Graphical Models

The idea of modelling systems using graph theory has its origin in several scientific areas: in statistical physics (the study of large particle systems), in genetics (studying inheritable properties of natural species), and in interactions in contingency tables. This new and extended edition of Graphical Models provides the basic mathematical and statistical theory of graphical models, incorporating the many advances that have been made in the field since the publication of the first edition in 1996. Lauritzen discusses basic graph theory and the fundamentals of conditional independence both in abstract form for conditional independence based on graphs and for probabilistic conditional independence. The associated Markov theory, forming the basis of all models in the book, 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 new and important chapter is devoted to structure estimation because this has become a dominating part of modern developments. Causal interpretation of models based on directed acyclic graphs and chain graphs are also discussed. The appendices contain 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 distributions and derived distributions, as well as a brief exposition of the theory of exponential families.