  Bayes' theorem incorporates distinct types of information through the
likelihood and prior. Direct observations of state variables enter the
likelihood and modify posterior probabilities through consistent updating.
Information in terms of expected values of state variables modify posterior
probabilities by constraining prior probabilities to be consistent with the
information. Constraints on the prior can be exact, limiting hypothetical
frequency distributions to only those that satisfy the constraints, or be
approximate, allowing residual deviations from the exact constraint to some
degree of tolerance. When the model parameters and constraint tolerances are
known, posterior probability follows directly from Bayes' theorem. When
parameters and tolerances are unknown a prior for them must be specified. When
the system is close to statistical equilibrium the computation of posterior
probabilities is simplified due to the concentration of the prior on the
maximum entropy hypothesis. The relationship between maximum entropy reasoning
and Bayes' theorem from this point of view is that maximum entropy reasoning is
a special case of Bayesian inference with a constrained entropy-favoring prior.
