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

authors

  • ZHU, WEIXUAN
  • LEISEN, FABRIZIO

publication date

  • January 2015

start page

  • 89

end page

  • 105

issue

  • 1

volume

  • 27

international standard serial number (ISSN)

  • 1048-5252

electronic international standard serial number (EISSN)

  • 1029-0311

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 Levy'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.

keywords

  • bayesian nonparametrics; pitman–yor process; lauricella function of fourth kind; multivariate lévy measure; partial exchangeability; exchangeable partition probability function