Consistent Markov Edge Processes and Random Graphs

We discuss Markov edge processes ₑ; e ∈ E\ defined on edges of a directed acyclic graph $(V, E)$ with the consistency property: PE'(Yₑ; e ∈ E') = PE(Yₑ; e ∈ E') for a large class of subgraphs $(V',E')$ of $(V,E)$ obtained through a mesh dismantling algorithm. The probability distribution PE of such edge process is a discrete version of consistent polygonal Markov graphs studied in {arakDS1993, arakDS1989}. The class of Markov edge processes is related to the class of Bayesian networks and may be of interest to causal inference and decision theory. On regular ν-dimensional lattices, consistent Markov edge processes have similar properties to Pickard random fields on Z², representing a far-reaching extension of the latter class. A particular case of binary consistent edge process on Z³ was disclosed by Arak in a private communication. We prove that symmetric binary Pickard model generates Arak model on Z² as a contour model.