A class of dependent Dirichlet processes via latent multinomial processes

TL;DR

This paper introduces a flexible framework for creating dependent Dirichlet processes using latent multinomial processes, enabling modeling of temporal and spatial dependencies with conjugacy properties for Bayesian inference.

Contribution

It presents a novel method to induce dependence among Dirichlet processes via latent processes, extending their applicability in Bayesian nonparametrics.

Findings

Dependence characterized through correlation between processes.

Posterior distributions derived in a Bayesian context.

Numerical example demonstrating the approach.

Abstract

We describe a procedure to introduce general dependence structures on a set of Dirichlet processes. Dependence can be in one direction to define a time series or in two directions to define spatial dependencies. More directions can also be considered. Dependence is induced via a set of latent processes and exploit the conjugacy property between the Dirichlet and the multinomial processes to ensure that the marginal law for each element of the set is a Dirichlet process. Dependence is characterised through the correlation between any two elements. Posterior distributions are obtained when we use the set of Dirichlet processes as prior distributions in a bayesian nonparametric context. Posterior predictive distributions induce partially exchangeable sequences defined by generalised P\'olya urs. A numerical example to illustrate is also included.