One of the most central ideas of statistical science is the assessment of dependencies among a set of stochastic variables. The familiar concepts of correlation, regression, and prediction are manifestations of this idea, and many aspects of causal relationships rest on representations of multivariate dependence.

Graphical Markov Models (GMM) use graphs, either directed, undirected, or mixed, to represent multivariate dependencies in an economical and computationally efficient way. A GMM is constructed by specifying local dependencies for each variable = node of the graph in terms of its immediate neighbours, parents, or both, yet represents a complex system of dependencies by means of the global structure of the graph. The local specification permits efficiencies in modeling, inference, and probabilistic calculations.

GMMs based on undirected graphs (= Markov random fields) are used to represent spatial dependencies in such applications as statistical mechanics and image analysis, while GMMs based on directed graphs (= path diagrams) occur as structural equation models (SEM) in psychometrics, econometrics, and similar fields. In statistics, the use of GMMs for both continuous and categorical data accelerated in the late 1970s, beginning with work by Darroch, Lauritzen, Speed, Wermuth and others on graphical log-linear models and recursive SEMs, then continued in work by Dawid, Spiegelhalter, Frydenberg, Cox and others with applications in medical diagnosis, epidemiology, etc. At the same time, separate but convergent developments of these ideas occurred in computer science, decision analysis, management science, and philosophy, where GMMs have been called influence diagrams or Bayesian belief networks and are used for the construction of expert systems, neural networks, and causal models. The application of GMMs to expert systems has proved hugely successful - the vibrant Uncertainty in Artificial Intelligence community currently focuses much of its effort on GMM methodology.

Prerequisites: Students are assumed to be acquainted with the basics in the following areas and subjects within mathematics and statistics: linear, algebra, group and group action, likelihood inference (estimation and test), probability theory, univariate distributions, conditional distributions, the multivariate normal distribution, multivariate analysis of variance (MANOVA), and contingency tables.