Particle filtering with pairwise Markov processes

The estimation of an unobservable process, x, from an observed process, y, is often performed in the framework of hidden Markov models (HMM). In the linear Gaussian case, the classical recursive solution is given by the Kalman filter. On the other hand, particle filters are Monte Carlo based methods which provide approximate solutions in more complex situations. We consider pairwise Markov models (PMM) by assuming that the pair (x, y) is Markovian. We show that this model is strictly more general than the HMM, and yet still enables particle filtering.