Forgetting properties for hidden Markov models

Hidden Markov models provide the opportunity to capture a number of nonlinear and/or nongaussian signal processing problems. This paper discusses the existence of results applicable to hidden Markov model filters and fixed-lag smoothers which parallel results applicable to Kalman filters and fixed-lag smoothers, in relation to forgetting of initial conditions, effect of round-off errors, and appropriate choices of the lag for a fixed-lag smoother. Some related problems on maximum a posteriori sequence estimation are also discussed. Tools for addressing these problems are provided by extensions of the Perron-Frobenius theory to inhomogenous products of positive matrices, and inhomogenous matrix products in the so-called max plus algebra.