# Likelihood ratio

https://arbital.com/p/likelihood_ratio

by Nate Soares Jul 7 2016 updated Jul 7 2016

[summary: Given a piece of evidence $e$ and two hypothsese $H_i$ and $H_j,$ the likelihood ratio between them is the ratio of the Likelihood each hypothesis assigns to $e$ For example, let $e$ = "Mr. Boddy was knifed", and say that Professor Plum is 25% likely to use a knife while Mrs. White is only 5% likely to use a knife. Then the likelihood ratio of $e$ between the hypotheses "Plum did it" and "Mrs. White did it" is 25/5 = 5/1. See also Relative likelihood.]

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.