Graphical Markov models provide a method of representing possibly complicated multivariate dependencies in such a way that the general qualitative features can be understood, that statistical independencies are highlighted, and that some properties can be derived directly. Variables are represented by the nodes of a graph. Pairs of nodes may be joined by an edge. Edges are directed if one variable is a response to the other variable considered as explanatory, but are undirected if the variables are on an equal footing. Absence of an edge typically implies statistical independence, conditional, or marginal depending on the kind of graph. The need for a number of types of graph arises because it is helpful to represent a number of different kinds of dependence structures. Of special importance are chain graphs in which variables are arranged in a sequence or chain of blocks, the variables in any one block being on an equal footing, some being possibly joint responses to variables in the past and some being jointly explanatory to variables in the future of the block considered. Some main properties of such systems are outlined, and recent research results are sketched. Suggestions for further reading are given. As an illustrative example, some analysis of data on the treatment of chronic pain is presented.
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