Dimensions Of Undirected Graphical Models

Graphical model is the product which combine probability theory and graph theory,it is a family of probability distributions defined in terms of a graph which is used to represent conditional independence relations among random variables.For nearly four decades,graphical models have been more and more widely used in some areas,such as bio-informatics,economics,sociology,causal inference,machine learning and statistics.Dimensions of graphical models are important for test,model selection.Given a discrete undirected graphical models?DUGM?,one can view the dimension of the corresponding toric ideal or the rank of the associated matrix as its dimension.The Vapnik-Chervonenkis?VC?dimension and Euclidean embedding dimension in machine learning are two indicators to measure the complexity of function classes.Further,given any DUGM,the VC dimension,the Euclidean embedding dimension of the concept class induced by it and its dimension are identical.VC dimension is very important in the evaluation of classifier performance.Pe?a?2009?gived a formula for computing the dimension of graph.Naturally,we wonder the relationship between the definition and VC dimension.In this thesis,beginning from two special undirected graphical models corresponding to undirected graph G_n,G_n+1,where,G_n is a loop,G_n+1 is based on G_n by adding a node X_n+1which is adjacent to each node in the G_n,each node corresponds to a binary random variable.The difference of the dimensional values between two definitions is obtained by calculating the rank of corresponding matrix.Then,we show that the difference of the two dimensional values is also 1 for any DUGMs.In fact,it is found that the two definitions are consistent in essence.In the field of statistical learning,faithfulness assumption is a basic hypothesis.Multivariate totally positive of order two?MTP₂?is a special form which implies most other definition of positive dependence.One of the basic problem between multivariate totally positive of order two distributions and undirected graphs is faithfulness.This thesis deals with the dimension of multivariate totally positive of order two undirected graphical model.Given an undirected graph G,we prove that the strictly positive discrete multivariate totally positive of order two distributions with fixed sample space that factorize according to G with dimension d have positive Lebesgue measure with respect to?^d.Namely,we extend the faithfulness results in Pe?a?2009,2011?and Li&Li?2017?to the Lebesgue measure case,that is,almost all positive discrete multivariate totally positive of order two distributions that factorize according to G are faithful to it.In this paper,we also consider regular Gaussian multivariate totally positive of order two distributions,and obtain the same faithfulness result as discrete cases.Finally,we summarize this thesis and look forward to the future work of the graphical model.