1 6 M ar 2 01 0 PROBABILITY DISTRIBUTIONS WITH SUMMARY GRAPH STRUCTURE

A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class are multivariate regression chain graphs. They describe the independences of stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that then result after possible marginalising or conditioning, we use summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences which remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to derive these properties of summary graphs and to translate them into special types of path in graphs. 1. Motivation, some previous and some of the new results. 1.1. Motivation. Graphical Markov models are probability distributions defined for a d V × 1 random vector variable Y V whose component variables may be discrete or continuous and whose joint density f V satisfies the independence statements specified directly as well as those implied by an associated graph. The set of all such statements is the independence structure captured by the graph. One such type of graph had been introduced for multivariate regression A multivariate regression chain graph consists of nodes in set, V , that represent random variables and of edges that couple node pairs such that a re-cursive order of the joint responses is reflected in the graph and each defining independence constraint respects the given ordering; see Marchetti and Lupparelli (2010). This distinguishes multivariate regression graphs from all other currently known types of chain graphs; see Drton (2009) for implications on associated discrete distributions. Because of this property, multivariate regression chain graphs are particularly well suited for studies of effects of hypothesized causes on joint responses, see Cox and Wermuth (2004), and more generally for modeling developmental processes, such as in panel studies. These provide data on a group of individuals, termed the 'panel', collected repeatedly, say over years or decades. Often one wants to compare corresponding analyses with results in other studies that have core sets of variables in common, …

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