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₋ given the empirical observation (1/n) ^n₈=₋h (X₈) =, where X₁, X₂, are independent identically distributed random variables with common density g converges to f_ (x) =e^^{th (X) }g (x) (Suitably normalized), where is chosen to satisfy f₋₀₌₁₃₀ (x) h (x) dx=. 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.