Official URL: 10.1109/CDC.1995.478570

Hidden Markov models have proved suitable for many interesting applications which can be modelled using some unobservable finite state Markov process, influencing measured signals. This can be used to describe bursty telecommunications traffic, or the faults in a complicated systems, for modelling the activity in neurons, for modelling speech patterns, etc. In all these applications, one has to estimate the unobservable underlying state of the Markov process, using the observed signals. Optimal recursive filters are well known for this estimation problem. Recently risk sensitive filters for the same problem have also been obtained. An important question in studying the quality of such filters is the rate at which arbitrarily assigned initial conditions are forgotten. In this paper we show that the effect of initial conditions on these filters dies out geometrically fast under very reasonable observability assumptions. The proof is given in the simplest case of finite state space and of a finite, quantised, observations spacc. However the method can be extended to more general models by continuity arguments.