A marginal sampler for $\sigma$-Stable Poisson-Kingman mixture models

TL;DR

This paper introduces a new efficient MCMC sampling method for a broad class of Bayesian nonparametric mixture models, improving inference for models like the Dirichlet and Pitman-Yor processes.

Contribution

It proposes a novel MCMC sampling scheme exploiting properties of $\sigma$-stable Poisson-Kingman RPMs, applicable to various popular Bayesian nonparametric models.

Findings

The new sampler is more efficient than existing methods.

It performs well on density estimation tasks.

It effectively handles multidimensional datasets.

Abstract

We investigate the class of $\sigma$-stable Poisson-Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs which encompasses most of the the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman-Yor process, the normalized inverse Gaussian process and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of $\sigma$-stable Poisson-Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for making inference in Bayesian nonparametric mixture modeling. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a fixed number of auxiliary variables per iteration. We apply our sampling scheme for a density estimation and clustering tasks with unidimensional and multidimensional datasets, and we compare it against competing sampling schemes.