Research On Conditional Independence Models And Related Problems

Conditional independence models draw the probabilistic relations between random variables. In practice, the causal relationship can be depicted by probabilistic distribution functions; hence it is suitable to describe probabilistic distributions over many graphs by conditional independence models. Presently, the research methods on conditional independence are mainly algebraic geometry methods and combinatorial methods. In this paper, one uses algebraic geometry methods to search conditional independence models, the content is belongs to algebraic statistics. The main works as follows:1. Give definitions of conditional independence models, prove many propositions about them. Study how conditional independence models translate into ideals and varieties. The meaning of primary decomposition is also given.2. The Bayesian networks are introduced. The relations between them and the so called distinguished component are given. Hidden variables and phylogenetic inference are also presented.3. Combining the ideal of algebraic statistics and the algebraic geometry of Bayesian networks and phylogenetic inference, one establishes a unity framework and introduces the concept of distinguished component and derived. In this stage, many questions are arisen.4. In this framework one studies a special class of feedback systems and defines the concept of global stability and iterative dimension in topology. They are proved to be meaning in some conditions.5. Finally, the summary of the full text is carried on; the directions of further research are pointed out.