The Structure, Decomposition And Collapsibility Of Graphical Models

Graphical models, which have their origin in statistical physics, are increasingly valuable in problems of higher dimension and greater complexity. For example, graph-ical models have been introduced into systems biology to explore gene expression data and describe gene association networks with thousands of variables. This paper mainly discuss multinomial graphical models, Gaussian graphical models and con-ditional Gaussian graphical models. Mutinomial and Gaussian graphical models are characterized by undirected graphs, and conditional Gaussian graphical models are represented by marked graphs.Collapsibility plays an important role in structural reductions and model selec-tions. By pointing out the collapsible variable set, we can avoid missing or unobserved variables. Estimating and testing directly on sub-models can greatly reduce the de-mand for sample size.The problem we concern about is that under multinomial, Gaussian and condi-tional Gaussian graphical models, how to find the minimal set B containing our in-terested set A such that the whole model can collapsible onto the sub-model induced by B. Both Asmussen, Edwards and Frydenberg pointed out some relationship between the collapsibility of models and the decomposition of graphs. Madigan and Mosurski proposed an algorithm to find the minimal set B under decomposable multinomial graphical models, and this algorithm is also valid for decomposable Gaus-sian graphical models. But for general undirected graphs this algorithm can not find the minimal set B. In this paper, we handle with the most general case. For general multinomial and Gaussian graphical models, we give an algorithm to find the minimal set B and generalize this method to condition Gaussian graphical models.Basing on the decomposition of graphs with statistical sense, this paper presents fine structures for graphical models, and furthermore study the collapsibility of graph-ical models. Except for the three graphical models mentioned above, we also discuss the collapsibility of hierarchical models and probability propagation computation of conditional Gaussian Bayesian network, and the fine structure of models is useful in these two problems.For doing research on the collapsibility of multinomial and Gaussian graphical models, we begin from the decomposition of undirected graphs. The existence and uniqueness of this decomposition are given by Leimer, and the corresponding al-gorithm is also proposed by him. In this paper, we show the set class consisting of prime blocks, generated by decompositions, can form the junction tree structure, and provide some equivalent representations for this structure. This junction tree structure can help us study the collapsibility of multinomial and Gaussian graphical models, and the method we proposed can also be applied to hierarchical models.For considering the collapsibility of conditional Gaussian graphical models, we build the relation between the m-decomposition of marked graphs and the decomposi-tion of star graphs. This relation can convert the collapsibility of conditional Gaussian graphical models to corresponding problems on undirected graphs for considering. Furthermore, we study m-prime blocks for marked graphs and construct a junction tree of these m-prime blocks, which is similar as general undirected graphs. By in-troducing the relation between the minimal m-triangulation of marked graphs and the minimal triangulation of star graphs, we can obtain a minimal m-triangulation of moral graphs of conditional Gaussian Bayesian networks, and build a m-junction tree. This tree can provide the conditional independence relations as much as possible for conditional Gaussian Bayesian networks. By using this tree, we can efficiently reduce the complexity of probability propagation computations.