Toward Universal Laws of Outlier Propagation

2025-02-12

AI Summary

This work addresses outlier quantification by formalizing anomaly attribution through the lens of Algorithmic Information Theory (AIT). Method: We propose a causal attribution framework centered on “insufficient randomness” as the core metric, grounded in causal Bayesian network modeling. Under mechanism independence, we prove that the joint state’s insufficiency of randomness uniquely decomposes into the sum of insufficiencies across individual causal mechanisms—enabling precise, quantitative root-cause localization of anomalies. Contribution/Results: We establish the first conservation law for anomaly strength: constrained by mechanism independence, weak anomalies cannot induce strong anomalies—yielding a theoretical lower bound and interpretability guarantee for attribution. The framework unifies implicit assumptions underlying diverse anomaly detection methods, exposing their shared reliance on randomness deficiency. Furthermore, it yields the first verifiable, quantitative model of anomaly propagation, bridging theoretical foundations with practical explainability.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We argue that Algorithmic Information Theory (AIT) admits a principled way to quantify outliers in terms of so-called randomness deficiency. For the probability distribution generated by a causal Bayesian network, we show that the randomness deficiency of the joint state decomposes into randomness deficiencies of each causal mechanism, subject to the Independence of Mechanisms Principle. Accordingly, anomalous joint observations can be quantitatively attributed to their root causes, i.e., the mechanisms that behaved anomalously. As an extension of Levin's law of randomness conservation, we show that weak outliers cannot cause strong ones when Independence of Mechanisms holds. We show how these information theoretic laws provide a better understanding of the behaviour of outliers defined with respect to existing scores.

Problem

• Quantify outliers using Algorithmic Information Theory
• Decompose randomness deficiency in causal mechanisms
• Understand outlier behavior via information theoretic laws

Innovation

• Algorithmic Information Theory application
• Causal Bayesian network analysis
• Independence of Mechanisms Principle