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

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  • Weixuan Zhu
  • Fabrizio Leisen

Abstract

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.

Suggested Citation

  • Weixuan Zhu & Fabrizio Leisen, 2015. "A multivariate extension of a vector of two-parameter Poisson-Dirichlet processes," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(1), pages 89-105, March.
  • Handle: RePEc:taf:gnstxx:v:27:y:2015:i:1:p:89-105
    DOI: 10.1080/10485252.2014.966103
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    References listed on IDEAS

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    1. De Iorio, Maria & Muller, Peter & Rosner, Gary L. & MacEachern, Steven N., 2004. "An ANOVA Model for Dependent Random Measures," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 205-215, January.
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    5. Hatjispyros, Spyridon J. & Nicoleris, Theodoros & Walker, Stephen G., 2011. "Dependent mixtures of Dirichlet processes," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2011-2025, June.
    6. Leisen, Fabrizio & Lijoi, Antonio, 2011. "Vectors of two-parameter Poisson-Dirichlet processes," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 482-495, March.
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    8. Stefano Favaro & Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2009. "Bayesian non‐parametric inference for species variety with a two‐parameter Poisson–Dirichlet process prior," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 993-1008, November.
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    Cited by:

    1. Jim E. Griffin & Fabrizio Leisen, 2017. "Compound random measures and their use in Bayesian non-parametrics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 525-545, March.
    2. Riva Palacio, Alan & Leisen, Fabrizio, 2018. "Integrability conditions for compound random measures," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 32-37.
    3. Camerlenghi, Federico & Lijoi, Antonio & Prünster, Igor, 2017. "Bayesian prediction with multiple-samples information," Journal of Multivariate Analysis, Elsevier, vol. 156(C), pages 18-28.

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