Conjugacy properties of time-evolving Dirichlet and gamma random measures

TL;DR

This paper extends conjugacy results for Dirichlet and gamma random measures to a dynamic setting, enabling Bayesian updating of time-evolving processes with explicit formulas and algorithms.

Contribution

It introduces a novel framework for Bayesian nonparametric models with time-dependent Dirichlet and gamma processes, providing closed-form posterior mixtures and recursive algorithms.

Findings

Posterior distributions are finite mixtures of Dirichlet and gamma measures.

Explicit formulas for mixture parameters and weights are derived.

Algorithms for recursive computation of posteriors are developed.

Abstract

We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming-Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson-Watanabe model, and the data are collected according to a Poisson point process with intensity given by the process state at discrete times. Both driving processes are diffusions taking values in the space of discrete measures whose support varies with time, and are stationary and reversible with respect to Dirichlet and gamma priors respectively. A common methodology is developed to obtain in closed form the time-marginal posteriors given past and present data. These are shown to belong to classes of finite mixtures of Dirichlet processes and gamma random measures for the two models respectively, yielding conjugacy of these classes to the type of data we consider. We provide explicit results on the parameters of the mixture components and on the mixing weights, which are time-varying and drive the mixtures towards the respective priors in absence of further data. Explicit algorithms are provided to recursively compute the parameters of the mixtures. Our results are based on the projective properties of the signals and on certain duality properties of their projections.