Large sample properties of Gibbs-type priors

In this paper we concisely summarize some recent findings that can be found in De Blasi, Lijoi and Pruenster (2012) and concern large sample properties of Gibbs-type priors. We shall specifically focus on consistency according to the frequentist approach which postulates the existence of a “ true ” distribution P_0 that generates the data. We show that the asymptotic behaviour of the posterior is completely determined by the probability of obtaining a new distinct observation. Exploiting the predictive structure of Gibbs-type priors, we are able to establish that consistency holds essentially always for discrete P0 , whereas inconsistency may occur for diffuse P_0. Such findings are further illustrated by means of three specific priors admitting closed form expressions and exhibiting a wide range of asymptotic behaviours.