The Pitman-Yor multinomial process for mixture modelling

Discrete nonparametric priors play a central role in a variety of Bayesian procedures, most notably when used to model latent features, such as in clustering, mixtures and curve fitting. They are effective and well-developed tools, though their infinite dimensionality is unsuited to some applications. If one restricts to a finite-dimensional simplex, very little is known beyond the traditional Dirichlet multinomial process, which is mainly motivated by conjugacy. This paper introduces an alternative based on the Pitman-Yor process, which provides greater flexibility while preserving analytical tractability. Urn schemes and posterior characterizations are obtained in closed form, leading to exact sampling methods. In addition, the proposed approach can be used to accurately approximate the infinite-dimensional Pitman-Yor process, yielding improvements over existing truncation-based approaches. An application to convex mixture regression for quantitative risk assessment illustrates the theoretical results and compares our approach with existing methods.