# Quantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information

## TL;DR

This paper presents a model-agnostic framework using multivariate mutual information extensions, especially the O-information, to quantify high-order interdependencies and synergy in complex systems, demonstrated through analysis of Baroque music scores.

## Contribution

It introduces the O-information metric for characterizing synergy and redundancy in high-order interactions, with analytical properties and applications to diverse fields.

## Key findings

- O-information effectively distinguishes synergy and redundancy.
- The framework reveals high-order dependencies in Baroque music.
- Analytical properties link O-information to existing metrics.

## Abstract

This article introduces a model-agnostic approach to study statistical synergy, a form of emergence in which patterns at large scales are not traceable from lower scales. Our framework leverages various multivariate extensions of Shannon's mutual information, and introduces the O-information as a metric capable of characterising synergy- and redundancy-dominated systems. We develop key analytical properties of the O-information, and study how it relates to other metrics of high-order interactions from the statistical mechanics and neuroscience literature. Finally, as a proof of concept, we use the proposed framework to explore the relevance of statistical synergy in Baroque music scores.