Ambiguities in the forms of the entropic functional and constraints in the maximum entropy formalism

Many important probability distributions in physics, biology, and other fields can be obtained by the constrained maximization of appropriate information-entropic functionals. The associated maximum entropy formalisms and their applications have been the focus of intense research efforts in recent years. It may seem that this (generalized) maximum entropy approach suffers from a basic ambiguity, in the sense that any probability distribution seems to be derivable from the maximization of any entropic measure if an appropriate constraint is used. Here we argue that, in general, the aforementioned ambiguity disappears when maximum entropy representations of mono-parametric families of probability distributions are considered, as contrasted to maximum entropy representations of just a single, isolated instance of a probability distribution.