Statistica Sinica 23 (2013),

AN ASYMPTOTIC ANALYSIS OF A CLASS OF DISCRETE

NONPARAMETRIC PRIORS

Pierpaolo De Blasi$^1$, Antonio Lijoi$^2$ and Igor Prünster$^1$

$^1$University of Torino and Collegio Carlo Alberto, Italy and

$^2$University of Pavia and Collegio Carlo Alberto, Italy

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 $P_0$. We provide a full taxonomy of their limiting behaviours: consistency holds essentially always for discrete $P_0$, whereas inconsistency may occur for diffuse $P_0$. 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 $P_0$.

Key words and phrases: Asymptotics, Bayesian consistency, Bayesian nonparametrics, Gibbs-type priors, foundations, species sampling.