Distribution theory for hierarchical processes

Hierarchies of discrete probability measures are remarkably popular as nonparametric priors in applications, arguably due to two key properties: (i) they naturally represent multiple heterogeneous populations; (ii) they produce ties across populations, resulting in a shrinkage property often described as “sharing of information.” In this paper, we establish a distribution theory for hierarchical random measures that are generated via normalization, thus encompassing both the hierarchical Dirichlet and hierarchical Pitman–Yor processes. These results provide a probabilistic characterization of the induced (partially exchangeable) partition structure, including the distribution and the asymptotics of the number of partition sets, and a complete posterior characterization. They are obtained by representing hierarchical processes in terms of completely random measures, and by applying a novel technique for deriving the associated distributions. Moreover, they also serve as building blocks for new simulation algorithms, and we derive marginal and conditional algorithms for Bayesian inference.