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On the support of MacEachern’s dependent Dirichlet processes and extensions. (English) Zbl 1330.60067

Summary: We study the support properties of Dirichlet process-based models for sets of predictor-dependent probability distributions. Exploiting the connection between copulas and stochastic processes, we provide an alternative definition of MacEachern’s dependent Dirichlet processes. Based on this definition, we provide sufficient conditions for the full weak support of different versions of the process. In particular, we show that under mild conditions on the copula functions, the version where only the support points or the weights are dependent on predictors have full weak support. In addition, we also characterize the Hellinger and Kullback-Leibler support of mixtures induced by the different versions of the dependent Dirichlet process. A generalization of the results for the general class of dependent stick-breaking processes is also provided.

MSC:

60G57 Random measures
62F15 Bayesian inference
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G99 Nonparametric inference
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