Elements for a New Information Theory Based on Fractional Calculus via Modified Riemann-Liouville Derivative

摘要

By generalizing the basic functional equation f(xy)=f(x) +f(y) in the form f~β (xy) ≥ f~β (x) + f~β (y), β > 1, one can derive a family of solutions which are exactly the inverse of the Mittag-Leffler function, referred to as Mittag-Leffler logarithm, or logarithm of fractional order. This result is used to define a new family of generalized informational entropies which are indexed by a parameter clearly related to fractals, via fractional calculus, and which is quite relevant in the presence of creation of uncertainty, via defects of the observation process. The relation with Shannon's entropy, Renyi's entropy and Tsallis' entropy is clarified, and it is shown that Tsallis' generalized logarithm has a direct significance in terms of fractional calculus. When β = 2, one comes across the probability amplitude useful in quantum mechanics, and one so arrives at an approach to the definition of "amplitude of informational entropy". One examines the kind of result that one can so expect to obtain when one uses Jaynes' maximum entropy principle in this framework. In the presence of uncertain definition (or fuzzy definition!) the Mittag-Leffler density function would be more relevant than the. Gaussian density. To some extent, this new formulation could be fully supported by the derivation of a new family of fractional Fisher information.