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With Bayesian statistics, we investigate the full parameter space of theconstrained "next-to-minimal" supersymmetric Standard Model (CNMSSM) withnaturalness priors, which were derived in a previous work. In the past, mostBayesian analyses of the CNMSSM ignored naturalness of the electroweak (EW)scale by making prejudicial assumptions for parameters defined at the EW scale.We test the CNMSSM against the CMSSM with Bayesian evidence, which, withnaturalness priors, incorporates a penalty for fine-tuning of the EW scale.With the evidence, we measure credibility with respect to all measurements,including the EW scale and LHC direct searches. We find that the evidence infavor of the CNMSSM versus the CMSSM is "positive" to "strong" but that if oneignores the $\mu$-problem, the evidence is "barely worth mentioning" to"positive." The $\mu$-problem significantly influences our findings. Unless oneconsiders the $\mu$-problem, the evidence in favor of the CNMSSM versus theCMSSM is at best "positive," which is two grades below "very strong." We,furthermore, identify the most probable regions of the CMSSM and CNMSSMparameter spaces and examine prospects for future discovery at hadroncolliders.

Is the CNMSSM more credible than the CMSSM?