Two-filter formulae for discrete-time non-linear bayesian smoothing

Two-filter formulas for the Bayes solution of the fixed-interval discrete-time non-linear smoothing problem are obtained. The smoothed a posteriori density is computed under the assumptions of a general Markov signal observed through a general memoryless noisy channel. The case where there is feedback from the observation to the signal is also considered. The derived algorithms complement a two-pass algorithm obtained by Askar and Derin (1981). Known smoothing results for the linear Gaussian case are interpreted in the light of the general Bayesian results.