图模型的对偶性
Duality in Graphical Models
摘要：
图形模型已被证明是表示随机变量的高维系统的强大工具。这种模型的一个例子是无向图，其中缺少边表示两个随机变量之间的条件独立性。另一个例子是双向图，其中边的缺失编码成对的边缘独立性。这两类图模型都被广泛研究，虽然它们被认为是彼此对偶的，但除了少数情况外，这种对偶性还没有被彻底研究。在本文中，我们演示了如何无向和双向模型之间的对偶可以用来运输一类图形模型的结果，以一种透明的方式的对偶模型。我们继续应用这种技术来扩展以前现有的结果，以及证明新的，在三个重要的领域。首先，我们讨论了无向和双向模型的两两和全局马尔可夫性，使用pseudographoid和反向pseudographoid规则，这是较弱的条件比通常使用的交集和组合规则。其次，我们调查这些pseudographoid和反向pseudographoid规则的背景下，概率分布，在这个过程中使用的对偶概念。对偶性使我们能够快速地将它们与更熟悉的交集和组合属性联系起来。第三，也是最后，我们运用二元论的方法来理解忠实的含义，这反过来又导致了一个更一般的形式的存在的结果。
英文摘要：
Graphical models have proven to be powerful tools for representing high-dimensional systems of random variables. One example of such a model is the undirected graph, in which lack of an edge represents conditional independence between two random variables given the rest. Another example is the bidirected graph, in which absence of edges encodes pairwise marginal independence. Both of these classes of graphical models have been extensively studied, and while they are considered to be dual to one another, except in a few instances this duality has not been thoroughly investigated. In this paper, we demonstrate how duality between undirected and bidirected models can be used to transport results for one class of graphical models to the dual model in a transparent manner. We proceed to apply this technique to extend previously existing results as well as to prove new ones, in three important domains. First, we discuss the pairwise and global Markov properties for undirected and bidirected models, using the pseudographoid and reverse-pseudographoid rules which are weaker conditions than the typically used intersection and composition rules. Second, we investigate these pseudographoid and reverse pseudographoid rules in the context of probability distributions, using the concept of duality in the process. Duality allows us to quickly relate them to the more familiar intersection and composition properties. Third and finally, we apply the dualization method to understand the implications of faithfulness, which in turn leads to a more general form of an existing result.
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