Legendre transform structure and extremal properties of the relative Fisher information

Abstract

Variational extremization of the relative Fisher information (RFI, hereafter) is performed. Reciprocity relations, akin to those of thermodynamics are derived, employing the extremal results of mthe RFI expressed in terms of probability amplitudes. A time independent Schrodinger-like equation (Schrodinger like link) for the RFI is derived. The concomitant Legendre transform structure (LTS hereafter) is developed by utilizing a generalized RFI-Euler theorem, which shows that the entire mathematical structure of htermodynamics translates into the RFI framework, both for equilibrium and non equilibrium cases. The qualitatevily distinct nature of the present results visd-a-vis those of prio studies utilizing the Shannon Entropy and/or the Fisher information mmeasure is discussed. A principled relationship between the RFI and the FIM ferameworks is derived. The utility of this relationship is demosnstrated by an example wherein the energy eigenvalues of the Schroedinger-like link for the RFI are inferred solely using the quantum mechanical virial theorem and the LTS of the RFI.