Functional dirichlet process

Dirichlet process mixture (DPM) model is one of the most important Bayesian nonparametric models owing to its efficiency of inference and flexibility for various applications. A fundamental assumption made by DPM model is that all data items are generated from a single, shared DP. This assumption, however, is restrictive in many practical settings where samples are generated from a collection of dependent DPs, each associated with a point in some covariate space. For example, documents in the proceedings of a conference are organized by year, or photos may be tagged and recorded with GPS locations. We present a general method for constructing dependent Dirichlet processes (DP) on arbitrary covariate space. The approach is based on restricting and projecting a DP defined on a space of continuous functions with different domains, which results in a collection of dependent random measures, each associated with a point in covariate space and is marginally DP distributed. The constructed collection of dependent DPs can be used as a nonparametric prior of infinite dynamic mixture models, which allow each mixture component to appear/disappear and vary in a subspace of covariate space. Furthermore, we discuss choices of base distributions of functions in a variety of settings as a flexible method to control dependencies. In addition, we develop an efficient Gibbs sampler for model inference where all underlying random measures are integrated out. Finally, experiment results on temporal modeling and spatial modeling datasets demonstrate the effectiveness of the method in modeling dynamic mixture models on different types of covariates.