On Dependent Dirichlet Processes for General Polish Spaces

TL;DR

This paper extends the theory of dependent Dirichlet processes to general Polish spaces, providing conditions for their continuity, association, and support properties, and analyzing mixture models for predictor-dependent distributions.

Contribution

It generalizes dependent Dirichlet processes from Euclidean to Polish spaces and establishes conditions for their desirable theoretical properties.

Findings

Dependent Dirichlet processes can be defined on general Polish spaces.

Conditions for continuity, association, and support are established.

Mixture models exhibit strong continuity and weak consistency under i.i.d. sampling.

Abstract

We study Dirichlet process-based models for sets of predictor-dependent probability distributions, where the domain and predictor space are general Polish spaces. We generalize the definition of dependent Dirichlet processes, originally constructed on Euclidean spaces, to more general Polish spaces. We provide sufficient conditions under which dependent Dirichlet processes have appealing properties regarding continuity (weak and strong), association structure, and support (under different topologies). We also provide sufficient conditions under which mixture models induced by dependent Dirichlet processes have appealing properties regarding strong continuity, association structure, support, and weak consistency under i.i.d. sampling of both responses and predictors. The results can be easily extended to more general dependent stick-breaking processes.