## The Annals of Statistics

### Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear Inverse Problems

Imre Csiszar

#### 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.

#### Article information

Source
Ann. Statist., Volume 19, Number 4 (1991), 2032-2066.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348385

Digital Object Identifier
doi:10.1214/aos/1176348385

Mathematical Reviews number (MathSciNet)
MR1135163

Zentralblatt MATH identifier
0753.62003

JSTOR