Annals of Statistics

Markov Fields and Log-Linear Interaction Models for Contingency Tables

J. N. Darroch, S. L. Lauritzen, and T. P. Speed

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We use a close connection between the theory of Markov fields and that of log-linear interaction models for contingency tables to define and investigate a new class of models for such tables, graphical models. These models are hierarchical models that can be represented by a simple, undirected graph on as many vertices as the dimension of the corresponding table. Further all these models can be given an interpretation in terms of conditional independence and the interpretation can be read directly off the graph in the form of a Markov property. The class of graphical models contains that of decomposable models and we give a simple criterion for decomposability of a given graphical model. To some extent we discuss estimation problems and give suggestions for further work.

Article information

Ann. Statist., Volume 8, Number 3 (1980), 522-539.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F99: None of the above, but in this section
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Contingency tables decomposability Gibbs states graphical models triangulated graphs


Darroch, J. N.; Lauritzen, S. L.; Speed, T. P. Markov Fields and Log-Linear Interaction Models for Contingency Tables. Ann. Statist. 8 (1980), no. 3, 522--539. doi:10.1214/aos/1176345006.

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