## What is Likelihood?

Aliases: *likelihood function*

In frequentist inference the likelihood is a quantity proportional to the probability that, from a population having a particular value of θ, a sample having the observed value x_{0}, should be obtained. Likelihood, being the outcome of a likelihood function thus defined, describes the plausibility, under a certain statistical model (the null hypothesis in hypothesis testing), of a certain parameter value after observing a particular outcome. Formally: L(θ; x_{0}) ∝ ƒ(x_{0}; θ), ∀θ∈Θ .

Likelihood is central to parametric statistical inference. The likelihood is a basis for the likelihood ratio test: a uniformly most powerful test for comparing two point hypotheses. It is also the basis for the maximum likelihood estimate.

In practice one often calculates the natural logarithm of the likelihood function (log-likelihood) as being more convenient (easier to differentiate). The fact that a logarithm is strictly increasing is useful when calculating maximum likelihood: log-likelihood reaches the maximum at the same point as the likelihood.

In Bayesian inference likelihood has a different meaning as the conditional probability P(E|H): the probability of evidence E given H, conditioning on an non-observable entity. The likelihood, being a conditional density can be multiplied by the prior probability density of the parameter and then normalized, producing the posterior probability.