The Limits of Information Breakdown

by

Author: Denis Avetisyan

New research reveals fundamental incompatibilities in how we decompose information, challenging core assumptions in fields like neuroscience and machine learning.

The structure of redundant information, as illustrated by the redundancy lattice $ (\mathcal{A}\_{3},\preceq)$, reveals a nested architecture where each level builds upon and contains the information of those preceding it, demonstrating an inherent order within complex systems.
The structure of redundant information, as illustrated by the redundancy lattice $ (\mathcal{A}\_{3},\preceq)$, reveals a nested architecture where each level builds upon and contains the information of those preceding it, demonstrating an inherent order within complex systems.

A rigorous analysis demonstrates that desirable properties for Partial Information Decomposition – non-negativity, re-encoding invariance, and adherence to the chain rule – cannot be simultaneously satisfied.

Despite decades of development, a universally accepted framework for Partial Information Decomposition (PID)-separating how multiple sources contribute redundant, unique, and synergistic information-remains elusive. This paper, ‘Novel Inconsistency Results for Partial Information Decomposition’, addresses this challenge by demonstrating fundamental incompatibilities between seemingly essential axioms for PID. Specifically, we prove that non-negativity, invariance under invertible transformations, and a chain rule cannot be simultaneously satisfied within any PID framework, forcing a critical re-evaluation of core principles. This raises the question: what foundational assumptions must we relinquish to achieve a consistent and practically useful theory of information decomposition?

The Echo of Information: Dissecting Complex Systems

The behavior of any complex system – be it a brain processing sensory input, an ecosystem responding to environmental changes, or a financial market reacting to global events – fundamentally hinges on how information from diverse sources integrates and influences its state. Understanding this process isn’t simply about knowing that information is combined, but quantifying how each contributing source shapes the overall outcome. A failure to account for these interactions can lead to inaccurate models and predictions, as even seemingly insignificant inputs can have disproportionate effects when combined with others. Consequently, discerning the origins and relative contributions of information streams is paramount for effectively analyzing and potentially controlling the dynamics of these intricate systems, offering insights that extend from neuroscience and ecology to economics and engineering.

Partial Information Decomposition, or PID, offers a rigorous method for dissecting the contributions of multiple sources to a complex signal, effectively answering the question of ‘who knows what’. Rather than simply measuring total information, PID quantifies the unique information each source provides – the knowledge absent if that source were removed – and the redundant information shared across sources. This decomposition isn’t merely about splitting data; it’s about understanding the functional roles of each component within a system. By applying principles like the Target Chain Rule, researchers can move beyond correlation and establish a causal understanding of information flow, revealing how individual elements contribute to the overall knowledge contained within a collective. This framework has implications for fields ranging from neuroscience, where it can map information processing in the brain, to machine learning, where it can optimize distributed sensor networks and enhance collaborative algorithms.

Partial Information Decomposition (PID) dissects complex signals by leveraging fundamental principles of information theory. At its heart lies the Target Chain Rule, a mathematical framework that allows researchers to quantify how much each source contributes to knowledge about a specific target variable. This isn’t simply additive; PID distinguishes between unique information – that which a source provides independently – and redundant information, shared across multiple sources. By identifying these components, the framework moves beyond simply knowing ‘who knows what’ to understanding how different sources collectively determine knowledge. This decomposition isn’t merely theoretical; it provides a quantifiable measure of information contribution, expressed in bits, enabling researchers to map the flow of information within complex systems and pinpoint the most critical sources for a given task, offering insights into everything from neural networks to ecological interactions.

The Foundations of Decomposition: Core Assumptions

The Predictive Information Decomposition (PID) framework relies on several core assumptions regarding the behavior of consistent information measures. Specifically, PID posits Local Positivity – that information transfer between any two variables is non-negative. Re-encoding Invariance dictates that the decomposition of information should remain consistent regardless of invertible transformations applied to the variables involved. Finally, the Identity Property states that combining information from multiple independent sources should yield the sum of their individual contributions. These properties are not merely mathematical conveniences; they are foundational requirements ensuring the framework’s ability to reliably quantify and decompose information transfer in complex systems, particularly when dealing with non-linear interactions and multiple sources of information.

The assumptions of Local Positivity, Re-encoding Invariance, and the Identity Property are not merely definitional within the PID framework; they fundamentally constrain how information measures are calculated and combined. Local Positivity dictates that information contributions from individual events or components must be non-negative. Re-encoding Invariance ensures that the measured information remains consistent regardless of how the underlying data is represented or transformed, preventing arbitrary fluctuations due to encoding choices. The Identity Property establishes a baseline for combining information from independent sources – the total information should be the sum of individual contributions. These properties collectively define a coherent system, enabling consistent and predictable information quantification even when dealing with complex data combinations and transformations, and forming the basis for deriving meaningful insights.

