Bayesian model selection: Application to the adjustment of fundamental physical constants

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 pe-riodic 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 con-sistent. 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 guid-ance even when the data comprise only a small number of measurements.