Enriched Pitman-Yor processes

In Bayesian nonparametrics there exists a rich variety of discrete priors, including the Dirichlet process and its generalizations, which are nowadays well-established tools. Despite the remarkable advances, few proposals are tailored for modeling observations lying on product spaces, such as $\mathbb{R}^p$. In this setting, most of the available priors lack of flexibility and they do not allow for separate partition structures among the spaces. We address these issues by introducing a discrete nonparametric prior termed enriched Pitman-Yor process (EPY). Theoretical properties of this novel prior are extensively investigated. Specifically, we discuss its formal link with the enriched Dirichlet process and normalized random measures, we describe a square-breaking representation and we obtain closed form expressions for the posterior law and the involved urn schemes. In second place, we show that several existing approaches, including Dirichlet processes with a spike and slab base measure and mixture of mixtures models, implicitly rely on special cases of the EPY, which therefore constitutes a unified probabilistic framework for many Bayesian nonparametric priors. Interestingly, our unifying formulation will allow to naturally extend these models, while preserving their analytical tractability. As an illustration, we employ the EPY for a species sampling problem in ecology.