Bayesian fixed-interval smoothing algorithms in singular state-space systems

Fixed-interval Bayesian smoothing in state-space systems has been addressed for a long time. However, as far as the measurement noise is concerned, only two cases have been addressed so far : the regular case, i.e. with positive definite covariance matrix; and the perfect measurement case, i.e. with zero measurement noise. In this paper we address the smoothing problem in the intermediate case where the measurement noise covariance is positive semi definite (p.s.d.) with arbitrary rank. We exploit the singularity of the model in order to transform the original state-space system into a pairwise Markov chain (PMC) with reduced state dimension. Finally, the a posteriori Markovianity of the reduced state enables us to propose a family of fixed-interval smoothing algorithms.