An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems

TL;DR

This paper develops an exact theoretical framework for causal emergence in linear stochastic systems, providing analytical tools to identify optimal coarse-graining strategies that enhance causal effects, validated through physical system examples.

Contribution

It introduces a novel analytical approach for causal emergence in continuous stochastic systems, linking maximal emergence to eigenvalues and eigenvectors of system matrices.

Findings

Maximal causal emergence depends on principal eigenvalues.

Optimal coarse-graining aligns with eigenvectors of system matrices.

Analytical results match numerical simulations in physical models.

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.