bayesian nonparametric theory.

Findings

Exact posterior distributions for NRMI means derived

Distributional results for two-parameter Poisson-Dirichlet process obtained

Extensions beyond Dirichlet process in Bayesian nonparametrics

Abstract

The study of properties of mean functionals of random probability measures is an important area of research in the theory of Bayesian nonparametric statistics. Many results are now known for random Dirichlet means, but little is known, especially in terms of posterior distributions, for classes of priors beyond the Dirichlet process. In this paper, we consider normalized random measures with independent increments (NRMI's) and mixtures of NRMI. In both cases, we are able to provide exact expressions for the posterior distribution of their means. These general results are then specialized, leading to distributional results for means of two important particular cases of NRMI's and also of the two-parameter Poisson--Dirichlet process.