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J.O. Berger 263others). If x is the overall data from the trial, and a total of m tests areultimately conducted by the procedure, all claimed to be rejections (i.e., allclaimed to correspond to the H i 0 being false), the Bayesian computesPr (at least one incorrect rejection |x)= 1− Pr(no incorrect rejections|x)m∏= 1− {1 − Pr(H0|x)} i , (23.5)i=1where Pr(H i 0|x) is the posterior probability that H i 0 is true given the data.Clearly, as m grows, (23.5) will go to 1 so that, if there are enough tests,the Bayesian becomes essentially sure that at least one of the rejections waswrong. From Section 23.5.3, recall that Bayesian testing can be exactly equivalentto conditional frequentist testing, so it should be possible to construct aconditional frequentist variant of (23.5). This will, however, be pursued elsewhere.While we assumed that the hypotheses are all aprioriindependent, itis more typical in the multiple endpoint scenario that they will be apriorirelated (e.g., different dosages of a drug). This can be handled within theBayesian approach (and will be explored elsewhere), but it is not clear howa frequentist could incorporate this information, since it is information aboutthe prior probabilities of hypotheses.23.5.5 True to false discovery oddsA very important paper in the history of genome wide association studies (theeffort to find which genes are associated with certain diseases) was Burtonet al. (2007). Consider testing H 0 : θ =0versusanalternativeH 1 : θ ≠ 0,with rejection region R and corresponding Type I and Type II errors α andβ(θ). Let p(θ) bethepriordensityofθ under H 1 , and define the average power∫1 − ¯β = {1 − β(θ)}p(θ)dθ.Frequentists would typically just pick some value θ ∗ at which to evaluate thepower; this is equivalent to choosing p(θ) to be a point mass at θ ∗ .The paper observed that, pre-experimentally, the odds of correctly rejectingH 0 to incorrectly rejecting areO pre = π 1π 0× 1 − ¯βα , (23.6)where π 0 and π 1 =1− π 0 are the prior probabilities of H 0 and H 1 .Thecorresponding false discovery rate would be (1 + O pre ) −1 .The paper went on to assess the prior odds π 1 /π 0 of a genome/diseaseassociation to be 1/100, 000, and estimated the average power of a GWAS