I’ll not discuss about Bayesian or Frequentist foundation (discussed by previous speakers) I’ll try to discuss some facts and my point of view. I think that we cannot solve here a century old debate.. We should keep in mind that with the two approaches we are answering two different questions (see previous talks). This statement simply translates into the fact that Bayesians give PDF and Frequentists give CL.  caveat for Bayesian is the prior dependence.  Caveat for the Frequentist is how to define CL (5%,32%) and how to treat systematic (often with a bayesian approach) and of theoretical errors which often have been already combined When we do a phenomenological work we should not forget physics A phenomenological work implies - to do predictions - to indicate which are the important things to do/ to measure/ to calculate - to use all the available informations.

SM predictions of  ms SM predictions of sin2  Our collaboration or protocollaboration is in read (CKMFitter in this figure is in blue) Our collaboration is in read or protocoll. is in blue (CKMfitter in this figure is in yellow) sin2  and  m s were predicted before these mesurements.  Crucial test of the SM  Important motivation to perform this measurement

Priors are not a problem. They are part of the Bayesian approach. We check that any time. If the measurement (likelihood) is starting to be precise, there is no dependence. In this case the two methods give similar answers. In other case the question is different the answer is different. Example of depence on prior even in presence of not very precise (30%) measurement (  from Dalitz technique ) using Cartesian or Polar coordinates Past comparison Freq/Bayes Test done with same inputs in 2002 for the first CKM workshop Frequentist/ Bayesian In the today situation the two approaches would given even more similar results

Statistics should not allow people to…forget physics. Example on  How it is possible ? Our analysis of  was criticised..and not very kindly..because of some argument like the prior dependence, the dependence of the result from different parametrization + the fact that we were not able to reproduce the 8 ambiguities…+ the fact that we do not have a solution at  ~0 Recall : hep-ph/ Utfit answer Please have a look at hep-ph/ submitted to PRD

Gronau London method requires some a priori MINIMAL ASSUPTION on strong interactions, namely - flavour blindness and CP conservation - negligible isospin symm. breaking. We believe that the strong coupling constant has a natural size of  QCD ~1GeV We do not expect : 1-CL Frequentist plot from CKMFitter ??  QCD

In addition.. the Baysian result does not depend on use of different parametrizations and does not depend on priors (cut on the upper value of |P|) as claimed..

More details on the anaysis of  from M. Bona presentation at SLAC

Some very instructive slides from a seminar given by R. Faccini

The frequentistic approach returns the region of the parameters for which the data have at least a given probability (1-C.L.) of being described by the model – –The true value is a fixed number – no distribution –Utilize toy MC CKMFitter (RFit option) The bayesian approach tries to calculate the probability distribution of the true parameters by assigning probability density functions to all unknown parameters –Utilize Bayes theorem –The true value has a distribution –P(H) : a-priori probability UTFit

How to read plots B A B AR 1-C.L.=probability of the data being in worse agreement than what observed with a given hypothesis on the true value The intercepts with 1-C.L.=0.32 are the boundary of the interval of hypotheses on the true value which are discarded by the data with a prob. of at least 32% (the so called “68% C.L. region”). Prob. Density (“a-posteriori”) is an estimate of the pdf of the true value projection of red area : region by which the true value is covered at 68% probability note that the UTFit convention in case of multiple peaks is to start from maximum of likelihood and integrate over equi-probable contours Otherwise the integration starts from the median 1-C.L. “Allowed” region

What they say of each other Frequentists of bayesian approach –A priori probabilities completely arbitrary –How can I trust a calculation based on something which is completely unknown? –Why should I try to calculate something imprecisely if I can already calculate something exactly? If I can trust the Bayesian result only when it agrees with the frequentist one, why shall I try it at all !?! –The integration of the likelihood to get the C.L. regions has degrees of arbitrariness Bayesians of frequentist approach –The probability of the true value being in a C.L. region is unknown (not necessarily above C.L.) The result has no really usefulness, in principle the true value could be anywhere with unknown probability –There is some freedom in the choice of the test statistics and of the definition of “worse agreement”.