抽象马尔可夫随机场
Abstract Markov Random Fields
AI总结：
本文研究了基于F函数的马尔可夫随机场，定义了F独立性和F马尔可夫随机场，并探讨了其在信息函数中的应用。
摘要：
马尔可夫随机场以其信息图的性质完全表征。最近，信息图被扩展为F图，适用于更广泛的函数F，如KL散度和交叉熵。本文将马尔可夫随机场的概念扩展到此类函数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F图进行表征。在此过程中，我们还定义了F对偶总相关性，并证明其消失等价于F互独立性。然后，我们将结果应用于概率质量函数的信息函数F。我们展示，如果一组随机变量的概率分布对于同一图是马尔可夫随机场，则正式恢复了该图的F马尔可夫随机场的概念。然后，我们研究了特定马尔可夫链上的KL图，导致热力学第二定律的可视化表示，并为扩散模型的证据下界分解提供了简单的显式推导。
英文摘要：
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.
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