  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 contains
regression graphs. Regression graphs are a type of so-called chain graph and
describe 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 result after possible marginalising or conditioning, we
introduce 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 that 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
paths in graphs.
