Decomposition And Collapsibility In Graphical Models

For nearly three decades, graphical models have been more and more widely used in some areas, such as bio-informatics, economics, sociology, causal infer-ence, artificial intelligence and statistics. Especially in statistics, graphical models are used most widely. Using graphical models not only allows the complex rela-tionships among random variables to be described visually, but also allows some statistical problems to be transformed into some corresponding graph-related problems, which can simplify the statistical problems to a certain extent.With improvement of science and technology in various application areas and rapid development of computer technology, there are more and more high-dimensional data. In this case, how to reduce the variable dimensions and the complexity of the problems has become a very practical, important and urgent problem. Studies of decomposition and collapsibility provide very competitive ways to this problem. Collapsibility refers to transforming a global problem into a corresponding local problem by ignoring the unrelated varibles, that is, "absorb-ing the essence and discarding the dross". Decomposition refers to decomposing a global problem into some corresponding local problems and then integrating the results of these local problems to solve the original global problem, that is, "divid-ing and conquering". For graphical models, decomposition and collapsibility are very important research topics, and interaction graph can be used to determine whether a problem can be decomposed or collapsed, which indicates that many effective methods in graph theory can provide strong support for treatment of the problem.Decomposition and collapsibility are very inornate and basic ideas, which are widely used in statistical inference or structural learning problems of graphical models. Decomposition and collapsibility can be defined from several different angles for different purposes, and there have been many good results about de-composition and collapsibility. However, with the production of new issues and new purposes in application areas, the study of decomposition and collapsibility also has been developing rapidly.The present paper concentrates on the following three topics in detail around the specific theme of decomposition and collapsibility in graphical models:de-composition for structural learning of Bayesian networks, collapsibility and de-composition of likelihood ratio tests in graphical models of undirected graphs, and collapsibility in conditional models of graphical models of undirected graphs. For structural learning of a Bayesian network, Section 3 proposed a definition of minimal d-scparation tree to improve the efficiency of the structural learning, and discussed characterization and construction of a minimal d-separation tree in detail. Using the constructed minimal d-separation tree, we could obtain a maximal efficiency in the process of the decomposition approach of structural learning based on a d-separation tree. In Section 4, we discussed collapsibil-ity and decomposition of likelihood ratio tests in graphical models of undirected graphs, and obtained some further results. We proposed a weaker sufficient con-dition for determining the collapsibility in the corresponding interaction graphs, and proposed a new way to decompose test statistic. In Section 5, we discussed collapsibility in conditional models of graphical models of undirected graphs for purely discrete and purely continuous cases. We introduced two different kinds of collapsibility in the conditional models, proved the equivalency between two kinds of collapsibility and proposed a necessary and sufficient condition for the equivalent collapsibility in term of the corresponding interaction graph.