Synergistic Information Emerges from the Failure of Distributivity in Random Variables

Core Concepts

Synergistic information arises as a direct consequence of the failure of distributivity in the operations on random variables, which is fundamentally distinct from the set-theoretic properties.

Abstract

The content explores the relationship between synergistic information and the failure of distributivity in random variables. It starts by highlighting the limitations of using set-theoretic intuition to understand synergistic behavior in information theory. The authors show that random variables, unlike sets, do not adhere to the distributivity axiom, leading to the emergence of synergistic information.
The key insights are:

The inclusion-exclusion principle for random variables contains a distributivity-breaking term, which corresponds to the synergistic information.
For a trivariate system, the amount of synergy is directly proportional to the extent of distributivity failure.
The authors construct a Venn-type diagram to represent the information atoms, including redundant, unique, and synergistic components.
The proposed framework resolves the self-contradictions that plagued previous partial information decomposition (PID) approaches and provides additional constraints on the solutions.
The authors generalize the results to an arbitrary number of variables, establishing a consistent theory of multivariate information decomposition.

The content demonstrates that synergistic behavior is not a violation of information conservation, but rather a consequence of the fundamental distinction between set theory and probability theory.

Stats

H(X1, X2, X3) = 2 bit
I3(O1; O2; O3) = -1 bit

Quotes

Πs = 1 bit
Πg = 1 bit

Key Insights Distilled From

Synergy as the failure of distributivity

by Ivan A. Sevo... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03455.pdf

Deeper Inquiries

How can the proposed framework be extended to handle continuous random variables

To extend the proposed framework to handle continuous random variables, we can leverage concepts from probability theory and information theory. One approach is to use probability density functions (PDFs) instead of probability mass functions (PMFs) for discrete random variables. The entropy of a continuous random variable can be defined using the differential entropy formula. Mutual information between continuous random variables can be calculated using integrals instead of sums. Additionally, the set-theoretic approach can be adapted to continuous random variables by considering intersections as regions in the joint probability density function space. The parthood table can be modified to include regions in the joint PDF space, and the covering numbers can be defined based on the volume of these regions.

What are the implications of the failure of distributivity in random variables for other fields beyond information theory, such as physics or biology

The failure of distributivity in random variables has implications beyond information theory and can be relevant in various fields such as physics and biology. In physics, the breakdown of distributivity can lead to the emergence of new phenomena that cannot be explained by the sum of individual components. This can be seen in quantum mechanics, where the behavior of composite systems cannot always be understood by analyzing the properties of individual particles. In biology, the failure of distributivity can manifest in complex interactions between biological entities, leading to emergent properties at the system level that cannot be predicted from the properties of individual components. Understanding the role of distributivity in these fields can provide insights into the nature of emergent phenomena and complex systems.

Can the insights from this work be used to develop new measures of emergence in complex systems

The insights from this work can be used to develop new measures of emergence in complex systems by focusing on the interplay between individual components and the system as a whole. By considering the failure of distributivity in random variables as a source of emergent behavior, new measures can be designed to quantify the extent of emergence in a system. These measures can capture the synergistic interactions between components that give rise to emergent properties. By analyzing the information decomposition of complex systems, researchers can identify the unique contributions of individual components, the redundant information shared between them, and the synergistic information that emerges from their interactions. This approach can provide a more nuanced understanding of emergence in complex systems and offer new ways to measure and characterize emergent phenomena.