Compound Dirichlet Processes

Abstract

The compound Poisson process and the Dirichlet process are the pillar structures of renewal theory and Bayesian nonparametric theory, respectively. Both processes have many useful extensions to fulfill the practitioners’ needs to model the particularities of data structures. Accordingly, in this contribution, we join their primal ideas to construct the compound Dirichlet process and the compound Dirichlet process mixture. As a consequence, these new processes have a rich structure to model the time occurrence among events, with also a flexible structure on the arrival variables. These models have a direct Bayesian interpretation of their posterior estimators and are easy to implement. We obtain expressions of posterior distribution, nonconditional distribution, and expected values. In particular, to find these formulas, we analyze sums of random variables with Dirichlet process priors. We assess our approach by applying our model on a real data example of a contagious zoonotic disease.