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

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.

Keywords:

• Bayesian nonparametrics
• Pitman–Yor process
• Lauricella function of fourth kind
• multivariate Lévy measure
• partial exchangeability
• exchangeable partition probability function

AMS Subject Classification::

• 62F15
• 62H05
• 60G57
• 60G51

Funding

The research leading to these results has received funding from the European Community's Seventh Framework Programme [FP7/2007-2013] under grant agreement no: 630677.

Notes

1. Recall that a sequence of measures (m_i)_i≥1 in converges, in the w^♯–topology, to a measure m in if and only if for any bounded set such that . See Daley and Vere-Jones (2003) for further details.