  Discrete random structures are important tools in Bayesian nonparametrics and
the resulting models have proven effective in density estimation, clustering,
topic modeling and prediction, among others. In this paper, we consider nested
processes and study the dependence structures they induce. Dependence ranges
between homogeneity, corresponding to full exchangeability, and maximum
heterogeneity, corresponding to (unconditional) independence across samples.
The popular nested Dirichlet process is shown to degenerate to the fully
exchangeable case when there are ties across samples at the observed or latent
level. To overcome this drawback, inherent to nesting general discrete random
measures, we introduce a novel class of latent nested processes. These are
obtained by adding common and group-specific completely random measures and,
then, normalising to yield dependent random probability measures. We provide
results on the partition distributions induced by latent nested processes, and
develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for
distributional homogeneity across groups is obtained as a by product. The
results and their inferential implications are showcased on synthetic and real
data.
