A link between the maximum entropy approach and the variational entropy form

TL;DR

This paper establishes a fundamental link between the maximum entropy principle and the variational entropy formulation, revealing a universal relation that clarifies previous ambiguities in the variational approach.

Contribution

It demonstrates that the variational formulation of entropic functionals can be derived directly from the maximum entropy principle, unifying two approaches.

Findings

Derived a universal relation between entropy rate and constraint functions.

Showed the variational formulation follows from the maximum entropy principle.

Resolved ambiguities in the variational approach to entropy.

Abstract

The maximum entropy approach operating with quite general entropy measure and constraint is considered. It is demonstrated that for a conditional or parametrized probability distribution $f(x|\mu)$ there is a "universal" relation among the entropy rate and the functions appearing in the constraint. It is shown that the recently proposed variational formulation of the entropic functional can be obtained as a consequence of this relation, that is from the maximum entropy principle. This resolves certain puzzling points appeared in the variational approach.