Editing Likelihood function (section)

{{Short description|Function related to statistics and probability theory}}
{{Bayesian statistics}}
A '''likelihood function''' (often simply called the '''likelihood''') measures how well a [[statistical model]] explains [[Realization (probability)|observed data]] by calculating the probability of seeing that data under different [[Statistical parameter|parameter]] values of the model. It is constructed from the [[joint probability distribution]] of the [[random variable]] that (presumably) generated the observations.<ref>{{cite book |first1=George |last1=Casella |first2=Roger L. |last2=Berger |title=Statistical Inference |location= |publisher=Duxbury |edition=2nd |year=2002 |isbn=0-534-24312-6 |page=290 }}</ref><ref>{{cite book |first=Jon |last=Wakefield |title=Frequentist and Bayesian Regression Methods |location= |publisher=Springer |edition=1st |year=2013 |isbn=978-1-4419-0925-1 |page=36 }}</ref><ref>{{cite book |first1 = Erich L. |last1=Lehmann | first2 = George |last2 = Casella |title=Theory of Point Estimation |location= |publisher=Springer |edition=2nd |year=1998 |isbn= 0-387-98502-6 |page=444 }}</ref> When evaluated on the actual data points, it becomes a function solely of the model parameters.

In [[maximum likelihood estimation]], the [[arg max|argument that maximizes]] the likelihood function serves as a [[Point estimation|point estimate]] for the unknown parameter, while the [[Fisher information]] (often approximated by the likelihood's [[Hessian matrix]] at the maximum) gives an indication of the estimate's [[Precision (statistics)|precision]].

In contrast, in [[Bayesian statistics]], the estimate of interest is the ''converse'' of the likelihood, the so-called [[posterior probability]] of the parameter given the observed data, which is calculated via [[Bayes' theorem|Bayes' rule]].<ref>{{cite book |first=Arnold |last=Zellner |title=An Introduction to Bayesian Inference in Econometrics |location=New York |publisher=Wiley |year=1971 |pages=13–14 |isbn=0-471-98165-6 }}</ref>