Observability and nonlinear filtering

TL;DR

This paper links the stability of nonlinear filters to a concept of observability in continuous-time hidden Markov models, showing that observability ensures filter stability and exploring related controllability conditions.

Contribution

It establishes a connection between observability and filter stability in continuous-time models, providing necessary and sufficient conditions for finite-state cases and extending results to non-compact spaces.

Findings

Observability implies filter stability in continuous-time models.

Complete characterization of filter stability for finite-state Markov signals.

Partial extension of results to non-compact state spaces.

Abstract

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces.