  The ratio of Bayesian evidences is a popular tool in cosmology to compare
different models. There are however several issues with this method: Bayes'
ratio depends on the prior even in the limit of non-informative priors, and
Jeffrey's scale, used to assess the test, is arbitrary. Moreover, the standard
use of Bayes' ratio is often criticized for being unable to reject models. In
this paper, we address these shortcoming by promoting evidences and evidence
ratios to frequentist statistics and deriving their sampling distributions. By
comparing the evidence ratios to their sampling distributions, poor fitting
models can now be rejected. Our method additionally does not depend on the
prior in the limit of very weak priors, thereby safeguarding the experimenter
against premature rejection of a theory with a uninformative prior, and
replaces the arbitrary Jeffrey's scale by probability thresholds for rejection.
We provide analytical solutions for some simplified cases (Gaussian data,
linear parameters, and nested models), and we apply the method to cosmological
supernovae Ia data. We dub our method the FB method, for Frequentist-Bayesian.
