Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components

TL;DR

This paper presents a new method for decomposing interventional causal effects into synergistic, redundant, and unique components, offering deeper insights into how causal influence is distributed among variables in complex systems.

Contribution

It introduces a formal interventional causal decomposition framework based on Partial Information Decomposition and M"obius inversion, extending prior observational decompositions to causal settings.

Findings

Decomposition applied to logic gates, cellular automata, and chemical networks.

Reveals context- and parameter-dependent distribution of causal power.

Provides insights into shared and combined causal influences.

Abstract

We introduce a novel framework for decomposing interventional causal effects into synergistic, redundant, and unique components, building on the intuition of Partial Information Decomposition (PID) and the principle of M\"obius inversion. While recent work has explored a similar decomposition of an observational measure, we argue that a proper causal decomposition must be interventional in nature. We develop a mathematical approach that systematically quantifies how causal power is distributed among variables in a system, using a recently derived closed-form expression for the M\"obius function of the redundancy lattice. The formalism is then illustrated by decomposing the causal power in logic gates, cellular automata, chemical reaction networks, and a transformer language model. Our results reveal how the distribution of causal power can be context- and parameter-dependent. The decomposition provides new insights into complex systems by revealing how causal influences are shared and combined among multiple variables, with potential applications ranging from attribution of responsibility in legal or AI systems, to the analysis of biological networks or climate models.