Editing Likelihood function (section)

===Relative likelihood function===
{{See also|Relative likelihood}}
Since the actual value of the likelihood function depends on the sample, it is often convenient to work with a standardized measure. Suppose that the [[maximum likelihood estimate]] for the parameter {{mvar|θ}} is <math display="inline">\hat{\theta}</math>.  Relative plausibilities of other {{mvar|θ}} values may be found by comparing the likelihoods of those other values with the likelihood of <math display="inline">\hat{\theta}</math>.  The ''relative likelihood'' of {{mvar|θ}} is defined to be<ref name='Kalbfleisch'>{{citation | author-link= James G. Kalbfleisch | last= Kalbfleisch | first= J. G. | year=1985 | title= Probability and Statistical Inference | publisher= Springer}} (§9.3).</ref><ref>{{citation| last= Azzalini | first= A. | title= Statistical Inference—Based on the likelihood | year= 1996 | publisher= [[Chapman & Hall]] | url= https://books.google.com/books?id=hyN6gXHvSo0C | isbn= 9780412606502 }} (§1.4.2).</ref><ref name='Sprott'>Sprott, D. A. (2000), ''Statistical Inference in Science'', Springer (chap.&nbsp;2).</ref><ref>Davison, A. C. (2008), ''Statistical Models'', [[Cambridge University Press]] (§4.1.2).</ref><ref>{{citation|first1= L. | last1= Held | first2= D. S. | last2= Sabanés Bové | title= Applied Statistical Inference—Likelihood and Bayes | year= 2014 | publisher= Springer}} (§2.1).</ref> 
<math display="block">R(\theta) = \frac{\mathcal{L}(\theta \mid x)}{\mathcal{L}(\hat{\theta} \mid x)}.</math>
Thus, the relative likelihood is the likelihood ratio (discussed above) with the fixed denominator <math display="inline"> \mathcal{L}(\hat{\theta})</math>. This corresponds to standardizing the likelihood to have a maximum of 1.

====Likelihood region====
A ''likelihood region'' is the set of all values of {{mvar|θ}} whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a ''{{mvar|p}}% likelihood region'' for {{mvar|θ}} is defined to be<ref name='Kalbfleisch'/><ref name='Sprott'/><ref name="Rossi2018">{{citation | last= Rossi | first= R. J. | year= 2018 | title= Mathematical Statistics | publisher= [[Wiley (publisher)|Wiley]] | page= 267 }}.</ref>

<math display="block">
\left\{\theta : R(\theta) \ge \frac p {100} \right\}.
</math>

If {{mvar|θ}} is a single real parameter, a {{mvar|p}}% likelihood region will usually comprise an [[Interval (mathematics)|interval]] of real values. If the region does comprise an interval, then it is called a ''likelihood interval''.<ref name='Kalbfleisch'/><ref name='Sprott'/><ref name=Hudson>{{Citation
| last1 = Hudson | first1 = D. J.
| title = Interval estimation from the likelihood function
| journal = [[Journal of the Royal Statistical Society, Series B]]
| volume = 33
| issue = 2
| pages = 256–262
| year = 1971 
| doi = 10.1111/j.2517-6161.1971.tb00877.x
}}.</ref>

Likelihood intervals, and more generally likelihood regions, are used for [[interval estimation]] within likelihoodist statistics: they are similar to [[confidence interval]]s in frequentist statistics and [[credible interval]]s in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of [[coverage probability]] (frequentism) or [[posterior probability]] (Bayesianism).

Given a model, likelihood intervals can be compared to confidence intervals. If {{mvar|θ}} is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for {{mvar|θ}} will be the same as a 95% confidence interval (19/20 coverage probability).<ref name='Kalbfleisch'/><ref name="Rossi2018"/> In a slightly different formulation suited to the use of log-likelihoods (see [[Likelihood-ratio test#Distribution: Wilks.27 theorem|Wilks' theorem]]), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a [[chi-squared distribution]] with degrees-of-freedom (df) equal to the difference in df's between the two models (therefore, the {{mvar|e}}<sup>&minus;2</sup> likelihood interval is the same as the 0.954 confidence interval; assuming difference in df's to be 1).<ref name="Rossi2018"/><ref name=Hudson/>