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# In other words the likelihood of getting a match as

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Unformatted text preview: f h1 match (or, the problem ed text. In other words, the likelihood of getting a match As Turing was aware, this formulation had two distinct instruct ethe logarithm of the independently by Wald shortly after ver h0 astions was developed ratio of these likelihoods: an (m )Evidence 1favoring h1 Pr(mahdefinite stoppingthan the advantages. First, it had | 1), was greater time that Pr(on m) h1| one given h , denoted / evidence favoring h0 sthe war (Wald, 1947). log ch likelihood of getting a match given hwhen to stop working Pr(Second instructed the codebreaking team , denoted Pr(m|h ). h0|m) 0 0 d As Turing was aware, thisPr(m|h1) formulation had two distinct he Accordingly, Turing defined the weight of evidence proon one of evidence 1) Weight pair of ciphers and turn their attention (to another. log follows n advantages. First, it had aPr(m|h0) stopping time that definite Second, it predicted the accuracy of the decision,hwhere be h0|m as Pr( rela 0). h vided by a match (or, similarly, athe weight ofto stop working instructed the codebreaking non-match) in favor ofcan team when evidence 1 ofollows. By Bayes’ theorem, probabilities a over h0 of the logarithm of the ratio of attention to another. Bayes probabilities, and probabilities: ea logarithmasrule:to several this quantity isthese likelihoods: on one pair Pr(h|m) =otherturn their measured of ciphers Pr(m|h)Pr(h)/Pr(m) s1 be related h pothesis after Pr( log ). Second, it predicted of accuracy of the decision, as ) and Pr(h 0s: units that depend on the basethethe logarithm (Good, Pr(h0 Pr( Pr(m of Pr( 1|m) Pr(hof o- In Banburismus, logarithms 1) base weight|hused follows.hBy Bayes’ theorem, the 10 were 1) evidence describe the p can 979). Weight oflog evidence lweight of evidence, (1) og (2) log Pr(m| actuPr(h0|m) be related to (the Banburismus processh other h1 were called “bans” severalPr(h0) probabilities:0) nd where evidence is sa (1) s: computed weights of evidence in units of 1/10th of lly probab prior probabili wherer(Pr(m|m) and Pr(h11|)m) are called the posterior P h1| h0of probabilities, this quantity is measured ) Pr(h As log logarithm ban—a adeciban—that the codebreakers considered log weight of evidence, hy(2) pothesi assumption m probabilities) and describe) the probability of each Pr(h0smalleston the baseevidence that is (Good, |m Pr(h0of of the logarithm be in units that depend the evidence has been sampled, and Pr(h a ed “abou...
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