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J.O. Berger 261conditional error probabilities (CEPs) are computed asα(s) =Pr(TypeIerror|S = s) ≡ P 0 {reject H 0 |S(X) =s},β(s) =Pr(TypeIIerror|S = s) ≡ P 1 {accept H 0 |S(X) =s},(23.1)where P 0 and P 1 refer to probability under H 0 and H 1 ,respectively.The corresponding conditional frequentist test is thenIf p 0 ≤ p 1 , reject H 0 and report Type I CEP α(s);If p 0 >p 1 , accept H 0 and report Type II CEP β(s);(23.2)where the CEPs are given in (23.1).These conditional error probabilities are fully frequentist and vary over therejection region as one would expect. In a sense, this procedure can be viewedas a way to turn p-values into actual error probabilities.It was mentioned in the introduction that, when a good conditional frequentistprocedure has been found, it often turns out to be numerically equivalentto a Bayesian procedure. That is the case here. Indeed, Berger et al.(1994) shows thatα(s) =Pr(H 0 |x) , β(s) =Pr(H 1 |x) , (23.3)where Pr(H 0 |x) and Pr(H 1 |x) are the Bayesian posterior probabilities of H 0and H 1 , respectively, assuming the hypotheses have equal prior probabilitiesof 1/2. Therefore, a conditional frequentist can simply compute the objectiveBayesian posterior probabilities of the hypotheses, and declare that they arethe conditional frequentist error probabilities; there is no need to formallyderive the conditioning statistic or perform the conditional frequentist computations.There are many generalizations of this beyond the simple versussimple testing.The practical import of switching to conditional frequentist testing (or theequivalent objective Bayesian testing) is startling. For instance, Sellke et al.(2001) uses a nonparametric setting to develop the following very general lowerbound on α(s), for a given p-value:1α(s) ≥11 −ep ln(p). (23.4)Some values of this lower bound for common p-values are given in Table 23.1.Thus p = .05, which many erroneously think implies strong evidence againstH 0 , actually corresponds to a conditional frequentist error probability at leastas large as .289, which is a rather large error probability. If scientists understoodthat a p-value of .05 corresponded to that large a potential errorprobability in rejection, the scientific world would be a quite different place.