Mutual Information (MI) within the Predictive Information Decomposition (PID) framework functions as a measure of the statistical dependence between random variables. Specifically, MI quantifies the amount of information one variable contains about another, calculated as $I(X;Y) = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)p(y)}$. In PID, MI is not simply a measure of association, but a necessary component for decomposing information into predictable, unique, and redundant parts, enabling a granular understanding of how information flows and is represented within a system. The accurate calculation and interpretation of MI are therefore foundational to applying the PID framework effectively.

The degree of redundancy of information atoms, ranging from synergistic (redundancy of 0) to genuinely redundant (redundancy ≥2), dictates their contribution-and sign-within the overall information measure as defined in Equations 29 and 31.
The degree of redundancy of information atoms, ranging from synergistic (redundancy of 0) to genuinely redundant (redundancy ≥2), dictates their contribution-and sign-within the overall information measure as defined in Equations 29 and 31.

A Fragile Framework: Inconsistencies Revealed

Recent theoretical work has established the mutual incompatibility of Local Positivity, the Target Chain Rule, Re-encoding Invariance, and the Identity Property within the Probabilistic Information Dispersal (PID) framework. This incompatibility has been rigorously demonstrated through proof by contradiction, indicating that a system cannot simultaneously satisfy all four properties under the axioms of PID. Specifically, the derivation reveals a logical inconsistency arising when attempting to uphold these properties concurrently, leading to a demonstrable contradiction within the established mathematical formalism of PID. This finding challenges the foundational coherence of the PID model and necessitates a re-evaluation of its core assumptions.

The XOR Source Copy Gate provides a specific instance demonstrating the incompatibility of Local Positivity, the Target Chain Rule, Re-encoding Invariance, and the Identity Property within the Probabilistic Ideal Data (PID) framework. This gate, when applied to a system adhering to these properties, generates a contradiction: specifically, the gate’s operation results in a non-zero probability for an impossible outcome, violating Local Positivity. Attempts to reconcile this violation by modifying the Target Chain Rule or Re-encoding Invariance invariably lead to further inconsistencies with the Identity Property, confirming that no single adjustment can simultaneously satisfy all four properties when the XOR Source Copy Gate is present. This counterexample is not a general failure of PID, but rather a precise demonstration of limitations within the stated assumptions.

The demonstrated incompatibility between Local Positivity, the Target Chain Rule, Re-encoding Invariance, and the Identity Property within the Probabilistic Information Dispersal (PID) framework constitutes a fundamental theoretical challenge. This is not merely a matter of refining existing models; the contradiction proves a core assumption underlying PID is flawed. Consequently, a comprehensive reassessment of the foundational principles governing PID is required, potentially necessitating the development of an entirely new theoretical basis to accommodate these conflicting properties and ensure the framework’s internal consistency. Further research must focus on identifying the precise origin of this inconsistency and exploring alternative axiomatic structures that avoid this problematic conflict.

The presented inconsistency results regarding Partial Information Decomposition highlight a fundamental truth about complex systems: elegance often demands compromise. This work demonstrates that striving for simultaneously satisfying properties – non-negativity, re-encoding invariance, and a chain rule – introduces inherent incompatibilities. As John McCarthy observed, “Every delay is the price of understanding.” This resonates with the challenges faced in defining a universally consistent framework for information decomposition; each attempted refinement, each desired property, necessitates a careful consideration of the trade-offs involved, and a willingness to accept limitations in pursuit of a more nuanced understanding of synergistic and redundant information.

What Lies Ahead?

The demonstrated incompatibilities regarding non-negativity, re-encoding invariance, and the chain rule within Partial Information Decomposition are not merely mathematical curiosities. They represent a fundamental friction-a systemic drag-inherent in any attempt to dissect complex information flows. Uptime for any decomposition scheme is, demonstrably, temporary. The pursuit of simultaneously satisfying these properties appears increasingly akin to chasing a receding horizon; each refinement of one axiom inevitably exacerbates the tension with another.

Future work will likely necessitate a pragmatic reassessment of these desiderata. Perhaps the focus should shift from seeking a universally ‘correct’ decomposition-a stable state that will never exist-to understanding the specific costs associated with prioritizing certain properties over others. Stability is an illusion cached by time, and the choice of which illusions to preserve will depend on the application, and the acceptable latency.

The revealed inconsistencies also suggest a need to examine the very foundations of information partitioning. The chain rule, long considered a cornerstone, may itself be a limiting factor-a constraint imposed by a particular, perhaps naive, view of information structure. Every request pays a tax in latency. The field now faces the task of determining whether that tax is unavoidable, or simply poorly accounted for.

Original article: https://arxiv.org/pdf/2512.16662.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-21 12:35