Bayesian Statistics at Work: the Troublesome Extraction of the CKM Phase α

In Bayesian statistics, one's prior beliefs about underlying model parameters are revised with the information content of observed data from which, using Bayes' rule, a posterior belief is obtained. A non-trivial example taken from the isospin analysis of B → PP (P = π or ρ) decays in heavy-flavor physics is chosen to illustrate the effect of the naive 'objective' choice of flat priors in a multi- dimensional parameter space in presence of mirror solutions. It is demonstrated that the posterior distribution for the parameter of interest, the phase α, strongly depends on the choice of the parameterization in which the priors are uniform, and on the validity range in which the (un-normalizable) priors are truncated. We prove that the most probable values found by the Bayesian treatment do not coincide with the explicit analytical solutions, in contrast to the frequentist approach. It is also shown in the appendix that the α → 0 limit cannot be consistently treated in the Bayesian paradigm, because the latter violates the physical symmetries of the problem. (authors)