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 the RFI expressed in terms of probability amplitudes. A time independent Schrödinger-like equation (Schrödinger-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 thermodynamics translates into the RFI framework, both for equilibrium and non-equilibrium cases. The qualitatively distinct nature of the present results vis-á-vis those of prior studies utilizing the Shannon entropy and/or the Fisher information measure (FIM, hereafter) is discussed. A principled relationship between the RFI and the FIM frameworks is derived. The utility of this relationship is demonstrated by an example wherein the energy eigenvalues of the Schrödinger-like link for the RFI is inferred solely using the quantum mechanical virial theorem and the LTS of the RFI.

Hellmann–Feynman connection for the relative Fisher information

“…(as is the case in [1]). The pertinent relationships from [1] remain unaffected, and are re-stated as…”

“…The relative Fisher information (RFI, hereafter) is a measure of uncertainty that is the focus of much attention in statistical physics, estimation theory, and allied disciplines (see [1]). The RFI is defined by [2,3] …”

“…Setting f (x) = ψ 2 (x) and performing a variational extremization of (3) with respect to ψ(x), subject to constraints, yields [1] …”

Fisher Information Properties

“…Here, it is important to point out the extreme physical information principle derived by Frieden and others in order to establish a general framework that explains physics [7,[17][18][19][20]. Of special interest has been the role of Fisher information to generate thermodynamical theory [7,[17][18][19][20][21][22]. It is very common in these approaches to use a special case of Fisher information where the estimated parameter is a location parameter.…”

“…As far as it was possible to determine, the first definition of the relative Fisher information was given by Otto and Villani [36], who defined it for the translationally-invariant case. Furthermore, this expression has been rediscovered or simply used in many applications thereafter in different problems and fields [22,[37][38][39][40][41][42][43][44]. Furthermore, it seems that the first general analysis of the relative Fisher information was presented by the author in [45].…”