The Annals of Statistics

Distribution theory for hierarchical processes

Federico Camerlenghi, Antonio Lijoi, Peter Orbanz, and Igor Prünster

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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.

Article information

Ann. Statist., Volume 47, Number 1 (2019), 67-92.

Received: July 2016
Revised: December 2017
First available in Project Euclid: 30 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 62G05: Estimation 62F15: Bayesian inference

Bayesian nonparametrics distribution theory hierarchical processes partition structure posterior distribution prediction random measures species sampling models


Camerlenghi, Federico; Lijoi, Antonio; Orbanz, Peter; Prünster, Igor. Distribution theory for hierarchical processes. Ann. Statist. 47 (2019), no. 1, 67--92. doi:10.1214/17-AOS1678.

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Supplemental materials

  • Distribution theory for hierarchical processes: Supplementary material. We provide the proofs of the theoretical results and specialize the Blackwell–MacQueen urn scheme of Section 6.1 to the case of hierarchies of Pitman–Yor processes.