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Title: Introducing dependence among Dirichlet processes: A copula based approach Authors:  Gianluca Mastrantonio - Politecnico of Turin (Italy) [presenting]
Clara Grazian - University of Oxford (United Kingdom)
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. A draw form a Dirichlet process is a random distribution which is almost surely discrete. The Dirichlet process may be extended in a hierarchical version, so that several processes share the same set of atoms with process-dependent weights. As the Dirichlet process, its hierarchical extension has an explicit representation, called stick-breaking, that generates the weights from independent Beta random variables. We propose a way to introduce dependence in the marginal distributions of the vectors of weights, by imposing a copula model with given dependence (for instance, implying spatial or temporal 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 may be used to produce a structured nonparametric clustering, where the weights of the Dirichlet process represents the probabilities to be allocated to each cluster and the spatial or temporal dependence is among the weights relative to the same cluster.