Exponential Filter Stability via Dobrushin's Coefficient

TL;DR

This paper introduces a new method for analyzing the stability of nonlinear filters in hidden Markov models using Dobrushin coefficients, allowing for more general conditions and exponential stability results.

Contribution

It presents a novel joint analysis using Dobrushin coefficients for transition and measurement kernels, relaxing previous restrictive mixing conditions for filter stability.

Findings

Filter stability is guaranteed if (1 - δ(T))(2 - δ(Q)) < 1.

The approach applies to controlled models.

Provides a concise stability analysis under relaxed conditions.

Abstract

Filter stability is a classical problem in the study of partially observed Markov processes (POMP), also known as hidden Markov models (HMM). For a POMP, an incorrectly initialized non-linear filter is said to be (asymptotically) stable if the filter eventually corrects itself as more measurements are collected. Filter stability results in the literature that provide rates of convergence typically rely on very restrictive mixing conditions on the transition kernel and measurement kernel pair, and do not consider their effects independently. In this paper, we introduce an alternative approach using the Dobrushin coefficients associated with both the transition kernel as well as the measurement channel. Such a joint study, which seems to have been unexplored, leads to a concise analysis that can be applied to more general system models under relaxed conditions: in particular, we show that if $(1 - \delta(T))(2-\delta(Q)) < 1$, where $\delta(T)$ and $\delta(Q)$ are the Dobrushin coefficients for the transition and the measurement kernels, then the filter is exponentially stable. Our findings are also applicable for controlled models.