Hellmann-Feynman connection for the relative Fisher information

TL;DR

This paper develops a thermodynamics-inspired mathematical framework for the relative Fisher information (RFI) using Hellmann-Feynman relations, deriving new reciprocity relations, a Legendre transform structure, and inferring probability densities and energies.

Contribution

It introduces a novel thermodynamics-like Legendre transform structure for RFI and derives new reciprocity relations from Hellmann-Feynman theorem, enabling inference of probability densities and energies.

Findings

Derived new reciprocity relations for RFI.

Established a Legendre transform structure for RFI.

Benchmark results for inferred energies match established data.

Abstract

The $(i)$ reciprocity relations for the relative Fisher information (RFI, hereafter) and $(ii)$ a generalized RFI-Euler theorem, are self-consistently derived from the Hellmann-Feynman theorem. These new reciprocity relations generalize the RFI-Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schr\"odinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI-LTS link and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf's, hereafter) and energy-eigenvalues of the above mentioned Schr\"odinger-like equation. The energy-eigenvalues obtained here via inference are benchmarked against established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided.