Triplet Partially Markov Chains and Trees

Hidden Markov models (HMM), like chains or trees considered in this paper, are widely used in different situations. Such models, in which the hidden process X is a Markov one, allow one estimating the latter from an observed process Y , which can be seen as a noisy version of X . This is possible once the distribution of X conditional on Y is a Markov distribution. These models have been recently generalized to Pairwise Markov models (PMM), in which one assumes the markovianity of ) , ( Y X , and Triplet Markov models (TMM), in which the distribution of ) , ( Y X is the marginal distribution of an Markov model ) , , ( Y U X . In this paper we propose further generalization of TMM by considering that ) , , ( Y U X is a Markov model with respect to ) , ( U X , but is not necessarily a Markov one with respect to Y . We show that in such models, called “partially Markov”, classical restoration algorithms remain valid.