On the stick-breaking representation of $\sigma$-stable Poisson-Kingman models

Stefano Favaro,Bernardo Nipoti,Yee Whye Teh,Maria Lomeli

doi:10.1214/14-ejs921

Abstract

In this paper we investigate the stick-breaking representation for the class of $\sigma$-stable Poisson-Kingman models, also known as Gibbs-type random probability measures. This class includes as special cases most of the discrete priors commonly used in Bayesian nonparametrics, such as the two parameter Poisson-Dirichlet process and the normalized generalized Gamma process. Under the assumption $\sigma=u/v$, for any coprime integers $1\leq u<v$ such that $u/v\leq1/2$, we show that a $\sigma$-stable Poisson-Kingman model admits an explicit stick-breaking representation in terms of random variables which are obtained by suitably transforming Gamma random variables and products of independent Beta and Gamma random variables.

Highlights

Random probability measures play a fundamental role in Bayesian nonparametrics as their distributions act as nonparametric priors
In this paper we focus on the stick-breaking representation of discrete random probability measures
Recent simulation algorithms developed in the context of hierarchical mixture modeling, such as the blocked Gibbs-sampler by Ishwaran and James [15], the slice sampling by Walker [43] and the retrospective sampling by Papaspiliopoulos and Roberts

Summary

Introduction

Random probability measures play a fundamental role in Bayesian nonparametrics as their distributions act as nonparametric priors. Apart from the two parameter Poisson-Dirichlet process, most of the discrete random probability measures do not admit a stick-breaking representation in terms of a collection of independent Vi’s. To the best of our knowledge the normalized inverse Gaussian process provides the first example of a prior admitting a stick-breaking representation in terms of dependent Vi’s, and such that for any i ≥ 1 the conditional distribution of Vi given

Results

Conclusion