Statistica Sinica 18(2008), 1653-1668

INVESTIGATING NONPARAMETRIC PRIORS WITH

GIBBS STRUCTURE

Antonio Lijoi$^1$, Igor Prünster$^2$ and Stephen G. Walker$^3$

$^1$Universit`a di Pavia and CNR-IMATI Milano,

$^2$Universit`a di Torino, Collegio Carlo Alberto and ICER

and $^3$University of Kent

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 $\sigma\in(0,1)$, 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.

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