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Likelihood ratio

Given a piece of evidence e0 and two hypothsese Hi and Hj, the likelihood ratio between them is the ratio of the likelihood each hypothesis assigns to e0.

For example, imagine the evidence is e = “Mr. Boddy was knifed”, and the hypotheses are HP = “Professor Plum did it” and HW = “Mrs. White did it.” Let’s say that, if Professor Plum were the killer, we’re 25% sure he would have used a knife. Let’s also say that, if Mrs. White were the killer, there’s only a 5% chance she would have used a knife. Then the likelihood ratio of e0 between HP and HW is 255 = 5, which says that HP assigns five times as much likelihood to e as does HW, which means that the evidence supports the “Plum did it” hypothesis five times as much as it supports the “Mrs. White did it” hypothesis.

A likelihood ratio of 5 denotes relative likelihoods of (5:1). Relative likelihoods can be multiplied by odds in order to update those odds, as per Bayes’ rule.

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