Stick-breaking representations of 1/2-stable Poisson-Kingman models

Bayesian nonparametrics has recently undergone a strong development in terms of both theory and practice. At the heart of the approach there is the concept of random probability measure whose distribution acts as a prior for Bayesian nonparametric inference, the most notable example being the Dirichlet process. Many well-known priors used in practice admit different, though equivalent in distribution, representations. Some of these are convenient if one wishes to thoroughly analyze the theoretical properties of the priors being used, others are more useful in terms of modelling and computation. In terms of the latter, the so-called stick-breaking constructions certainly stand out. Indeed, they allow to define efficient simulation algorithms and, furthermore, represent a useful tool for the construction of priors capable of incorporating certain forms of dependence for the observables and hence, go beyond the exchangeability setting. In this paper we focus on the class of $\frac{1}{2}$-stable Poisson-Kingman models introduced by Pitman (2003), which include as special cases several nonparametric priors currently used in Bayesian nonparametrics. This class of random probability measures is defined hierarchically in terms of an underlying normalized $\frac{1}{2}$-stable completely random measure which is suitably mixed with respect to the normalizing total mass. By exploiting such a hierarchical representation, we introduce a novel constructive definition for the class of $\frac{1}{2}$-stable Poisson-Kingman models by providing a completely explicit stick-breaking representation of the conditional underlying random probability measure given the normalizing total mass. The proposed constructive definition allows us to recover as special cases the stick-breaking representation of some well-known random probability measures such as, the two parameter Poisson-Dirichlet process with discount parameter $\frac{1}{2}$ and the normalized inverse Gaussian process. But new priors could be obtained by a particular choice of the mixing distribution for the total mass. Our result is interesting from both theoretical, modelling and computational points of view. Firstly, it completes the study of the well-known class of $\frac{1}{2}$-stable Poisson-Kingman models, by providing a constructive definition in terms of a latent random variable and a collection of dependent stick-breaking weights defined by means of Gamma and inverse Gamma random variables. Secondly, it suggests a simple way to define new Bayesian nonparametric models based on the class of $\frac{1}{2}$-stable Poisson-Kingman models by modifying well-established models based on the stick-breaking definition of the Dirichlet process. Finally, our representation allows to extend to the class of $\frac{1}{2}$-stable Poisson-Kingman models various recently proposed simulation algorithms such as, for instance, the slice sampling and the retrospective sampling. Indeed, both these simulation algorithms assume to have access to a stick-breaking representation of the underlying random probability measure