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 ²⁵⁄₅ = 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.