  Markov random fields are known to be fully characterized by properties of
their information diagrams, or I-diagrams. In particular, for Markov random
fields, regions in the I-diagram corresponding to disconnected vertex sets in
the graph vanish. Recently, I-diagrams have been generalized to F-diagrams, for
a larger class of functions F satisfying the chain rule beyond Shannon entropy,
such as Kullback-Leibler divergence and cross-entropy. In this work, we
generalize the notion and characterization of Markov random fields to this
larger class of functions F and investigate preliminary applications.
  We define F-independences, F-mutual independences, and F-Markov random fields
and characterize them by their F-diagram. In the process, we also define F-dual
total correlation and prove that its vanishing is equivalent to F-mutual
independence. We then apply our results to information functions F that are
applied to probability distributions. We show that if the probability
distributions are Markov random fields for the same graph, then we formally
recover the notion of an F-Markov random field for that graph. We then study
the Kullback-Leibler divergence on specific Markov chains, leading to a visual
representation of the second law of thermodynamics.
