Filtering hidden Markov measures

TL;DR

This paper develops a unified filtering methodology for two classes of time-evolving random measures, enabling explicit computation of posterior distributions in models involving Fleming--Viot and Dawson--Watanabe diffusions.

Contribution

It introduces a common approach to derive explicit filtering distributions for dynamic Fleming--Viot and Dawson--Watanabe models, extending classic static Bayesian updates.

Findings

Filtering distributions are mixtures of Dirichlet and gamma measures.

An explicit algorithm computes mixture parameters.

Extends static Bayesian posterior characterizations to dynamic settings.

Abstract

We consider the problem of learning two families of time-evolving random measures from indirect observations. In the first model, the signal is a Fleming--Viot diffusion, which is reversible with respect to the law of a Dirichlet process, and the data is a sequence of random samples from the state at discrete times. In the second model, the signal is a Dawson--Watanabe diffusion, which is reversible with respect to the law of a gamma random measure, and the data is a sequence of Poisson point configurations whose intensity is given by the state at discrete times. A common methodology is developed to obtain the filtering distributions in a computable form, which is based on the projective properties of the signals and duality properties of their projections. The filtering distributions take the form of mixtures of Dirichlet processes and gamma random measures for each of the two families respectively, and an explicit algorithm is provided to compute the parameters of the mixtures. Hence, our results extend classic characterisations of the posterior distribution under Dirichlet process and gamma random measures priors to a dynamic framework.