Summary: An attempt is made to determine the logically consistent rules for selecting a vector from any feasible set defined by linear constraints, when either all -vectors or those with positive components or the probability vectors are permissible. Some basic postulates are satisfied if and only if the selection rule is to minimize a certain function which, if a “ prior guess” is available, is a measure of distance from the prior guess. Two further natural postulates restrict the permissible distances to the author’s -divergences and Bregman’s divergences [L. M. Bregman, USSR Comput. Math. Math. Phys. 7, No. 3, 200-217 (1969); translation from Zh. Vychisl. Mat. Mat. Fiz. 7, 620-631 (1967; Zbl 0186.238)], respectively.
As corollaries, axiomatic characterizations of the methods of least squares and minimum discrimination information are arrived at. Alternatively, the latter are also characterized by a postulate of composition consistency. As a special case, a derivation of the method of maximum entropy from a small set of natural axioms is obtained.