Nested Compound Random Measures

Nested nonparametric processes are vectors of random probability measures widely used in the Bayesian literature to model the dependence across distinct, though related, groups of observations. These processes allow a two-level clustering, both at the observational and group levels. However, most available models are either not computationally efficient or mathematically tractable. In the present paper, we introduce a range of nested processes that overcome these issues. Our proposal builds upon Compound Random Measures, introduced early by Griffin and Leisen. We provide a complete investigation of the theoretical properties of our model, along with the posterior characterization for vectors of Compound Random Measures, which is interesting per se and still not available in the current literature. We develop the first Ferguson & Klass algorithm for nested nonparametric processes. Finally, we test the model’s performance on different simulated scenarios and we exploit the construction to study air pollution in various provinces of an Italian region (Lombardy). We empirically show how nested processes based on Compound Random Measures outperform other Bayesian competitors.