A marginal sampler for $σ$-Stable Poisson-Kingman mixture models

Bayesian nonparametric mixture models reposed on random probability measures like the Dirichlet process allow for flexible modelling of densities and for clustering applications where the number of clusters is not fixed a priori. Our understanding of these models has grown significantly over the last decade: there is an increasing realization that while these models are nonparametric in nature and allow an arbitrary number of components to be used, they do impose significant prior assumptions regarding the clustering structure. In recent years there is growing interest in extending modelling flexibility beyond the classical Dirichlet process. Examples include Pitman-Yor processes, normalized inverse Gaussian processes, and normalized random measures. With different processes, additional flexibility and characterizations are gained and have been used to develop tractable Markov chain Monte Carlo posterior simulation algorithms. In this paper we explore the use of a very wide class of random probability measures, called $\sigma$-stable Poisson-Kingman processes for Bayesian nonparametric mixture modelling. This class of processes encompasses most known tractable random probability measures proposed in the literature so far, and we argue that it forms a natural class to study. We review certain characterizations and propose a tractable posterior inference algorithm for the whole class.