Newton, Einstein, Jeffreys and Bayesian model selection

摘要

In Jefferys and Berger apply Bayesian model selection to the problem of choosing between rival theories, in particular between Einstein's theory of general relativity (GR) and Newtonian gravity (NG). [1] presents a debate between Harold Jeffreys and Charles Poor regarding the observed 43/century anomalous perhelion precession of Mercury. GR made a precise prediction of 42.98/century while proponents of NG suggested several physical mechanisms that were eventually refuted, with the exception of a modified inverse square law. Using Bayes Factors (BF) and data available in 1921, shows that GR is preferable to NG by a factor of about 25 to 1. A scale for BF used by Jeffreys, suggests that this is positive to strong evidence for GR over modified NG but it is not very strong or even overwhelming.In this work we calculate the BF from the period 1921 till 1993. By 1960 we see that the BF, due to better data gathering techniques and advances in technology, had reached a factor of greater than 100 to 1, making GR strongly preferable to NG and by 1990 the BF reached 1000:1. Ironically while BF had reached a state of near certainty even in 1960 rival theories of gravitation were on the rise — notably the Brans-Dicke (BD) scalar-tensor theory of gravity. The BD theory is postulated in such a way that for small positive values of a scalar parameter , the BF would favor GR while the BF would approach unity with certainty as grows larger, at which point either theory would be prefered, i.e., it is a theory that cannot lose. Does this mean Bayesian model selection needs to be overthrown? This points to the need for cogent prior information guided by physics and physical experiment.