Characterization of the Maximum Probability Fixed Marginals R×C Contingency Tables

Abstract

In this paper operators i[j] and [j]k are defined, whose effects on an r×c contingency table X are to subtract 1 from xij and to add 1 to xkj , respectively, so that the composition i[j]k of the two operators changes the j-th column of the contingency table without altering its total. Also a loop is defined as a composition of such operators that leaves unchanged both row and column totals. This is used to characterize the r×c contingency tables of maximum probability over the fixed marginals reference set (under the hypothesis of row and column independence). Another characterization of such maximum probability tables is given using the concept of associated U tables, a U = {uij} table being defined as a table such that uij > 0, 1 ≤ i ≤ r and 1 ≤ j ≤ c, and for a given set of values rh, 1 ≤ h < r, uh+1,j = rhuhj for all j. Finally, a necessary and sufficient condition for the uniqueness of a maximum probability table in the fixed marginals reference set is provided.