Open Access
Translator Disclaimer
February 2019 Distribution theory for hierarchical processes
Federico Camerlenghi, Antonio Lijoi, Peter Orbanz, Igor Prünster
Ann. Statist. 47(1): 67-92 (February 2019). DOI: 10.1214/17-AOS1678

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.

Citation

Download Citation

Federico Camerlenghi. Antonio Lijoi. Peter Orbanz. Igor Prünster. "Distribution theory for hierarchical processes." Ann. Statist. 47 (1) 67 - 92, February 2019. https://doi.org/10.1214/17-AOS1678

Information

Received: 1 July 2016; Revised: 1 December 2017; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036195
MathSciNet: MR3909927
Digital Object Identifier: 10.1214/17-AOS1678

Subjects:
Primary: 60G57 , 62F15 , 62G05

Keywords: Bayesian nonparametrics , distribution theory , hierarchical processes , partition structure , posterior distribution , prediction , Random measures , species sampling models

Rights: Copyright © 2019 Institute of Mathematical Statistics

JOURNAL ARTICLE
26 PAGES


SHARE
Vol.47 • No. 1 • February 2019
Back to Top