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August 2011 Probability distributions with summary graph structure
Nanny Wermuth
Bernoulli 17(3): 845-879 (August 2011). DOI: 10.3150/10-BEJ309


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.


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Nanny Wermuth. "Probability distributions with summary graph structure." Bernoulli 17 (3) 845 - 879, August 2011.


Published: August 2011
First available in Project Euclid: 7 July 2011

zbMATH: 1245.62062
MathSciNet: MR2817608
Digital Object Identifier: 10.3150/10-BEJ309

Keywords: concentration graph , Directed acyclic graph , endogenous variables , graphical Markov model , independence graph , multivariate regression chain , partial closure , partial inversion , triangular system

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability


Vol.17 • No. 3 • August 2011
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