  The Bayesian approach to inference stands out for naturally allowing
borrowing information across heterogeneous populations, with different samples
possibly sharing the same distribution. A popular Bayesian nonparametric model
for clustering probability distributions is the nested Dirichlet process, which
however has the drawback of grouping distributions in a single cluster when
ties are observed across samples. With the goal of achieving a flexible and
effective clustering method for both samples and observations, we investigate a
nonparametric prior that arises as the composition of two different discrete
random structures and derive a closed-form expression for the induced
distribution of the random partition, the fundamental tool regulating the
clustering behavior of the model. On the one hand, this allows to gain a deeper
insight into the theoretical properties of the model and, on the other hand, it
yields an MCMC algorithm for evaluating Bayesian inferences of interest.
Moreover, we single out limitations of this algorithm when working with more
than two populations and, consequently, devise an alternative more efficient
sampling scheme, which as a by-product, allows testing homogeneity between
different populations. Finally, we perform a comparison with the nested
Dirichlet process and provide illustrative examples of both synthetic and real
data.
