Abstract Markov Random Fields

TL;DR

This paper extends the concept of Markov random fields to a broader class of information functions beyond Shannon entropy, providing new characterizations and applications in probabilistic models and thermodynamics.

Contribution

It generalizes Markov random fields using F-diagrams for functions satisfying the chain rule, introduces F-independences, and links these to thermodynamic principles and diffusion models.

Findings

F-mutual independence characterized by F-dual total correlation

Recovery of traditional Markov properties for probability distributions

Visualization of thermodynamic laws via Kullback-Leibler diagrams

Abstract

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 mass functions. We show that if the probability distributions of a set of random variables 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 diagrams on specific Markov chains, leading to a visual representation of the second law of thermodynamics and a simple explicit derivation of the decomposition of the evidence lower bound for diffusion models.