# Likelihood ratio

Given a piece of evidence $$e_0$$ and two hypothsese $$H_i$$ and $$H_j,$$ the likelihood ratio between them is the ratio of the likelihood each hypothesis assigns to $$e_0.$$

For example, imagine the evidence is $$e$$ = “Mr. Boddy was knifed”, and the hypotheses are $$H_P$$ = “Professor Plum did it” and $$H_W$$ = “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 $$e_0$$ between $$H_P$$ and $$H_W$$ is 255 = 5, which says that $$H_P$$ assigns five times as much likelihood to $$e$$ as does $$H_W,$$ 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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