Second order Markov mesh models described as Markov Random Fields

We consider the problem of formulating Markov Mesh models as Markov
Random field. Since Markov Mesh models are a subclass of Markov Random
Fields, this can in principle always be done. In these notes we explore the details
of the parameter translation for the case of a 1D stationary Markov Mesh model
consisting of a homogeneous external field and general two-particle interactions.
We show theoretically that the corresponding Markov Random Field includes
also higher order interactions, the complexity of the interactions being limited
by the neighbourhood structure. Explicite formulas and recursive algorithms expressing
the MRF parameters in terms of the independent Markov Mesh parameters
are provided. A matlab implementation of the parameter translation is also
described. Using simulations we explore to which extent the higher order interactions
in the MRF formulation are necessary in order to reproduce the statistics
of the considered 2nd order Markov Mesh models. We find that in general the
higher order interactions have significant impact.