Skip navigation
Skip navigation

A Bayesian view of the Poisson-Dirichlet Process

Buntine, Wray; Hutter, Marcus

Description

The two parameter Poisson-Dirichlet Process (PDP), a generalisation of the Dirichlet Process, is increasingly being used for probabilistic modelling in discrete areas such as language technology, bioinformatics, and image analysis. There is a rich literature about the PDP and its derivative distributions such as the Chinese Restaurant Process (CRP). This article reviews some of the basic theory and then the major results needed for Bayesian modelling of discrete problems including details of...[Show more]

dc.contributor.authorBuntine, Wray
dc.contributor.authorHutter, Marcus
dc.date.accessioned2015-08-25T05:36:25Z
dc.date.available2015-08-25T05:36:25Z
dc.identifier.urihttp://hdl.handle.net/1885/14917
dc.description.abstractThe two parameter Poisson-Dirichlet Process (PDP), a generalisation of the Dirichlet Process, is increasingly being used for probabilistic modelling in discrete areas such as language technology, bioinformatics, and image analysis. There is a rich literature about the PDP and its derivative distributions such as the Chinese Restaurant Process (CRP). This article reviews some of the basic theory and then the major results needed for Bayesian modelling of discrete problems including details of priors, posteriors and computation. The PDP allows one to build distributions over countable partitions. The PDP has two other remarkable properties: first it is partially conjugate to itself, which allows one to build hierarchies of PDPs, and second using a marginalised relative the CRP, one gets fragmentation and clustering properties that lets one layer partitions to build trees. This article presents the basic theory for understanding the notion of partitions and distributions over them, the PDP and the CRP, and the important properties of conjugacy, fragmentation and clustering, as well as some key related properties such as consistency and convergence. This article also presents a Bayesian interpretation of the Poisson-Dirichlet process based on an improper and infinite dimensional Dirichlet distribution. This means we can understand the process as just another Dirichlet and thus all its sampling properties emerge naturally. The theory of PDPs is usually presented for continuous distributions (more generally referred to as non-atomic distributions), however, when applied to discrete distributions its remarkable conjugacy property emerges. This context and basic results are also presented, as well as techniques for computing the second order Stirling numbers that occur in the posteriors for discrete distributions.
dc.rights© The Author(s)
dc.source.urihttp://arxiv.org/abs/1007.0296
dc.titleA Bayesian view of the Poisson-Dirichlet Process
dc.typeJournal article
dc.date.issued2010-07
local.publisher.urlhttp://arxiv.org/
local.type.statusSubmitted Version
local.contributor.affiliationHutter, M., Research School of Computer Science, The Australian National University
local.bibliographicCitation.startpage1
local.bibliographicCitation.lastpage50
dcterms.accessRightsOpen Access
dcterms.licenseThe URI http://arxiv.org/licenses/nonexclusive-distrib/1.0/ is used to record the fact that the submitter granted the following license to arXiv.org on submission of an article: I grant arXiv.org a perpetual, non-exclusive license to distribute this article. I certify that I have the right to grant this license. I understand that submissions cannot be completely removed once accepted. I understand that arXiv.org reserves the right to reclassify or reject any submission. Revision history 2004-01-16 - License above introduced as part of arXiv submission process 2007-06-21 - This HTML page created
CollectionsANU Research Publications

Download

File Description SizeFormat Image
1007.0296.pdf1.06 MBAdobe PDFThumbnail


Items in Open Research are protected by copyright, with all rights reserved, unless otherwise indicated.

Updated:  19 May 2020/ Responsible Officer:  University Librarian/ Page Contact:  Library Systems & Web Coordinator