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
Model building
Weak prior ... Strong prior Conjugate prior Linear regression Empirical Bayes Hierarchical model
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
Bayes factor Model averaging Posterior predictive
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The likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data, but viewed as a function of the parameters of a statistical model.^[1][2][3] That is, the likelihood function ${\textstyle {\mathcal {L}}(\theta \mid x)}$, which gives the likelihood of a vector of parameters ${\textstyle \theta }$ under the assumption that a set of observed data ${\textstyle x}$ is true, is numerically the same as the probability function ${\textstyle f(x\mid \theta )}$, which gives the probability of a set of data ${\textstyle x}$ under the assumption that a vector of parameters ${\textstyle \theta }$ is true.

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

In contrast, in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, via a posterior probability ${\textstyle f(\theta \mid x)}$ calculated via Bayes' rule.^[4]