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.

翻译：马尔可夫随机场已知可以由其信息图（即I-图）的性质完全刻画。具体而言，对于马尔可夫随机场，I-图中对应图中不连通顶点集的区域会消失。近期，I-图被推广到F-图，适用于满足链式法则的更大一类函数F（如Kullback-Leibler散度和交叉熵），而不仅限于香农熵。本文我们将马尔可夫随机场的概念和刻画推广到这一类更大的函数F，并探讨初步应用。我们定义了F-独立性、F-互独立性和F-马尔可夫随机场，并通过其F-图对其进行刻画。在此过程中，我们还定义了F-对偶总相关，并证明其消失等价于F-互独立性。随后，我们将结果应用于作用于概率质量函数的信息函数F。我们证明：如果一组随机变量的概率分布是同一图的马尔可夫随机场，那么该图就正式恢复了F-马尔可夫随机场的概念。最后，我们研究特定马尔可夫链上的Kullback-Leibler图，这给出了热力学第二定律的可视化表示，以及扩散模型中证据下界分解的简单显式推导。