Recursive Kalman-type optimal estimation and detection of Hidden Markov chains

Efficient algorithms for computing the ‘a posterior? probabilities (APPs) of discrete-index finite-state hidden Markov sequences are proposed. They are obtained by reducing the APPs computation to the optimal nonlinear minimum mean square error (MMSE) estimation of the noisily observed sequences of the indicator functions associated with the chain states. Following an innovations approach, finite-dimensional and recursive Kalman-like ‘filter’ and ‘smoothers’ for the Markov chain state sequence are thus obtained, and exact expressions of their MSE performance are given. The filtered and smoothed state estimates coincide with the corresponding APP sequences. Finite-dimensional MMSE nonlinear filter and smoothers are also given for the so-called ‘number of jumps’ and for the ‘occupation time’ processes associated with the Markov state sequence.