Likelihood function
| Part of a series on |
| Bayesian statistics |
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| Posterior = Likelihood × Prior ÷ Evidence |
| Background |
| Model building |
| Posterior approximation |
| Estimators |
| Evidence approximation |
| Model evaluation |
A likelihood function (often simply called the likelihood) gives the relative merit of various statistical models for describing a data set. Often the models being compared are parameterized by a parameter, with the parameter often written as θ, or they are parameterized by multiple parameters given as the components of a vector. For a probability function (or probability density function) Pr[x | θ] that gives the probability (or probability density) of data x for a given model-specifying parameter θ, the likelihood is any function of θ equal to cPr[x | θ] for some positive value c.
In maximum likelihood estimation, the model parameter(s) or argument that maximizes the likelihood function serves as a 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.
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' rule.