Particle Filtering in Pairwise and Triplet Markov Chains

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 provide approximate solutions in more complex situations. In this paper, we propose two successive generalizations of the classical HMM. We first 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. We next consider Triplet Markov Models (TMM) by assuming the Markovianity of a triplet (x, r,y), in which r is some additional auxiliary process. We show that the Triplet model is strictly more general than the Pairwise one, and yet still enables particle filtering.