Maximum entropy and conditional probability

It is well-known that maximum entropy distributions, subject to appropriate moment constraints, arise in physics and mathematics. In an attempt to find a physical reason for the appearance of maximum entropy distributions, the following theorem is offered. The conditional distribution of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Xₗ</tex> given the empirical observation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1/n)∑ⁿᵢ=ₗh(Xᵢ)=α</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X₁,X₂, ⋯</tex> are independent identically distributed random variables with common density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</tex> converges to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fλ(x)=e^{λᵗh(X)}g(x)</tex> (Suitably normalized), where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</tex> is chosen to satisfy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∫ flambda(x)h(x)dx= α</tex> . Thus the conditional distribution of a given random variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> is the (normalized) product of the maximum entropy distribution and the initial distribution. This distribution is the maximum entropy distribution when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</tex> is uniform. The proof of this and related results relies heavily on the work of Zabell and Lanford.