Reply to: 'Improved Determination of the CKM Angle α from B → ππ decays'

In reply to a paper by M. Bona et al. (UTfit Collaboration, arXiv:hep-ph/0701204) we demonstrate why the arguments made therein do not address the criticism exposed in our earlier paper (arXiv:hep-ph/0607246) on the fundamental shortcomings of the Bayesian approach when it comes to the extraction of parameters of Nature from experimental data. As for the isospin analysis and the CKM angle α it is shown that the use of uniform priors for the observed quantities in the Explicit Solution parametrization is equivalent to a frequentist construction resulting from a change of variables, and thus relies neither on prior PDFs nor on Bayes' theorem. This procedure provides in this particular case results that are similar to the Confidence Level approach, but the treatment of mirror solutions remains incorrect and it is far from being general. In a second part it is shown that important differences subsist between the Bayesian and frequentist approaches, when following the proposal of paper by Bona M. et al. and inserting additional information on the hadronic amplitudes beyond isospin invariance. In particular the frequentist result preserves the exact degeneracy that is expected from the remaining symmetries of the problem while the Bayesian procedure does not. Moreover, in the Bayesian approach reducing inference to the 68% or 95% credible interval is a misconception of the meaning of the posterior PDF, which in turn implies that the significant dependence of the latter to the chosen parametrization cannot be viewed as a minor effect, contrary to the claim in the paper by M. Bona et al. (authors)