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October 1996 Markov properties of nonrecursive causal models
J. T. A. Koster
Ann. Statist. 24(5): 2148-2177 (October 1996). DOI: 10.1214/aos/1069362315


This paper aims to solve an often noted incompatibility between graphical chain models which elucidate the conditional independence structure of a set of random variables and simultaneous equations systems which focus on direct linear interactions and correlations between random variables. Various authors have argued that the incompatibility arises mainly from the fact that in a simultaneous equations system (e.g., a LISREL model) reciprocal causality is possible whereas this is not so in the case of graphical chain models. In this article it is shown that this view is not correct. In fact, the definition of the Markov property embodied in a graph can be generalized to a wider class of graphs which includes certain nonrecursive graphs. The resulting class of reciprocal graph probability models strictly includes the class of chain graph probability models. The class of lattice conditional independence probability models is also strictly included. It is shown that the resulting methodology is directly applicable to quite general simultaneous equations systems that are subject to mild restrictions only. Provided some adjustments are made, general simultaneous equations systems can be handled as well. In all cases, consistency with the LISREL methodology is maintained.


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J. T. A. Koster. "Markov properties of nonrecursive causal models." Ann. Statist. 24 (5) 2148 - 2177, October 1996.


Published: October 1996
First available in Project Euclid: 20 November 2003

zbMATH: 0867.62056
MathSciNet: MR1421166
Digital Object Identifier: 10.1214/aos/1069362315

Primary: 62H99
Secondary: 62J99

Keywords: chain graph , Conditional independence , finite distributive lattice , Gibbs factorization , global Markov property , Graphical model , lattice conditional independence model , LISREL model , Nonrecursive causal model , reciprocal graph , simultaneous equations system , undirected graph

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 5 • October 1996
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