PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

A Bayesian Review of the Poisson-Dirichlet Process
Wray Buntine and Marcus Hutter
(2010) Technical Report. Cornell UNiversity Library, USA.

Abstract

The two parameter Poisson-Dirichlet process is also known as the Pitman-Yor Process and related to the Chinese Restaurant Process, is a generalisation of the Dirichlet Process, and is increasingly being used for probabilistic modelling in discrete areas such as language and images. This article reviews the theory of the Poisson-Dirichlet process in terms of its consistency for estimation, the convergence rates and the posteriors of data. This theory has been well developed for continuous distributions (more generally referred to as non-atomic distributions). This article then presents a Bayesian interpretation of the Poisson-Dirichlet process: it is a mixture using an improper and infinite dimensional Dirichlet distribution. This interpretation requires technicalities of priors, posteriors and Hilbert spaces, but conceptually, this means we can understand the process as just another Dirichlet and thus all its sampling properties fit naturally. Finally, this article also presents results for the discrete case which is the case seeing widespread use now in computer science, but which has received less attention in the literature.

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EPrint Type:Monograph (Technical Report)
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Learning/Statistics & Optimisation
ID Code:5774
Deposited By:Wray Buntine
Deposited On:12 August 2010