Kalman filtering using pairwise Gaussian models

An important problem in signal processing consists in recursively estimating an unobservable process x={x_n}_n∈IN from an observed process y={y_n}_n∈IN. This is done classically in the framework of hidden Markov models (HMM). In the linear Gaussian case, the classical recursive solution is given by the well-known Kalman filter. We consider pairwise Gaussian models by assuming that the pair (x, y) is Markovian and Gaussian. We show that this model is strictly more general than the HMM, and yet still enables Kalman-like filtering.