  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
$\mathbb{R}^p$. Indeed, for multivariate random measures, most available priors
lack flexibility and do not allow for separate partition structures among the
spaces. We introduce a discrete nonparametric prior, termed enriched Pitman-Yor
process (EPY), aimed at addressing these issues. Theoretical properties of this
novel prior are extensively investigated. 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 us to naturally extend these models while preserving their
analytical tractability. As an illustration, we employ the EPY for a species
sampling problem in ecology and for functional clustering in an e-commerce
application.
