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
X_{l}
given the empirical observation
(1/n)\sum^{n}_{i}=_{l}h(X_{i})=\alpha
, where
X_{1},X_{2}, \cdots
are independent identically distributed random variables with common density
g
converges to
f_{\lambda}(x)=e^{\lambda^{t}h(X)}g(x)
(Suitably normalized), where
\lambda
is chosen to satisfy
\int f_{lambda}(x)h(x)dx= \alpha
. Thus the conditional distribution of a given random variable
X
is the (normalized) product of the maximum entropy distribution and the initial distribution. This distribution is the maximum entropy distribution when
g
is uniform. The proof of this and related results relies heavily on the work of Zabell and Lanford.