Bayesian smoothing algorithms in partially observed Markov Chains

Let x = x_nn∈N be a hidden process, y = y _nn∈N an observed process and r = r_{n n}∈N some auxiliary process. We assume that t = t _nn∈N with t_n = (x_n, r _n, y_n-1) is a (Triplet) Markov Chain (TMC). TMC are more general than Hidden Markov Chains (HMC) and yet enable the development of efficient restoration and parameter estimation algorithms. This paper is devoted to Bayesian smoothing algorithms for TMC. We first propose twelve algorithms for general TMC. In the Gaussian case, these smoothers reduce to a set of algorithms which include, among other solutions, extensions to TMC of classical Kalman-like smoothing algorithms (originally designed for HMC) such as the RTS algorithms, the Two-Filter algorithms or the Bryson and Frazier algorithm.