Graphical Markov Models

Graphical Markov models are multivariate statistical models which are currently under vigorous development and which combine two simple but most powerful notions, generating processes in single and joint response variables and conditional independences captured by graphs. The development of graphical Markov started with work by Wermuth (1976, 1980) and Darroch, Lauritzen and Speed (1980) which built on early results in 1920 to 1930 by geneticist Sewall Wright and probabilist Andrej Markov as well as on results for log-linear models by Birch (1963), Goodman (1970), Bishop, Fienberg and Holland (1973) and for covariance selection by Dempster (1972). Wright used graphs, in which nodes represent variables and arrows indicate linear dependence, to describe hypotheses about stepwise processes in single responses that could have generated his data. He developed a method, called path analysis, to estimate linear dependences and to judge whether the hypotheses are well compatible with his data which he summarized in terms of simple and partial correlations. With this approach he was far ahead of his time, since corresponding formal statistical methods for estimation and tests of goodness of fit were developed much later and graphs that capture independences even much later than tests of goodness of fit. It remains a primary objective of graphical Markov models to uncover graphical representations that lead to an understanding of data generating processes. Such processes are no longer restricted to linear relations but contain linear dependences as special cases. A probabilistic data generating process is a recursive sequence of conditional distributions in which response variables may be vector variables that contain discrete or continuous components. Thereby, each conditional distribution specifies both the dependence of response Ya, say, on an explanatory variable vector Yb and the undirected associations of the components of Ya. Graphical Markov models also generalize sequences in single responses and single explanatory variables that have been named Markov chains, after probabilist Markov. He recognized at the beginning of the 29th century that seemingly complex joint probability distributions may be radically simplified by using the notion of conditional independence and defined what are now called Markov chains. In a Markov chain of random variables Y1, . . . , Yi, . . . , Yd, the joint distribution is built up by starting with the density of fd of Ydand generating next fd−1|d. Then, conditional independence of Yd−2 from Yd given Yd−1 is taken into account with fd−2|d−1,d = fd−2|d−1. One continues such that, with fi|i+1,...d = fi|i+1, response Yi is conditionally independent of Yi+2, . . . , Yd given Yi+1, written compactly