Optimal filtering and the dual process

TL;DR

This paper establishes a connection between optimal filtering for hidden Markov models and Markov process duality, enabling efficient computation of filtering distributions for certain processes, including new duals for the Cox-Ingersoll-Ross model.

Contribution

It introduces a novel duality framework for optimal filtering, generalizing known filters and deriving a new dual process for the Cox-Ingersoll-Ross model.

Findings

Filtering distributions form finite mixtures with polynomial computational cost

Special cases include Kalman, Cox-Ingersoll-Ross, and Wright-Fisher filters

New dual process for Cox-Ingersoll-Ross model derived

Abstract

We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Special cases of our framework include the Kalman filter, and computable filters for the Cox-Ingersoll-Ross process and the one-dimensional Wright-Fisher process, which have been investigated before. The dual we obtain for the Cox-Ingersoll-Ross process appears to be new in the literature.