The Annals of Statistics

Markov properties of nonrecursive causal models

J. T. A. Koster

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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.

Article information

Ann. Statist., Volume 24, Number 5 (1996), 2148-2177.

First available in Project Euclid: 20 November 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section
Secondary: 62J99: None of the above, but in this section

Nonrecursive causal model simultaneous equations system LISREL model graphical model undirected graph chain graph reciprocal graph conditional independence global Markov property Gibbs factorization finite distributive lattice lattice conditional independence model


Koster, J. T. A. Markov properties of nonrecursive causal models. Ann. Statist. 24 (1996), no. 5, 2148--2177. doi:10.1214/aos/1069362315.

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