The Annals of Mathematical Statistics

Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables

I. J. Good

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Abstract

The principle of maximum entropy, together with some generalizations, is interpreted as a heuristic principle for the generation of null hypotheses. The main application is to $m$-dimensional population contingency tables, with the marginal totals given down to dimension $m - r$ ("restraints of the $r$th order"). The principle then leads to the null hypothesis of no "$r$th-order interaction." Significance tests are given for testing the hypothesis of no $r$th-order or higher-order interaction within the wider hypothesis of no $s$th-order or higher-order interaction, some cases of which have been treated by Bartlett and by Roy and Kastenbaum. It is shown that, if a complete set of $r$th-order restraints are given, then the hypothesis of the vanishing of all $r$th-order and higher-order interactions leads to a unique set of cell probabilities, if the restraints are consistent, but not only just consistent. This confirms and generalizes a recent conjecture due to Darroch. A kind of duality between maximum entropy and maximum likelihood is proved. Some relationships between maximum entropy, interactions, and Markov chains are proved.

Article information

Source
Ann. Math. Statist., Volume 34, Number 3 (1963), 911-934.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177704014

Digital Object Identifier
doi:10.1214/aoms/1177704014

Mathematical Reviews number (MathSciNet)
MR150880

Zentralblatt MATH identifier
0143.40705

JSTOR
links.jstor.org

Citation

Good, I. J. Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables. Ann. Math. Statist. 34 (1963), no. 3, 911--934. doi:10.1214/aoms/1177704014. https://projecteuclid.org/euclid.aoms/1177704014


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