Backward Map for Filter Stability Analysis

TL;DR

This paper introduces a backward map for analyzing nonlinear stochastic filter stability, linking variance decay properties to model assumptions like detectability through a Poincaré Inequality in hidden Markov models.

Contribution

It proposes a novel backward map approach for filter stability analysis and establishes the connection between Poincaré Inequality and detectability in HMMs.

Findings

Backward map enables stability analysis via variance decay.

Poincaré constant is positive iff the HMM is detectable.

Stability results depend on ergodicity and observability conditions.

Abstract

In this paper, a backward map is introduced for the purposes of analysis of the nonlinear (stochastic) filter stability. The backward map is important because the filter-stability in the sense of $\chisq$-divergence follows from showing a certain variance decay property for the backward map. To show this property requires additional assumptions on the model properties of the hidden Markov model (HMM). The analysis in this paper is based on introducing a Poincar\'e Inequality (PI) for HMMs with white noise observations. In finite state-space settings, PI is related to both the ergodicity of the Markov process as well as the observability of the HMM. It is shown that the Poincar\'e constant is positive if and only if the HMM is detectable.