Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components (2501.11447v1)

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, and chemical reaction networks. Our results reveal how the distribution of causal power can be context- and parameter-dependent. This 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.

• The paper introduces a novel framework decomposing interventional causal effects into synergistic, redundant, and unique components using Möbius inversion.
• It employs a robust mathematical formalism based on Partial Information Decomposition to quantify causal power distribution in systems like logic gates and cellular automata.
• The framework is applied to domains such as chemical networks, yielding actionable insights into parameter-dependent causal interactions and intervention strategies.

Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components

The paper presents a novel framework for decomposing interventional causal effects into three fundamental components: synergistic, redundant, and unique. This method builds on the principles of Partial Information Decomposition (PID) and utilizes M\"obius inversion to delineate the intricacies of causal interactions within complex systems. The authors argue that a proper causal decomposition should pivot around interventional, rather than merely observational, measures as advocated by Pearl's definition of causal quantities.

Mathematical Foundation

The cornerstone of this framework is a comprehensive mathematical formalism that leverages the M\"obius function of the redundancy lattice. The decomposition is applied to interventional causal effects, providing a structured approach to quantify how causal power is distributed among variables in a system. This is a notable advancement from previous efforts, which have primarily focused on observational measures. The paper meticulously outlines the mathematical prerequisites, explaining relevant concepts like partially ordered sets and chains/antichains, leading up to the application of the M\"obius inversion theorem.

The framework's mathematical robustness is further demonstrated by applying these decompositional methods across several domains, such as logic gates, cellular automata, and chemical reaction networks. These examples illustrate the context- and parameter-dependent nature of causal power distribution, revealing nuanced insights into how causal influences are shared and combined among multiple variables.

Examples and Application

1. Logic Gates: The paper examines how causal power operates within logical structures, demonstrating that even in seemingly simple systems, there can be complex synergistic interactions. The decomposition shows varied distributions of causal power, dependent on input probabilities.
2. Cellular Automata: The paper of 1D cellular automata illustrates the context-dependent nature of this decomposition. By varying initial states, the authors reveal how synergistic and redundant causal structures emerge in dynamic systems, showcasing the applicability of their framework in a temporal setting.
3. Chemical Networks: A model chemical network is explored to show how parameter changes, specifically reaction rates, can influence the decomposition of causal power. This case underscores the practical utility of the framework in systems biology, offering insights into regulatory patterns and potential sites for intervention within biochemical pathways.

Implications and Future Directions

The implications of this work are multifaceted. Practically, the framework can be applied in various scientific domains, from biology to artificial intelligence, to better attribute causal responsibility. Theoretically, it invites further exploration of causal structures, particularly in systems with feedback loops and complex interactions.

The authors also speculate that this formalism can be adapted to other definitions of causal power beyond the average treatment effects (ATE) considered. Such adaptations could broaden the framework's versatility, potentially integrating alternative measures like those based on the Kullback-Leibler divergence.

In essence, the paper sets the stage for further research into the decomposition of causal effects, promising deeper insights into the causal mechanisms governing complex systems. By focusing on interventional causality, this work aligns with a more actionable understanding of causal inference, ultimately aiming to enhance the precision and efficacy of interventions in multi-variable systems. This decomposition thus holds promise for advancing our understanding of the causal relationships in both natural and engineered systems.