  We prove that information-theoretic maximum entropy (MaxEnt) approach to
canonical ensemble is mathematically equivalent to the classic approach of
Boltzmann, Gibbs and Darwin-Fowler. The two approaches, however, "interpret" a
same mathematical theorem differently; most notably observing mean-energy in
the former and energy conservation in the latter. However, applying the same
MaxEnt method to grand canonical ensemble fails; while carefully following the
classic approach based on Boltzmann's microcanonical {\em equal probability a
priori} produces the correct statistics: One does not need to invoke quantum
mechanics; and there is no Gibbs paradox. MaxEnt and related minimum relative
entropy principle are based on the mathematical theorem concerning large
deviations of rare fluctuations. As a scientific method, it requires classic
mechanics, or some other assumptions, to provide meaningful {\em prior
distributions} for the expected-value based statistical inference. A naive
assumption of uniform prior is not valid in statistical mechanics.

Statistical Mechanics