Title: Dirichlet processes and copulas
Authors: Clara Grazian - University of Oxford (United Kingdom) [presenting]
Gianluca Mastrantonio - Romatre (Italy)
Enrico Bibbona - University of Torino (Italy)
Abstract: The Dirichlet process is a stochastic process defined on a space of distribution functions and which depends on a scaling parameter and a base distribution. It has an explicit representation called stick breaking representation, which is almost surely discrete, i.e. depends on some weighted atoms generated from the base distribution. The Dirichlet process may be extended in a hierarchical version, so that several processes share the same set of atoms with process dependent weights, which has a stick breaking representation. The original construction of the hierarchical Dirichlet process considers weights that are independent for different processes. We propose a way to introduce a dependence in the marginal distribution of the vectors of weights, by imposing a Gaussian copula whose correlation matrix has a given dependence structure (for instance, implying spatial dependence). We also prove that the dependence structure imposed on the (independent) components of the stick breaking representation is automatically transferred to the vectors of weights and that the order in which the components are taken does not matter. This representation of the hierarchical Dirichlet process may be used to produce a nonparametric clustering, where the weights of the Dirichlet process represents the probabilities to be allocated to each cluster and the dependence is among the weights relative to the same cluster.