Learning from Complex Systems: On the Roles of Entropy and Fisher Information in Pairwise Isotropic Gaussian Markov Random Fields

TL;DR

This paper explores the relationship between entropy, Fisher information, and complex system behavior using Gaussian Markov random fields, introducing Fisher curves to analyze information dynamics and asymmetries in state transitions.

Contribution

It introduces analytical tools based on Fisher information and entropy to characterize complex systems via Fisher curves, revealing directional asymmetries in information evolution.

Findings

Fisher curves provide a geometric representation of system behavior.

Moving towards higher entropy states differs from moving towards lower entropy states.

The tools effectively extract relevant information from complex patterns.

Abstract

Markov Random Field models are powerful tools for the study of complex systems. However, little is known about how the interactions between the elements of such systems are encoded, especially from an information-theoretic perspective. In this paper, our goal is to enlight the connection between Fisher information, Shannon entropy, information geometry and the behavior of complex systems modeled by isotropic pairwise Gaussian Markov random fields. We propose analytical expressions to compute local and global versions of these measures using Besag's pseudo-likelihood function, characterizing the system's behavior through its \emph{Fisher curve}, a parametric trajectory accross the information space that provides a geometric representation for the study of complex systems. Computational experiments show how the proposed tools can be useful in extrating relevant information from complex patterns. The obtained results quantify and support our main conclusion, which is: in terms of information, moving towards higher entropy states (A --> B) is different from moving towards lower entropy states (B --> A), since the \emph{Fisher curves} are not the same given a natural orientation (the direction of time).