  With Bayesian statistics, we investigate the full parameter space of the
constrained "next-to-minimal" supersymmetric Standard Model (CNMSSM) with
naturalness priors, which were derived in a previous work. In the past, most
Bayesian 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, with
naturalness 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 in
favor of the CNMSSM versus the CMSSM is "positive" to "strong" but that if one
ignores the $\mu$-problem, the evidence is "barely worth mentioning" to
"positive." The $\mu$-problem significantly influences our findings. Unless one
considers the $\mu$-problem, the evidence in favor of the CNMSSM versus the
CMSSM is at best "positive," which is two grades below "very strong." We,
furthermore, identify the most probable regions of the CMSSM and CNMSSM
parameter spaces and examine prospects for future discovery at hadron
colliders.
