Slice Sampling σ-Stable Poisson-Kingman Mixture Models

Abstract

The article is concerned with the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models. In particular, we consider the problem of slice sampling mixture models for a large class of mixing measures generalizing the celebrated Dirichlet process. Such a class of measures, known in the literature as σ -stable Poisson-Kingman models, includes as special cases most of the discrete priors currently known in Bayesian nonparametrics, for example, the twoparameter Poisson-Dirichlet process and the normalized generalized Gamma process. The proposed approach is illustrated on some simulated data examples. This article has online supplementary material.

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

“…We refer to such a model as a σ-stable Poisson-Kingman mixture model. A conditional MCMC method for σ-stable Poisson-Kingman mixture model has been recently introduced in Favaro & Walker (2012). The class of σ-stable Poisson-Kingman RPMs forms a large class of discrete RPMs which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, e.g., the Pitman-Yor and the normalized generalized Gamma processes.…”

“…Our main contribution is to provide a general purpose framework for performing posterior inference with any member of this class of priors. In contrast to Favaro & Walker (2012), we exploit marginal characterisations of σ-stable Poisson-Kingman RPMs in order to remove the infinite dimensionality of the sampling problem. Efficient algorithms often rely upon simplifying properties of the priors just as inference algorithms for graphical models rely upon the conditional independencies encoded by the graph.…”

Slice sampler algorithm for generalized Pareto distribution

“…Such a class of measures, known in the literature as σ-stable PoissonKingman models, includes as special cases most of the discrete priors currently known in Bayesian nonparametrics, e.g., the two parameter Poisson-Dirichlet process and the normalized generalized Gamma process. Favaro and Walker [42] show how to slice sample a class of mixture models which includes all of the popular choices of mixing measures. Based on the results, with standard stick-breaking models the stick-breaking variables are independent, even as they appear in the full conditional distribution sampled in the posterior MCMC algorithm.…”

“…However, the savings are in the prerunning work where setting up a slice sampler is far easier than setting up a retrospective sampler. In addition, Favaro and Walker [42] consider the problem of slice sampler mixture models for a large class of mixing measures generalizing the celebrated Dirichlet process. Such a class of measures, known in the literature as σ-stable PoissonKingman models, includes as special cases most of the discrete priors currently known in Bayesian nonparametrics, e.g., the two parameter Poisson-Dirichlet process and the normalized generalized Gamma process.…”

A blocked Gibbs sampler for NGG-mixture models via a priori truncation

“…The slice sampler has been extended to NRMI mixtures in Griffin and Walker (2011). See also Favaro and Walker (2013).…”