Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear Inverse Problems
The Annals of Statistics · 1991 · Volume 19 · Issue 4
Abstract
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, 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.
Keywords
62A99; 68T01; 92C55; 94A17; Image reconstruction; linear constraitns; logically consistent inference; minimum discrimination information; nonlinear projection; nonsymmetric distance; Selection rules;
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