Directed Graphical Models

The application of conditional independence models associated with variants of directed graphs has gained prominence in data analysis, in particular in machine learning and artificial intelligence, but also in more traditional statistical approaches. This chapter describes the most important concepts and some of the results of this field. Of course, only a relatively small part of the existing knowledge can be presented here. The concepts of directed pairwise, local, and global Markov properties are introduced and their relationships, including their equivalence under the assumption of (strict) positivity, are discussed. Related factorization results are proved. Then, the notion of dependence separation is introduced and its equivalence with the separation condition of the directed global Markov property is shown. Markov models associated with DAGs are also called Bayesian networks, and one of the algorithms that is used for what in that context is often called belief propagation is described. Finally, the various Markov interpretations associated with chain graphs are discussed. Further relevant results for the models presented here will be given in Chap. 10 , in the generality of marginal models.