Authors: Fabrice J.P.R. Pautot
Comments: 17 Pages.
We present a simple, fully probabilistic, Bayesian solution to -sample omnibus tests for comparison, with the Behrens-Fisher problem as a special case, which is free from the many defects found in the standard, classical, frequentist, likelihoodist and Bayesian approaches to those problems. We solve the main measure-theoretic difficulty for degenerate problems with continuous parameters of interest and Lebesgue-negligible point null hypothesis by approximating the corresponding continuous random variables by sequences of discrete ones defined on partitions of the parameter spaces and by taking the limit of the prior-to-posterior ratios of the probability of the null hypothesis for the corresponding discrete problems. Those limits are well defined under proper technicalities thanks to the Henstock-Kurzweil integral that is as powerful as the Lebesgue integral but still relies on Riemann sums, which are essential in the present approach. The solutions to the relative continuous problems take the form of Bayes-Poincaré factors that are new objects in Bayesian probability theory and should play a key role in the general theory of point null hypothesis testing, including other important problems such as the Jeffreys-Lindley paradox.