Likelihood function

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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 parlance, "likelihood" is usually a synonym for "probability" but, in statistical usage, a clear technical distinction is made: the probability of some observed outcomes given a set of parameter values is referred to as the likelihood of the set of parameter values given the observed outcomes. For example in technical parlance, one may ask "If I were to flip a fair coin 100 times, what is the probability of it landing heads-up every time?" or "Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?" but it would be improper to switch "likelihood" and "probability" in the two sentences.

Mathematically, writing X for the set of observed data and Θ for the set of parameter values, the expression P(X | Θ), the probability of X given Θ, can be interpreted as the expression L(Θ | X), the likelihood of Θ given X. The interpretation of L(Θ | X) as a function of Θ is especially obvious when X is fixed and Θ is allowed to vary.

Generally, L(Θ | X) is permitted to be any positive multiple of P(X | Θ). More precisely then, a likelihood function is any representative from an equivalence class of functions,

where the constant of proportionality α > 0 is not permitted to depend upon Θ. In particular, the numerical value L(Θ | X) alone is immaterial; all that matters are likelihood ratios, such as those of the form

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

A. W. F. Edwards defined support to be the natural logarithm of the likelihood ratio, and the support function as the natural logarithm of the likelihood function (i.e. same as the log-likelihood, see below).[1] However, there is potential for confusion with the mathematical meaning of 'support', 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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