  We show that the naive application of the maximum entropy principle can yield
answers which depend on the level of description, i.e. the result is not
invariant under coarse-graining. We demonstrate that the correct approach, even
for discrete systems, requires maximization of the relative entropy with a
suitable reference probability, which in some instances can be deduced from the
symmetry properties of the dynamics. We present simple illustrations of this
crucial yet surprising feature in examples of classical and quantum statistical
mechanics, as well as in the field of ecology.
