Two tales about Bayesian nonparametric modeling

Gibbs–type priors represent a natural generalization of the Dirichlet process: indeed, they select discrete distributions and can be characterized in terms of a system of predictive distributions that admit a simple and intuitive interpretation. By resorting to specific members of this wide class we address two important issues: (i) frequentist consistency; (ii) estimation of the discovery probability in species sampling problems. As for (i) we point out that Gibbs–type priors are consistent at discrete distributions whereas inconsistency may arise if the true distribution is diffuse thus highlighting that discrete nonparametric priors are typically consistent for the models they are designed for, namely data arising from discrete distributions. As for (ii), we emphasize how discrete nonparametric priors are well suited for addressing species sampling issues and display Bayesian nonparametric estimators of the probability of “discovering” a species that has appeared with any given frequency. These estimators can be compared with the frequentist counterparts, whenever the latter are available. Their practical use allows one appreciate the merits of the Bayesian nonparametric approach, which ensures that all objects of potential interest are modeled jointly and coherently.