In statistics, the likelihood function (often simply the likelihood) is a function of the parameters of a statistical model that plays a key role in statistical inference. In non-technical usage, "likelihood" is a synonym for "probability", but throughout this article only the technical definition is used. Informally, if "probability" allows us to predict unknown outcomes based on known parameters, then "likelihood" allows us to estimate unknown parameters based on known outcomes.

In a sense, likelihood works backwards from conditional probability. Forward reasoning: given parameter B, use the conditional probability P(A|B) to reason about outcome A. This is formalized in Bayes' theorem:

Backward reasoning: given outcome A, use the likelihood function L(B|A) to reason about parameter B. Formally, a likelihood function is a conditional probability function considered as a function of its second argument, with its first argument held fixed:

and also any other function proportional to such a function. That is, the likelihood function for B is the equivalence class of functions

for any constant of proportionality α > 0. The numerical value L(b | A) alone is immaterial; all that matters are likelihood ratios of the form

which are invariant with respect to the constant of proportionality α.

A. W. F. Edwards defined support as the natural logarithm of the likelihood ratio, and the support function as the natural logarithm of the likelihood function.[1] There is potential for confusion with the mathematical meaning of 'support', however, and this terminology is not widely used outside Edwards' main applied field of phylogenetics.

For more about making inferences via likelihood functions, see also the method of maximum likelihood, and likelihood-ratio testing.

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