A simple proof of Pitman–Yor’s Chinese restaurant process from its stick-breaking representation

Caroline Lawless,Julyan Arbel

doi:10.1515/demo-2019-0003

Abstract

Abstract For a long time, the Dirichlet process has been the gold standard discrete random measure in Bayesian nonparametrics. The Pitman-Yor process provides a simple and mathematically tractable generalization, allowing for a very flexible control of the clustering behaviour. Two commonly used representations of the Pitman-Yor process are the stick-breaking process and the Chinese restaurant process. The former is a constructive representation of the process which turns out very handy for practical implementation, while the latter describes the partition distribution induced. Obtaining one from the other is usually done indirectly with use of measure theory. In contrast, we propose here an elementary proof of Pitman-Yor’s Chinese Restaurant process from its stick-breaking representation.

Highlights

The Pitman–Yor process de nes a rich and exible class of random probability measures which was developed by Perman et al [26] and further investigated by Pitman [28], Pitman and Yor [30]
It is a simple generalization of the Dirichlet process [15], whose mathematical tractability contributed to its popularity in machine learning theory [7], probabilistic models for linguistic applications [35, 37], excursion theory [26, 30], measure-valued di usions in population genetics [14, 27], combinatorics [19, 36] and statistical physics [11]
Applications in this setting embrace a variety of inferential problems, including species sampling [3, 12, 24], survival analysis and graphical models in genetics [18, 25], image segmentation [34], curve estimation [6], exchangeable feature allocations [5] and time-series and econometrics [4, 7]

Summary

Introduction

The Pitman–Yor process de nes a rich and exible class of random probability measures which was developed by Perman et al [26] and further investigated by Pitman [28], Pitman and Yor [30] It is a simple generalization of the Dirichlet process [15], whose mathematical tractability contributed to its popularity in machine learning theory [7], probabilistic models for linguistic applications [35, 37], excursion theory [26, 30], measure-valued di usions in population genetics [14, 27], combinatorics [19, 36] and statistical physics [11].

Caroline Lawless and Julyan Arbel

Proofs of the technical results