Dynamics of the Fisher Information Metric

TL;DR

This paper introduces a dynamical framework for the Fisher information metric by deriving partial differential equations from a variational principle, enabling systematic symmetry imposition in statistical systems.

Contribution

It proposes a novel variational method to generate Fisher metrics satisfying PDEs, advancing the dynamical understanding of information geometry.

Findings

Derivation of PDEs governing Fisher metrics from a variational principle

Method to impose symmetries systematically on statistical models

Application motivated by entropy-based nonmonotonic reasoning

Abstract

We present a method to generate probability distributions that correspond to metrics obeying partial differential equations generated by extremizing a functional $J[g^{\mu\nu}(\theta^i)]$, where $g^{\mu\nu}(\theta^i)$ is the Fisher metric. We postulate that this functional of the dynamical variable $g^{\mu\nu}(\theta^i)$ is stationary with respect to small variations of these variables. Our approach enables a dynamical approach to Fisher information metric. It allows to impose symmetries on a statistical system in a systematic way. This work is mainly motivated by the entropy approach to nonmonotonic reasoning.