Density Estimation via Hierarchies of Nonparametric Priors

In Bayesian Nonparametrics partial exchangeability is a useful assumption tailored for heterogeneous, though related, groups of observations. Recent contributions in Bayesian literature have focused on the construction of dependent nonparametric priors to accommodate for partially exchangeable sequences of observations. In the present paper we concentrate on vectors of hierarchical Pitman-Yor processes, in which the dependence is created by choosing a common random base measure for each group of observations. These hierarchical processes are then used to define dependent hierarchical mixtures. We finally apply the model to estimate densities arising from multiple groups of observations by performing a suitable Gibbs sampling algorithm.