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Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. (English) Zbl 0753.62003
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 $n$-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 $f$-divergences and Bregman’s divergences [{\it 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.

62B10Statistical information theory
62A01Foundations and philosophical topics in statistics
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