An asymptotic analysis of a class of discrete nonparametric priors

Abstract

In this paper we analyze the asymptotic behaviour of Gibbs-type priors, that represent a natural generalization of the Dirichlet process.After determining their topological support, we investigate their consistency according to the "what if", or frequentist, approach, that postulates the existence of a "true" distribution P0.We provide a full taxonomy of their limiting behaviours: consistency holds essentially always for discrete P0, whereas inconsistency may occur for diffuse P0.Such findings are further illustrated by means of three special cases admitting closed form expressions and exhibiting a wide range of asymptotic behaviours.For both Gibbs-type priors and discrete nonparametric priors in general, the possible inconsistency should not be interpreted as evidence against their use tout court.It rather represents an indication that they are designed for modeling discrete distributions and evidence against their use in the case of diffuse P0.