Implementing Errors on Errors: Bayesian vs Frequentist

TL;DR

This paper compares Bayesian and frequentist methods for handling errors on errors in experimental data, showing their structural equivalence and unifying their probabilistic interpretations.

Contribution

It demonstrates the equivalence between Bayesian and frequentist approaches using gamma-distributed variables, clarifying their relationship.

Findings

Bayesian and frequentist methods are structurally equivalent.

The approaches can be interpreted within a unified probabilistic framework.

Prior choices relate to sampling assumptions.

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.