Conjugacy properties of time-evolving Dirichlet and gamma random measures

We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first...

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Authors: Papaspiliopoulos, Omiros, Ruggiero, Matteo, Spanò, Dario
Format: article
Status:Published version
Publication Date:2016
Country:España
Institution:Universitat Pompeu Fabra
Repository:Repositorio Digital de la UPF
OAI Identifier:oai:repositori.upf.edu:10230/44904
Online Access:http://hdl.handle.net/10230/44904
http://dx.doi.org/10.1214/16-EJS1194
Access Level:Open access
Keyword:Bayesian nonparametrics
Dawson–Watanabe process
Dirichlet process
Duality
Fleming–Viot process
Gamma random measure
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spelling Conjugacy properties of time-evolving Dirichlet and gamma random measuresPapaspiliopoulos, OmirosRuggiero, MatteoSpanò, DarioBayesian nonparametricsDawson–Watanabe processDirichlet processDualityFleming–Viot processGamma random measureWe extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming–Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson–Watanabe model, and the data are collected according to a Poisson point process with intensity given by the process state at discrete times. Both driving processes are diffusions taking values in the space of discrete measures whose support varies with time, and are stationary and reversible with respect to Dirichlet and gamma priors respectively. A common methodology is developed to obtain in closed form the time-marginal posteriors given past and present data. These are shown to belong to classes of finite mixtures of Dirichlet processes and gamma random measures for the two models respectively, yielding conjugacy of these classes to the type of data we consider. We provide explicit results on the parameters of the mixture components and on the mixing weights, which are time-varying and drive the mixtures towards the respective priors in absence of further data. Explicit algorithms are provided to recursively compute the parameters of the mixtures. Our results are based on the projective properties of the signals and on certain duality properties of their projections.Supported by the MINECO/FEDER via grant MTM2015-67304-P. Supported by the European Research Council (ERC) through StG “N-BNP” 306406.The Institute of Mathematical Statistics and the Bernoulli Society202020202016info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/10230/44904http://dx.doi.org/10.1214/16-EJS1194reponame:Repositorio Digital de la UPFinstname:Universitat Pompeu FabraInglésElectronic Journal of Statistics. 2016;10(2):3452-89info:eu-repo/grantAgreement/EC/FP7/306406info:eu-repo/grantAgreement/ES/1PE/MTM2015-67304-PCopyright for all articles in EJP is CC BY 4.0.https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:repositori.upf.edu:10230/449042026-06-12T07:21:37Z
dc.title.none.fl_str_mv Conjugacy properties of time-evolving Dirichlet and gamma random measures
title Conjugacy properties of time-evolving Dirichlet and gamma random measures
spellingShingle Conjugacy properties of time-evolving Dirichlet and gamma random measures
Papaspiliopoulos, Omiros
Bayesian nonparametrics
Dawson–Watanabe process
Dirichlet process
Duality
Fleming–Viot process
Gamma random measure
title_short Conjugacy properties of time-evolving Dirichlet and gamma random measures
title_full Conjugacy properties of time-evolving Dirichlet and gamma random measures
title_fullStr Conjugacy properties of time-evolving Dirichlet and gamma random measures
title_full_unstemmed Conjugacy properties of time-evolving Dirichlet and gamma random measures
title_sort Conjugacy properties of time-evolving Dirichlet and gamma random measures
dc.creator.none.fl_str_mv Papaspiliopoulos, Omiros
Ruggiero, Matteo
Spanò, Dario
author Papaspiliopoulos, Omiros
author_facet Papaspiliopoulos, Omiros
Ruggiero, Matteo
Spanò, Dario
author_role author
author2 Ruggiero, Matteo
Spanò, Dario
author2_role author
author
dc.subject.none.fl_str_mv Bayesian nonparametrics
Dawson–Watanabe process
Dirichlet process
Duality
Fleming–Viot process
Gamma random measure
topic Bayesian nonparametrics
Dawson–Watanabe process
Dirichlet process
Duality
Fleming–Viot process
Gamma random measure
description We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming–Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson–Watanabe model, and the data are collected according to a Poisson point process with intensity given by the process state at discrete times. Both driving processes are diffusions taking values in the space of discrete measures whose support varies with time, and are stationary and reversible with respect to Dirichlet and gamma priors respectively. A common methodology is developed to obtain in closed form the time-marginal posteriors given past and present data. These are shown to belong to classes of finite mixtures of Dirichlet processes and gamma random measures for the two models respectively, yielding conjugacy of these classes to the type of data we consider. We provide explicit results on the parameters of the mixture components and on the mixing weights, which are time-varying and drive the mixtures towards the respective priors in absence of further data. Explicit algorithms are provided to recursively compute the parameters of the mixtures. Our results are based on the projective properties of the signals and on certain duality properties of their projections.
publishDate 2016
dc.date.none.fl_str_mv 2016
2020
2020
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10230/44904
http://dx.doi.org/10.1214/16-EJS1194
url http://hdl.handle.net/10230/44904
http://dx.doi.org/10.1214/16-EJS1194
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Electronic Journal of Statistics. 2016;10(2):3452-89
info:eu-repo/grantAgreement/EC/FP7/306406
info:eu-repo/grantAgreement/ES/1PE/MTM2015-67304-P
dc.rights.none.fl_str_mv Copyright for all articles in EJP is CC BY 4.0.
https://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Copyright for all articles in EJP is CC BY 4.0.
https://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv The Institute of Mathematical Statistics and the Bernoulli Society
publisher.none.fl_str_mv The Institute of Mathematical Statistics and the Bernoulli Society
dc.source.none.fl_str_mv reponame:Repositorio Digital de la UPF
instname:Universitat Pompeu Fabra
instname_str Universitat Pompeu Fabra
reponame_str Repositorio Digital de la UPF
collection Repositorio Digital de la UPF
repository.name.fl_str_mv
repository.mail.fl_str_mv
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