无圈有向图模型的Markov等价类的计数
Enumerating Markov Equivalence Classes of Acyclic Digraph Models
摘要：
由无环有向图（ADG）（也称为有向无环图（DAG））确定的图形马尔可夫模型在统计学、计算机科学（如贝叶斯网络）、运筹学（如影响图）和许多相关领域中被广泛研究。由于不同的ADG可以确定相同的马尔可夫等价类，它长期以来一直感兴趣的是，通过直接与ADG的马尔可夫等价类，而不是与ADG本身工作，以确定在模型规范和搜索中获得的效率。编写了一个计算机程序来枚举Pearl & Verma等效准则所指定的ADG模型的等效类。该程序计算了模型的等价类，最多包括10个顶点。类别数与平均日增重之比似乎接近约0.267的渐近线。根据边缘数和类大小分析类。通过边缘，类的数量的分布接近高斯形状。按班级规模计算，规模为1的班级最常见，规模较大的班级的比例最初会下降，但随后会出现更不规则的模式。任何无向图生成的类的最大数量被发现增加约factorially。该计划还包括一个新的变化有序算法生成无向图。
英文摘要：
Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of number of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.
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