Abstract: 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 , are equivalent to -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 -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.
Key words and phrases: Bayesian nonparametrics, Gibbs exchangeable partitions, generalized gamma process, normalized random measures with independent increments, recursive equation, stable distribution.