Editing Likelihood principle (section)

==<span class="anchor" id="law of likelihood"></span>The law of likelihood==
A related concept is the '''law of likelihood''', the notion that the extent to which the evidence supports one parameter value or hypothesis against another is indicated by the ratio of their likelihoods, their [[likelihood ratio]]. That is,
:<math>\Lambda = {\mathcal L(a\mid X=x) \over \mathcal L(b\mid X=x)} = {P(X=x\mid a) \over P(X=x\mid b)}</math>
is the degree to which the observation {{mvar|x}} supports parameter value or hypothesis {{mvar|a}} against {{mvar|b}}. If this ratio is 1, the evidence is indifferent; if greater than 1, the evidence supports the value {{mvar|a}} against {{mvar|b}}; or if less, then vice versa.

In [[Bayesian statistics]], this ratio is known as the [[Bayes factor]], and [[Bayes' rule]] can be seen as the application of the law of likelihood to inference.

In [[frequentist inference]], the likelihood ratio is used in the [[likelihood-ratio test]], but other non-likelihood tests are used as well. The [[Neyman–Pearson lemma]] states the likelihood-ratio test is equally [[statistical power|statistically powerful]] as the most powerful test for comparing two [[simple hypothesis|simple hypotheses]] at a given [[significance level]], which gives a frequentist justification for the law of likelihood.

Combining the likelihood principle with the law of likelihood yields the consequence that the parameter value which maximizes the likelihood function is the value which is most strongly supported by the evidence. This is the basis for the widely used method of [[maximum likelihood]].