Hierarchical Quantification of Synergy in Channels

TL;DR

This paper introduces a hierarchical information geometric approach to decompose mutual information in multi-input channels, enabling quantification of synergies of different orders through a novel projection method.

Contribution

It proposes a new hierarchical framework using information geometry to decompose mutual information into synergy components, providing a clear geometric interpretation and practical computation method.

Findings

Decomposition of mutual information into single, pairwise, and higher-order interactions.

Introduction of synergy measures based on exponential family projections.

Efficient iterative scaling algorithm for synergy evaluation.

Abstract

The decomposition of channel information into synergies of different order is an open, active problem in the theory of complex systems. Most approaches to the problem are based on information theory, and propose decompositions of mutual information between inputs and outputs in se\-veral ways, none of which is generally accepted yet. We propose a new point of view on the topic. We model a multi-input channel as a Markov kernel. We can project the channel onto a series of exponential families which form a hierarchical structure. This is carried out with tools from information geometry, in a way analogous to the projections of probability distributions introduced by Amari. A Pythagorean relation leads naturally to a decomposition of the mutual information between inputs and outputs into terms which represent single node information, pairwise interactions, and in general n-node interactions. The synergy measures introduced in this paper can be easily evaluated by an iterative scaling algorithm, which is a standard procedure in information geometry.