Marginalization And Collapsibility Of Graphical Models Based On Variable Elimination

In recent years,with the rapid development of computer science,the era of big data has followed.In the fields of biomedicine,financial stock market and network communication,data sets are becoming more and more high-dimensional and complex.In this background,how to reduce the variable dimension and computational complexity of high-dimensional data sets is a problem that needs to be studied and solved urgently.Graphical models,which originated from physics,has been more and more researched and applied because it can handle high-dimensional data sets well.Graphical models can make various complex relationships between random variables simple and intuitive,thereby it can transform complex statistical inference problems into simple problems corresponding to graphs.For a large-scale graphical model,we usually focus on partial variables but not all the concerned variables.Then,how to find sub-models of variables of interest? It is natural to consider the marginalization and collapsibility for graphical models.The marginalization of graphical models is to solve the global statistical inference problem by considering local statistical inference problem.Collapsibility of graphical models is a special case of marginalization,which means that the same statistical inference results can be obtained before and after the marginalization of certain variables.This thesis uses variable elimination method to study the marginalization and collapsibility of Markov networks.Firstly,using the variable elimination method of the undirected graph,the minimal independent graphs of Markov networks are studied when marginalizing over a single variable.Subsequently,the univariate elimination method is extended to multiple variables,and the minimum independent graph algorithm for marginal models of Markov networks are designed.By introducing the concept of information path,we can describe the minimal independent graph of marginal models more precisely.Secondly,the issue of collapsibility of Markov networks is deeply studied.From the perspective of variable elimination,several equivalent conditions for the collapsibility of Markov network are proposed.We prove that the model collapsibility and estimated collapsibility are equivalent under certain conditions.Finally,we give and prove some equivalent conditions for judging the collapsibility of Markov networks from the viewpoint of graph theory.The results of this thesis can intuitively judge whether Markov networks are collapsible,which provides a new way of studying the marginalization and collapsibility for Markov networks.