Gibbs sampling methods for Pitman-Yor mixture models

We introduce a new sampling strategy for the two-parameter Poisson-Dirichlet process mixture model, also known as Pitman-Yor process mixture model (PYM). Our sampler is therefore applicable to the well-known Dirichlet process mixture model (DPM). Inference in DPM and PYM is usually performed via Markov Chain Monte Carlo (MCMC) methods, specifi cally the Gibbs sampler. These sampling methods are usually divided in two classes: marginal and conditional algorithms. Each method has its merits and limitations. The aim of this paper is to propose a new sampler that combines the main advantages of each class. The key idea of the proposed sampler consists in replacing the standard posterior updating of the mixing measure based on the stick-breaking representation, with a posterior updating of Pitman(1996) which represents the posterior law under a Pitman-Yor process as the sum of a jump part and a continuous one. We sample the continuous part in two ways, leading to two variants of the proposed sampler. We also propose a threshold to improve mixing in the first variant of our algorithm. The two variants of our sampler are compared with a marginal method, that is the celebrated Algorithm 8 of Neal(2000), and two conditional algorithms based on the stick-breaking representation, namely the efficient slice sampler of Kalli et al. (2011) and the truncated blocked Gibbs sampler of Ishwaran and James (2001). We also investigate e ffects of removing the proposed threshold in the first variant of our algorithm and introducing the threshold in the efficient slice sampler of Kalli et al. (2011). Results on real and simulated data sets illustrate that our algorithms outperform the other conditionals in terms of mixing properties