Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach

TL;DR

This paper establishes conditions under which the filter in general state-space hidden Markov models forgets initial conditions exponentially, using a coupling approach, which aids in filtering and estimator analysis.

Contribution

It provides simple, general conditions for exponential forgetting of initial states in hidden Markov models using a coupling method, applicable to broad model classes.

Findings

Exponential forgetting of initial conditions is guaranteed under certain conditions.

The coupling approach applies to models like Gaussian state-space without Kalman filter structure.

Results are useful for filtering approximation methods and asymptotic estimator properties.

Abstract

We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.