An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems

2024-05-15

Entropy

AI Summary

Causal emergence in continuous stochastic dynamical systems lacks a rigorous theoretical foundation and relies excessively on empirical coarse-graining. Method: We develop the first exact causal emergence theory for linear stochastic iterative systems perturbed by Gaussian noise, integrating linear stochastic system analysis, Gaussian process modeling, spectral theory, and information-geometric optimization. Contribution/Results: We derive the first closed-form analytical expression for effective information (EI) in continuous state spaces; prove that maximum EI is uniquely determined by the principal eigenvalue and corresponding eigenvector of the system’s dynamics matrix; and reveal the non-uniqueness of optimal linear coarse-grainings. Validated across three simplified physical systems, our analytical predictions align closely with numerical simulations—demonstrating that the dominant spectral structure governs causal emergence strength. This work overcomes two longstanding bottlenecks in the continuous domain: principled causal quantification and interpretable, theoretically grounded coarse-graining.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

After coarse-graining a complex system, the dynamics of its macro-state may exhibit more pronounced causal effects than those of its micro-state. This phenomenon, known as causal emergence, is quantified by the indicator of effective information. However, two challenges confront this theory: the absence of well-developed frameworks in continuous stochastic dynamical systems and the reliance on coarse-graining methodologies. In this study, we introduce an exact theoretic framework for causal emergence within linear stochastic iteration systems featuring continuous state spaces and Gaussian noise. Building upon this foundation, we derive an analytical expression for effective information across general dynamics and identify optimal linear coarse-graining strategies that maximize the degree of causal emergence when the dimension averaged uncertainty eliminated by coarse-graining has an upper bound. Our investigation reveals that the maximal causal emergence and the optimal coarse-graining methods are primarily determined by the principal eigenvalues and eigenvectors of the dynamic system’s parameter matrix, with the latter not being unique. To validate our propositions, we apply our analytical models to three simplified physical systems, comparing the outcomes with numerical simulations, and consistently achieve congruent results.

Problem

• Develop exact theory for causal emergence
• Quantify effective information in stochastic systems
• Identify optimal linear coarse-graining strategies

Innovation

• Exact theory for causal emergence
• Linear stochastic iteration systems
• Optimal linear coarse-graining strategies