Exponential stability of filters and smoothers for hidden markov models

In this paper, we address the problem of filtering and fixed-lag smoothing for discrete-time and discrete-state hidden Markov models (HMM's), with the intention of extending some important results in Kaiman filtering, notably the property of exponential stability. By appealing to a generalized Perron-Frobenius result for non-negative matrices, we are able to demonstrate exponential forgetting for both the recursive filters and smoothers; furthermore, methods for deriving overbounds on the convergence rate are indicated. Simulation studies for a two-state and two-output HMM verify qualitatively some of the theoretical predictions, and the observed convergence rate is shown to be bounded in accordance with the theoretical predictions.