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

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.