Bayesian nonparametrics is a relatively young area of research which has recently received abundant attention in the statistical literature. The considerable degree of flexibility it ensures if compared to standard parametric alternatives, and the development of new and efficient computational tools pushed both its theoretical development and its concrete use in a number of complex real world problems. This talk is divided in two parts. In the first, I will briefly review a milestone in Bayesian nonparametrics: the Dirichlet process, with particular emphasis on the Dirichlet process mixture models for density estimation.
Although Bayesian nonparametric models for continuous variables are well developed, the literature on related approaches for counts is limited. For this reason in the second part of the talk, I will discuss a recent contribution to model count variables (Canale and Dunson, 2011, JASA). The main idea is to induce prior distributions on the spaces of probability mass functions via priors on the space of continuous densities and suitable mapping functions. The procedures enjoys important theoretical properties such as large support of the prior and strong posterior consistency. Efficient Gibbs samplers are developed for posterior computation, and the approach is illustrated via simulations and applications to developmental toxicity and marketing studies.