# 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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