Contingency tables with prescribed marginals

For an adjustment of contingency tables to prescribed marginal frequencies Deming and Stephan (1940) minimize a Chi-square expression. Asymptotically equivalently, Ireland and Kullback (1968) minimize a Leibler-Kullback divergence, where the probabilistical arguments for both methods remain vague. Here we deduce a probabilistical model based on observed contingency tables. It shows that the two above methods and the maximum likelihood approach in Smith (1947) yield asymptotically the ‘most probable’ adjustment under prescribed marginal frequencies. The fundamental hypothesis of statistical mechanics relates observations to ‘most probable’ realizations. ‘Most probable’ is going to be used in the sense of so-called large deviations. The proposed adjustment has a significant product form and will be generalized to contingency tables with infinitely many cells.