Bayesian smoothing algorithms in pairwise and triplet markov chains

An important problem in signal processing consists in estimating an unobservable process x={x_n}_nisinN from an observed process y={y_n}_nisinN. In linear Gaussian hidden Markov chains (LGHMC), recursive solutions are given by Kalman-like Bayesian restoration algorithms. In this paper, we consider the more general framework of linear Gaussian triplet Markov chains (LGTMC), i.e. of models in which the triplet (x, r, y) (where r={r_n }_nisinN is some additional process) is Markovian and Gaussian. We address fixed-interval smoothing algorithms, and we extend to LGTMC the RTS algorithm by Rauch, Tung and Striebel, as well as the two-filter algorithm by Mayne and Fraser and Potter