Graphical Models

Abstract This book provides the basic mathematical and statistical theory of graphical models, incorporating some of the many advances that have been made since the appearance of the first edition. It contains the basic graph theory, the fundamentals of conditional independence both for probabilistic conditional independence and in abstract form, as well as conditional independence based on graph separation. The associated Markov theory is treated in some detail. The statistical theory based on likelihood methods and conjugate Bayesian analysis is developed for log-linear and Gaussian graphical models, as well as for graphical models involving mixed discrete and continuous data. A separate chapter is devoted to structure estimation as this has become a dominating part of modern developments. There are appendices collecting some of the general mathematical results needed as background for the main contents of the book, including basic measure theory and the theory of Markov kernels, convex optimization, properties of the multivariate Gaussian and derived distributions, as well as a brief exposition of the theory of exponential families.