Confounding Ghost Channels and Causality: A New Approach to Causal Information Flows

TL;DR

This paper introduces a novel information-theoretic framework that better captures causal information flows by using sigma algebras aligned with channels, addressing limitations of traditional mutual information measures.

Contribution

It proposes a new version of mutual information based on coupled sigma algebras, enabling causal interpretation of information flows in channels.

Findings

Develops a causal chain rule for the new mutual information

Shows that traditional mutual information may not reflect causality

Provides a mathematical foundation for causal information flow analysis

Abstract

Information theory provides a fundamental framework for the quantification of information flows through channels, formally Markov kernels. However, quantities such as mutual information and conditional mutual information do not necessarily reflect the causal nature of such flows. We argue that this is often the result of conditioning based on sigma algebras that are not associated with the given channels. We propose a version of the (conditional) mutual information based on families of sigma algebras that are coupled with the underlying channel. This leads to filtrations which allow us to prove a corresponding causal chain rule as a basic requirement within the presented approach.