Investigating nonparametric priors with Gibbs structure

This paper investigates nonparametric priors that induce infinite Gibbs-type partitions; such a feature is desirable both from a conceptual and a mathematical point of view. Recently it has been shown that Gibbs-type priors, with sigma is an element of (0, 1), are equivalent to sigma-stable Poisson-Kingman models. By looking at solutions to a recursive equation arising through Gibbs partitions, we provide an alternative proof of this fundamental result. Since practical implementation of general sigma-stable Poisson-Kingman models is difficult, we focus on a related class of priors, namely normalized random measures with independent increments; these are easily implementable in complex Bayesian models. We establish the result that the only Gibbs-type priors within this class are those based on a generalized gamma random measure.