# Compound Dirichlet Processes

## TL;DR

This paper introduces the compound Dirichlet process and its mixture, combining renewal and Bayesian nonparametric ideas to model event times and arrival variables with flexible, interpretable Bayesian structures.

## Contribution

It develops the compound Dirichlet process and mixture, providing new models with explicit posterior formulas and applications to real disease data.

## Key findings

- Models effectively capture event timing and arrival variability.
- Explicit posterior and distribution formulas derived.
- Successful application to zoonotic disease data.

## 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 joined their primal ideas to construct the compound Dirichlet process and the compound Dirichlet process mixture. As a consequence, these new processes had a fruitful 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 the posterior distribution, nonconditional distribution and expected values. In particular to find these formulas we analyze sums of random variables with Dirichlet process priors. We assessed our approach by applying our model on a real data example of a contagious zoonotic disease.