September 2023 Bayesian model selection: Application to the adjustment of fundamental physical constants
Olha Bodnar, Viktor Eriksson
Author Affiliations +
Ann. Appl. Stat. 17(3): 2118-2138 (September 2023). DOI: 10.1214/22-AOAS1710


A method originally suggested by Raymond Birge, using what came to be known as the Birge ratio, has been widely used in metrology and physics for the adjustment of fundamental physical constants, particularly in the periodic reevaluation carried out by the Task Group on Fundamental Physical Constants of CODATA (the Committee on Data of the International Science Council). The method involves increasing the reported uncertainties by a multiplicative factor large enough to make the measurement results mutually consistent. An alternative approach, predominant in the meta-analysis of medical studies, involves inflating the reported uncertainties by combining them, using the root sum of squares, with a sufficiently large constant (often dubbed dark uncertainty) that is estimated from the data.

In this contribution we establish a connection between the method based on the Birge ratio and the location-scale model, which allows one to combine the results of various studies, while the additive adjustment is reviewed in the usual context of random-effects models. Framing these alternative approaches as statistical models facilitates a quantitative comparison of them using statistical tools for model comparison. The intrinsic Bayes factor (IBF) is derived for the Berger and Bernardo reference prior, and then it is used to select a model for a set of measurements of the Newtonian constant of gravitation (“Big G”) to estimate a consensus value for this constant and to evaluate the associated uncertainty. Our empirical findings support the method based on the Birge ratio. The same conclusion is reached when the IBF corresponding to the Jeffreys prior is used and also when the comparison is based on the Akaike information criterion (AIC). Finally, the results of a simulation study indicate that the suggested procedure for model selection provides clear guidance, even when the data comprise only a small number of measurements.

Funding Statement

Olha Bodnar acknowledges valuable support from an internal grant (Rörlig resurs) from Örebro University.
This research was partially supported by the National Institute of Standards and Technology (NIST) Exchange Visitor Program.


The authors would like to thank Professor Kafadar and two anonymous referees for their constructive comments which improved the quality of this paper.

The first author is grateful to the Statistical Engineering Division of the National Institute of Standards and Technology (NIST) for providing an excellent and inspiring environment for research.


Download Citation

Olha Bodnar. Viktor Eriksson. "Bayesian model selection: Application to the adjustment of fundamental physical constants." Ann. Appl. Stat. 17 (3) 2118 - 2138, September 2023.


Received: 1 August 2021; Revised: 1 October 2022; Published: September 2023
First available in Project Euclid: 7 September 2023

MathSciNet: MR4637660
Digital Object Identifier: 10.1214/22-AOAS1710

Keywords: Birge ratio method , interlaboratory comparison study , intrinsic Bayes factor , location-scale model , Meta-analysis , Newtonian constant of gravitation , random-effects model , reference prior

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.17 • No. 3 • September 2023
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