On consistency of Gibbs-type priors

Most of the currently used discrete nonparametric priors are, with the exception of the Dirichlet process, inconsistent when used to model directly continuous data. On the other hand, when specified as basic building blocks within hierarchical mixture models, they generally lead to consistent density estimation. In this paper we announce several asymptotic results for a large class of discrete nonparametric priors, namely the class of Gibbs-type priors, which will be extensively presented and proved in De Blasi et al. (2011). Specifically, we provide sufficient conditions for consistency and present two examples within this class which exhibit completely opposite asymptotic posterior behaviours when the ''true'' distribution is continuous.