Likelihood function

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Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
Background
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Principle of indifference Principle of maximum entropy
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Posterior approximation
Markov chain Monte Carlo Laplace's approximation Integrated nested Laplace approximations Variational inference Approximate Bayesian computation
Estimators
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Evidence approximation
Evidence lower bound Nested sampling
Model evaluation
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The likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data viewed as a function of the parameters of a statistical model.^[1][2][3] Intuitively, the likelihood function ${\displaystyle {\mathcal {L}}(\theta \mid x)}$ is the probability of observing data ${\displaystyle x}$ assuming ${\displaystyle \theta }$ is the actual parameter.

In maximum likelihood estimation, the arg max (over the parameter ${\displaystyle \theta }$) of the likelihood function serves as a point estimate for ${\displaystyle \theta }$, while the Fisher information (often approximated by the likelihood's Hessian matrix) indicates the estimate's precision.

In contrast, in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' rule.^[4]