  The method of maximum entropy (ME) is extended to address the following
problem: Once one accepts that the ME distribution is to be preferred over all
others, the question is to what extent are distributions with lower entropy
supposed to be ruled out. Two applications are given. The first is to the
theory of thermodynamic fluctuations. The formulation is exact, covariant under
changes of coordinates, and allows fluctuations of both the extensive and the
conjugate intensive variables. The second application is to the construction of
an objective prior for Bayesian inference. The prior obtained by following the
ME method to its inevitable conclusion turns out to be a special case of what
are currently known under the name of entropic priors.
