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In the simple vs composite hypothesis test with a proper prior, the BayesFactor (BF) is shown to be the posterior mean of the Likelihood Ratio (LR).Therefore, the posterior standard deviation of the LR or rather its posteriorcumulative density function can be used to indicate the significativity of adetection by the BF and this detection procedure can be computed from a singleMarkov Chain. It is applied and compared for exoplanet detection.  The previous statistics can be expressed from the Fractional BF (FBF)\cite{ohagan95} and the Probability distribution of the LR (PLR)\cite{aitkin97}. Two properties of the PLR related to the GLRT are noted and aprocedure to optimize the PLR and the FBF two-parameters detectors according totheir ROC curves is proposed. The performances of all tests are compared.

Posterior probability of the Likelihood Ratio and (Fractional) Bayes Factor: new theoretical relations and practical uses