Exact and approximate Bayesian smoothing algorithms in partially observed Markov Chains

Let x = {x_n}_n∈IN be a hidden process, y = {y_n}_n∈IN an observed process and r = {r _n}_n∈IN some auxiliary process. We assume that t = {t_n}_n∈IN 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, they reduce to a set of algorithms which includes, among other solutions, extensions to TMC of classical Kalman-like smoothing algorithms such as the RTS algorithms, the Two-Filter algorithm or the Bryson and Frazier algorithm. We finally propose particle filtering (PF) approximations for the general case.