Distribution theory for hierarchical processes
Federico Camerlenghi, Antonio Lijoi, Peter Orbanz, Igor Prünster
The Annals of Statistics · 2019 · Volume 47 · Issue 1
Abstract
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.
Keywords
60G57, 62F15, 62G05, Bayesian nonparametrics, distribution theory, hierarchical processes, partition structure, posterior distribution, prediction, Random measures, species sampling models
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