Parameters Estimates Of Chain Graphical Models With Missing Data

In recent years,a graphical model is rapidly developed in a new field of study, it isan interdisciplin in probability statistics and graph theory.We use a graph to representa traditional multivariate statistics of describing the relationships among severalvariables, a graphical model is a powerful tool for inferring the uncertainty andcomplexity of the random variables. Especially a large number of variables in themodel,such as medical and genetic research, involved dozens or even hundreds ofvariables maybe, purely mathematical or statistical methods is hard to make sense ofso many variables,also it is difficult to visually represent the relationship betweenthese variables, making it difficult to exchange with researchers in the other field.Therefore,the graphical model is as a tool for inferring complex problems,using adecomposition of the graph, many high-dimensional problems can be broken downinto low-dimensional problems, and analysis the complex independent relationship,the timing relationship or causal relationship between the random variables with theintuitive structure of graph.So far,a graphical model has been widely applied instatistical physics,artificial intelligence,economics,engineering reliability, biostatistics,medicine and other fields.In parameter estimation or statistical inference of multivariate statistical analysis,maximum likelihood method is one of the most important and widely used methods.This parameter estimation method is a common method with complete data in thesaturated model, if the data pattern is incomplete, and the random variablescorrespond to a graph structure,that is for the case with unsaturated model, we shoulduse the EM algorithm to estimate the parameters.So far, the existing literature at home and abroad discussed the parameterestimation by EM algorithm and PIEM algorithm for mixed graphical models withmissing data, and factorization of posteriors distribution and parameter estimation byPI algorithm for the undirected graphical models with missing data, about parameterestimating for the chain graphical models with missing data, not every graphstructures are studied. In this paper, we will analysis a special graph structure, concrete to make estimation of unknown parameters from the following sections.In the first chapter,we introduce the main study results of graphical models,and alsointroduce the premise of this study in this paper and the knowledge.In the second chapter, giving the basic concept of the graph, Include an undirectedgraph, directed graph and chain graph. Make research that comparing on theinformation content on nodes of undirected graph, we generalized a conclusion whichis information on adjacent nodes are more than not adjacent nodes of undirectedgraph.In the third chapter,giving a special graph structure, for the case with full data,derive the Maximum Likelihood Estimates of unknown parameters,at the sametime,derive the parameters estimation with missing data under different conditions byEM algorithm, this process has proved some lemmas and theorems.In the fourth chapter, we summary the innovation of this paper and the issueswhich need to be further explored.