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September, 1963 Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables
I. J. Good
Ann. Math. Statist. 34(3): 911-934 (September, 1963). DOI: 10.1214/aoms/1177704014


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.


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I. J. Good. "Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables." Ann. Math. Statist. 34 (3) 911 - 934, September, 1963.


Published: September, 1963
First available in Project Euclid: 27 April 2007

zbMATH: 0143.40705
MathSciNet: MR150880
Digital Object Identifier: 10.1214/aoms/1177704014

Rights: Copyright © 1963 Institute of Mathematical Statistics

Vol.34 • No. 3 • September, 1963
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