Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables

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 rth order"). The principle then leads to the null hypothesis of no "rth-order interaction. " Significance tests are given for testing the hypothesis of no rth-order or higher-order interaction within the wider hypothesis of no sth-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 rth-order restraints are given, then the hypothesis of the vanishing of all rth-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.