The Annals of Statistics

Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models

A. P. Dawid and S. L. Lauritzen

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Abstract

This paper introduces and investigates the notion of a hyper Markov law, which is a probability distribution over the set of probability measures on a multivariate space that (i) is concentrated on the set of Markov probabilities over some decomposable graph, and (ii) satisfies certain conditional independence restrictions related to that graph. A stronger version of this hyper Markov property is also studied. Our analysis starts by reconsidering the properties of Markov probabilities, using an abstract approach which thereafter proves equally applicable to the hyper Markov case. Next, it is shown constructively that hyper Markov laws exist, that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models, and also that they form natural conjugate prior distributions for a Bayesian analysis of these models. As examples we construct a range of specific hyper Markov laws, including the hyper multinomial, hyper Dirichlet and the hyper Wishart and inverse Wishart laws. These laws occur naturally in connection with the analysis of decomposable log-linear and covariance selection models.

Article information

Source
Ann. Statist. Volume 21, Number 3 (1993), 1272-1317.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176349260

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176349260

Mathematical Reviews number (MathSciNet)
MR1241267

Zentralblatt MATH identifier
0815.62038

Subjects
Primary: 62H99: None of the above, but in this section
Secondary: 62E15: Exact distribution theory

Keywords
Bayesian statistics covariance selection collapsibility contingency tables cut decomposable graphs Dirichlet distribution expert systems graphical models hyper Dirichlet law hyper inverse Wishart law hyper matrix $F$ Law hyper matrix $t$ law hyper Normal law hyper Multinomial law hyper Wishart law inverse Wishart distribution $\log$-linear models matrix $F$ distribution matrix $t$ distribution multivariate analysis triangulated graphs Wishart distribution

Citation

Dawid, A. P.; Lauritzen, S. L. Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models. The Annals of Statistics 21 (1993), no. 3, 1272--1317. doi:10.1214/aos/1176349260. http://projecteuclid.org/euclid.aos/1176349260.


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See also

    Corrections

    • See Correction: A. P. Dawid, S. L. Lauritzen. Correction: Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models. Ann. Statist., vol. 23, no. 5 (1995), 1864.