An information geometric perspective on the complexity of macroscopic predictions arising from incomplete information

TL;DR

This paper explores how information geometry can quantify the complexity of macroscopic predictions in systems with incomplete microscopic information, using the Information Geometric Entropy as a key measure.

Contribution

It introduces the IGAC framework combining information geometry and inductive inference to analyze complex systems with limited data, highlighting the role of the IGE.

Findings

IGE effectively measures complexity of geodesic paths

IGAC provides probabilistic descriptions with partial information

Illustrative examples demonstrate the approach's applicability

Abstract

Motivated by the presence of deep connections among dynamical equations, experimental data, physical systems, and statistical modeling, we report on a series of findings uncovered by the Authors and collaborators during the last decade within the framework of the so-called Information Geometric Approach to Chaos (IGAC). The IGAC is a theoretical modeling scheme that combines methods of information geometry with inductive inference techniques to furnish probabilistic descriptions of complex systems in presence of limited information. In addition to relying on curvature and Jacobi field computations, a suitable indicator of complexity within the IGAC framework is given by the so-called Information Geometric Entropy (IGE). The IGE is an information geometric measure of complexity of geodesic paths on curved statistical manifolds underlying the entropic dynamics of systems specified in terms of probability distributions. In this manuscript, we discuss several illustrative examples wherein our modeling scheme is employed to infer macroscopic predictions when only partial knowledge of the microscopic nature of a given system is available. Finally, we include comments on the strengths and weaknesses of the current version of our proposed theoretical scheme in our concluding remarks.