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.

翻译：已知马尔可夫随机场完全由其信息图（I-图）的性质所刻画。具体而言，对于马尔可夫随机场，图中对应于不连通顶点集的I-图区域为零。最近，I-图被推广到F-图，适用于一类更广泛的、满足链式法则且超越香农熵的函数F，例如Kullback-Leibler散度和交叉熵。在本工作中，我们将马尔可夫随机场的概念和刻画推广到这类更广泛的函数F，并探讨其初步应用。我们定义了F-独立性、F-相互独立性和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，并证明其为零等价于F-相互独立性。随后，我们将结果应用于作用于概率分布的信息函数F。我们证明，如果概率分布对于同一图是马尔可夫随机场，那么我们形式上就恢复了该图的F-马尔可夫随机场概念。最后，我们研究了特定马尔可夫链上的Kullback-Leibler散度，从而得到了热力学第二定律的一种可视化表示。