On Conditional Independence Structures

Conditional independence (CI) is an important concept, which has been widely used in many fields in artificial intelligence and statistics. CI structures defined in different fields are different. In general, a CI structure satisfies four formal properties, and we call this CI structure a semi-graphoid. CI models were first described by means of graphs. Structural imset is another tool for describing CI structures, and we call each CI structure induced by a structural imset a structural model. The CI structure of a common probability measure is a structural model.In the former part of this thesis, we study decomposition and inference of CI struc-tures. These results can be used to simplify statistical inference and construct effective statistical model. Firstly, we resolve one open problem in Theme9(P.206) posed by Studeny (2005). A new approach is proposed to decomposing a directed acyclic graph, its optimal properties are also studied. Then, we view this approach from decomposi-tion of the corresponding CI model, and provide an algorithm for identifying maximal prime subgraphs of a directed acyclic graph. Secondly, we consider semi-graphoid inference and independence implication of a set of CI statements. Through viewing each elementary CI statement as an indeterminate in a polynomial ring and utilizing tools from computational algebraic geometry, we provide complete characterizations of these two problems in theory and present two algorithms. Thirdly, based on our previous work in this thesis on decomposition of two classes of structural models and CI inference, and the idea provided by Studeny (2005), we present an approach to decomposing a structural model, show one can decompose a structural model into its maximal prime convex sub-models exactly, and provide an algorithm. Fourthly, we present a revised definition for decomposing a structural model, and show this defi-nition is equivalent to the first one we offered with respect to CI models induced by undirected graphs and directed acyclic graphs.Then, based on CI structures, we consider the problem of statistical inference. Ⅳ Fifthly, we clarified a conjecture posed by Sturmfels&Uhler (2010), that is, a com-plete characterization of vanishing ideals of a class of undirected graphs. Sixthly, we show that the sufficient condition for χ2-asymptotics for the likelihood ratio tests in one-factor analysis offered by Drton (2009), is equivalent to the one-factor case of a condition provided by Anderson&Rubin(1956). This result deepens our understand-ing of the condition provided by Anderson&Rubin.The last result concerns the faithfulness of undirected graphs with respect to dis-crete measures. Based on complex algebraic geometry, we provided a new proof of the strengthened version of conjecture1posed by Geiger&Pearl in1993.