Editing Likelihood function (section)

===Likelihood ratio===
{{About|the likelihood ratio in general|the use of likelihood ratios in interpreting diagnostic tests|Likelihood ratios in diagnostic testing|the statistical test to compare goodness of fit|Likelihood-ratio test|section=yes}}
A ''likelihood ratio'' is the ratio of any two specified likelihoods, frequently written as:
<math display="block">\Lambda(\theta_1:\theta_2 \mid x) = \frac{\mathcal{L}(\theta_1 \mid x)}{\mathcal{L}(\theta_2 \mid x)}.</math>

The likelihood ratio is central to [[likelihoodist statistics]]: the ''[[law of likelihood]]'' states that the degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio.

In [[frequentist inference]], the likelihood ratio is the basis for a [[test statistic]], the so-called [[likelihood-ratio test]]. By the [[Neyman–Pearson lemma]], this is the most [[Statistical power|powerful]] test for comparing two [[simple hypothesis|simple hypotheses]] at a given [[significance level]]. Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof.<ref>{{cite journal |first=A. |last=Buse |title=The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note |journal=[[The American Statistician]] |volume=36 |issue=3a |year=1982 |pages=153–157 |doi=10.1080/00031305.1982.10482817 }}</ref> The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by [[Wilks' theorem]].

The likelihood ratio is also of central importance in [[Bayesian inference]], where it is known as the [[Bayes factor]], and is used in [[Bayes' rule]]. Stated in terms of [[odds]], Bayes' rule states that the ''posterior'' odds of two alternatives, {{tmath|A_1}} and {{tmath|A_2}}, given an event {{tmath|B}}, is the ''prior'' odds, times the likelihood ratio. As an equation:
<math display="block">O(A_1:A_2 \mid B) = O(A_1:A_2) \cdot \Lambda(A_1:A_2 \mid B).</math>

The likelihood ratio is not directly used in AIC-based statistics. Instead, what is used is the relative likelihood of models (see below).

In [[evidence-based medicine]], likelihood ratios [[Likelihood ratios in diagnostic testing|are used in diagnostic testing]] to assess the value of performing a [[diagnostic test]].