Title: Geometry of Conditional Independence
Description: This thesis investigates geometric aspects of the notions of conditional independence and conditional probability. In Chapter 2, the connection between conditional independence models and polyhedral fans is developed. The main result uses algebraic techniques and the permutohedron, a polytope that plays an important role in the geometry of conditional independence. The results are applied to define a class of rank tests useful for exploratory data analysis. In Chapter 3, this class of rank tests, called topographical models, for use in analyzing microarray data have been developed. The necessary algorithms and counting theorems required to make this test practical have been applied to two data sets. In Chapter 4, the machinery of Chapter 2 is used to settle three open theoretical questions about conditional independence models. with exploring a more algebraic perspective on semigraphoids. In Chapter 5, a question raised by the work of Besag on the relations among conditional probabilities is answered that accomplished via tonic geometry, moment map, the space of conditional probability distributions to generalized permutohedra etc.
Subjects: Probabilities., Conditional probabilities, Random variables, Polyhedral functions, Abstract algebra, Topology, moment map, Discrete geometry, Mathematical statistics., Mathematics-Combinatorics, Graphical modeling (Statistics)