Markov Cell Processes

We define a class of Markov cell processes, PX, on a finite set, X, as a product of conditional probabilities on cells (subsets of X forming a partially directed intersection graph). The class of Markov cell processes includes Bayesian networks and Markov edge processes. A nested conditional independence (NCI) condition is introduced that allows an explicit expression of a joint probability distribution through its marginals. The NCI condition is used in our construction of consistent Markov cell processes PX whose marginals coincide with PX′ on smaller sets X′⊂X. We discuss three classes of consistent Markov cell processes on rectangular domains X⊂Z3 equipped with ‘cubic’ cells, which include the Arak model and 3D Pickard model.