The Annals of Statistics

Logical and Algorithmic Properties of Conditional Independence and Graphical Models

Dan Geiger and Judea Pearl

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This article develops an axiomatic basis for the relationship between conditional independence and graphical models in statistical analysis. In particular, the following relationships are established: (1) every axiom for conditional independence is an axiom for graph separation, (2) every graph represents a consistent set of independence and dependence constraints, (3) all binary factorizations of strictly positive probability models can be encoded and determined in polynomial time using their correspondence to graph separation, (4) binary factorizations of non-strictly positive probability models can also be derived in polynomial time albeit less efficiently and (5) unconditional independence relative to normal models can be axiomatized with a finite set of axioms.

Article information

Ann. Statist., Volume 21, Number 4 (1993), 2001-2021.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60A05: Axioms; other general questions
Secondary: 60J99: None of the above, but in this section 60G60: Random fields 62A15 62H25: Factor analysis and principal components; correspondence analysis

Conditional independence Markov fields Markov networks graphical models


Geiger, Dan; Pearl, Judea. Logical and Algorithmic Properties of Conditional Independence and Graphical Models. Ann. Statist. 21 (1993), no. 4, 2001--2021. doi:10.1214/aos/1176349407.

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