A multivariate extension of a vector of two-parameter Poisson-Dirichlet processes

In the big data era there is a growing need to model the main features of large and non-trivial data sets. This paper proposes a Bayesian nonparametric prior for modelling situations where data are divided into different units with different densities, allowing information pooling across the groups....

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Bibliographic Details
Published in:Journal of nonparametric statistics 2015-01, Vol.27 (1), p.89-105
Main Authors: Zhu, Weixuan, Leisen, Fabrizio
Format: Article
Language:English
Subjects:
Algorithms
Bayesian nonparametrics
Data collection
Density
Exchange
exchangeable partition probability function
Laplace transforms
Lauricella function of fourth kind
Mathematical analysis
Mathematical models
Monte Carlo methods
multivariate Lévy measure
partial exchangeability
Pitman-Yor process
Probability distribution
Vector space
Vectors (mathematics)
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Summary:In the big data era there is a growing need to model the main features of large and non-trivial data sets. This paper proposes a Bayesian nonparametric prior for modelling situations where data are divided into different units with different densities, allowing information pooling across the groups. Leisen and Lijoi [(2011), 'Vectors of Poisson-Dirichlet processes', J. Multivariate Anal., 102, 482-495] introduced a bivariate vector of random probability measures with Poisson-Dirichlet marginals where the dependence is induced through a Lévy's Copula. In this paper the same approach is used for generalising such a vector to the multivariate setting. A first important contribution is the derivation of the Laplace functional transform which is non-trivial in the multivariate setting. The Laplace transform is the basis to derive the exchangeable partition probability function (EPPF) and, as a second contribution, we provide an expression of the EPPF for the multivariate setting. Finally, a novel Markov Chain Monte Carlo algorithm for evaluating the EPPF is introduced and tested. In particular, numerical illustrations of the clustering behaviour of the new prior are provided.
ISSN:1048-5252
1029-0311
DOI:10.1080/10485252.2014.966103