We take the point of view that a Markov random field is a collection of so-called full conditional mutual independencies. Using the theory of -Measure, we have obtained a number of fundamental characterizations related to conditional mutual independence and Markov random fields. We show that many aspects of conditional mutual independence and Markov random fields have very simple set-theoretic ...

Graph covers and the Bethe free energy (BFE) have been useful theoretical tools for producing lower bounds on a variety of counting problems in graphical models, including permanent ferromagnetic Ising model. Here, we investigate weighted homomorphism over bipartite graphs that are related to conjecture Sidorenko. We show BFE does yield bound natural settings, when it bound, necessarily improve...