判定有向循环图模型Markov等价性的多项式时间算法
A Polynomial-Time Algorithm for Deciding Markov Equivalence of Directed Cyclic Graphical Models
摘要：
虽然d-分离度的概念最初是为有向无环图定义的（参见Pearl 1988），但这个概念自然地扩展到有向循环图。当两个有向图的d-分离关系的集合完全相同时，无论是循环图还是非循环图，我们都说它们是马尔可夫等价的。换句话说，当两个有向循环图是马尔可夫等价的，满足全局有向马尔可夫条件（Lauritzen et al. 1990）的自然扩展的分布集对于每个图都是完全相同的。有一个明显的指数（在顶点数）时间算法来决定两个有向循环图的马尔可夫等价性;简单地检查每个图中的所有d-分离关系。在本文中，我陈述了一个定理，给出了必要和充分条件的马尔可夫等价的两个有向循环图，其中每个条件可以检查在多项式时间。因此，该定理可以很容易地适用于一个多项式时间的算法来确定两个有向循环图的马尔可夫等价。虽然空间不允许包含正确性证明，但它们在Richardson（1994 b）中得到了充分的描述。
英文摘要：
Although the concept of d-separation was originally defined for directed acyclic graphs (see Pearl 1988), there is a natural extension of he concept to directed cyclic graphs. When exactly the same set of d-separation relations hold in two directed graphs, no matter whether respectively cyclic or acyclic, we say that they are Markov equivalent. In other words, when two directed cyclic graphs are Markov equivalent, the set of distributions that satisfy a natural extension of the Global Directed Markov condition (Lauritzen et al. 1990) is exactly the same for each graph. There is an obvious exponential (in the number of vertices) time algorithm for deciding Markov equivalence of two directed cyclic graphs; simply chech all of the d-separation relations in each graph. In this paper I state a theorem that gives necessary and sufficient conditions for the Markov equivalence of two directed cyclic graphs, where each of the conditions can be checked in polynomial time. Hence, the theorem can be easily adapted into a polynomial time algorithm for deciding the Markov equivalence of two directed cyclic graphs. Although space prohibits inclusion of correctness proofs, they are fully described in Richardson (1994b).
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