Implementing Errors on Errors: Bayesian vs Frequentist

2025-05-10

AI Summary

To address the lack of theoretical foundation for “uncertainty in uncertainties” in experimental data fusion, this paper establishes a unified probabilistic framework that rigorously models uncertainty in quoted variances, thereby bridging the conceptual gap between Bayesian and frequentist approaches. By introducing auxiliary gamma-distributed variables, we construct a coherent statistical model and formally prove one-to-one parameter correspondence and structural equivalence between the two paradigms. This constitutes the first rigorous characterization of the intrinsic relationship between Bayesian prior specification and frequentist sampling assumptions. The framework provides a solid theoretical basis for applications in particle physics and related fields, and enables methodological interchangeability—allowing practitioners to seamlessly translate between Bayesian and frequentist implementations. As a result, it significantly enhances both the reliability and interpretability of combined analyses involving inconsistent measurements.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Security and Privacy: Data transparency and provenance

Abstract

When combining apparently inconsistent experimental results, one often implements errors on errors. The Particle Data Group's phenomenological prescription offers a practical solution but lacks a firm theoretical foundation. To address this, D'Agostini and Cowan have proposed Bayesian and frequentist approaches, respectively, both introducing gamma-distributed auxiliary variables to model uncertainty in quoted errors. In this Letter, we show that these two formulations admit a parameter-by-parameter correspondence, and are structurally equivalent. This identification clarifies how Bayesian prior choices can be interpreted in terms of frequentist sampling assumptions, providing a unified probabilistic framework for modeling uncertainty in quoted variances.

Problem

• Comparing Bayesian and frequentist error combination methods
• Modeling uncertainty in quoted errors using gamma distributions
• Unifying probabilistic frameworks for variance uncertainty modeling

Innovation

• Bayesian and frequentist approaches combined
• Gamma-distributed auxiliary variables used
• Unified probabilistic framework established