  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.
