An application of the colored adjacency matrix for understanding the graphical model structures

Abstract

The operations, marginalization and conditionalization, on a probability model affect the probability model in a variety of ways. If we denote the probability model before one of the operations by M and by M`` that after the operation, M and M`` may belong to the same family of probability models or not. For example, marginalization on a Gaussian model (or a multinomial model) yields another Gaussian model (or a multinomial model) while it may not be the case for some other models such as a mixture of Gaussian models. If we interpret the model structure of a probability model as a graphical representation of the Markov properties which are latent in the probability model, then different probability models may share a model structure. In this thesis we will investigate, in the context of model structure, the relationship between the models before and after each of the two operations. Consider a set of random variables, $X_1, … , X_n$ where $X_i$ (i=2, … , n) has a set of possibly explanatory variables, $X_1, … , X_{i-1}$ in the form of a linear regression model. Such cause-effect relationships among the X variables can be represented in a directed acyclic graph (DAG) and can also be represented in a linear triangular system. Let G be a DAG of the n random variables. Then G can be represented in an adjacency matrix, which we will denote by A(G). The (i,j)-entry of the matrix equals 1 if there is an arrow from node i to node j, or i → j, in G. We will propose a method of finding the new model structure of a DAG, G, by using matrix operations, which is created by applying each of marginalization and conditionalization on the model of G. We will also explore properties of the matrix operations.