Exact and Approximate Hidden Markov Chain Filters Based on Discrete Observations

TL;DR

This paper develops exact and approximate filtering formulas for a hidden Markov model observed at discrete times with noise, focusing on the two-state case and proposing three approximation methods for larger state spaces.

Contribution

It provides the first exact formulas for the densities in the two-state case and introduces three novel approximation techniques for larger state spaces.

Findings

Exact formulas derived for two-state HMM densities.

Three approximation methods compared numerically.

Approximate methods perform well in simulations.

Abstract

We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically and the third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study.