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

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.