Abstract Markov Random Fields
<|reference_start|>Abstract Markov Random Fields: 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.<|reference_end|>
@article{lang2024abstract,
    title={Abstract Markov Random Fields},
    author={Leon Lang, Cl'elia de Mulatier, Rick Quax, Patrick Forr'e},
    journal={arXiv preprint arXiv:2407.02134},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.02134},
    primaryClass={cs.IT math.IT}
}