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Likelihood function
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<p>A likelihood function (often simply called the likelihood) measures how well a <a href="/facts/Statistical_model/jKkT5ftm">statistical model</a> explains <a href="/facts/Realization_(probability)/lDklUzqX">observed data</a> by calculating the probability of seeing that data under different <a href="/facts/Statistical_parameter/fkJju25M">parameter</a> values of the model. It is constructed from the <a href="/facts/Joint_probability_distribution/klX2ksGY">joint probability distribution</a> of the <a href="/facts/Random_variable/TwTBXnLT">random variable</a> that (presumably) generated the observations. When evaluated on the actual data points, it becomes a function solely of the model parameters. </p><p>In <a href="/facts/Maximum_likelihood_estimation/0Yq2dpQD">maximum likelihood estimation</a>, the <a href="/facts/Arg_max/TfSepkFV">argument that maximizes</a> the likelihood function serves as a <a href="/facts/Point_estimation/AYsuptFp">point estimate</a> for the unknown parameter, while the <a href="/facts/Fisher_information/Q2JLexN9">Fisher information</a> (often approximated by the likelihood's <a href="/facts/Hessian_matrix/RguMNr3m">Hessian matrix</a> at the maximum) gives an indication of the estimate's <a href="/facts/Precision_(statistics)/xsMYkkjA">precision</a>. </p><p>In contrast, in <a href="/facts/Bayesian_statistics/9w7b1Bw4">Bayesian statistics</a>, the estimate of interest is the <i>converse</i> of the likelihood, the so-called <a href="/facts/Posterior_probability/fojsMD0D">posterior probability</a> of the parameter given the observed data, which is calculated via <a href="/facts/Bayes%2527_theorem/wQqFB2Dt">Bayes' rule</a>. </p>