Bayesian approach and Naturalness in MSSM analyses for the LHC

The start of LHC has motivated an effort to determine the relative probability of the different regions of the MSSM parameter space, taking into account the present, theoretical and experimental, wisdom about the model. Since the present experimental data are not powerful enough to select a small region of the MSSM parameter space, the choice of a judicious prior probability for the parameters becomes most relevant. Previous studies have proposed theoretical priors that incorporate some (conventional) measure of the fine-tuning, to penalize unnatural possibilities. However, we show that such penalization arises from the Bayesian analysis itself (with no ad hoc assumptions), upon the marginalization of the mu-parameter. Furthermore the resulting effective prior contains precisely the Barbieri-Giudice measure, which is very satisfactory. On the other hand we carry on a rigorous treatment of the Yukawa couplings, showing in particular that the usual practice of taking the Yukawas "as required", approximately corresponds to taking logarithmically flat priors in the Yukawa couplings. Finally, we use an efficient set of variables to scan the MSSM parameter space, trading in particular B by tan beta, giving the effective prior in the new parameters. Beside the numerical results, we give accurate analytic expressions for the effective priors in all cases. Whatever experimental information one may use in the future, it is to be weighted by the Bayesian factors worked out here.