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.