Pairwise and Uniformly Hidden Markov Fields

In Hidden Markov Fields (HMF) models there are two random fields : the hidden Markov field X and the observed field Y. In Pairwise Markov Fields (PMF) models one directly assumes the Markovianity of the couple (X, Y). PMF are more general than HMF; in fact, in PMF X is not necessarily Markovian. The aim of the paper is to provide some necessary and sufficient conditions under which PMF are HMF. We introduce the notion of "uniformly" HMF (UHMF) and we provide a general condition under which a PMF is an UHMF. Some interest of the presented results in the frame of Triplet Markov Fields (TMF) models, in which a third auxiliary random field is added and one considers the Markovianity of (X, U, Y), is also briefly discussed.