In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model. Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics. In informal contexts, "likelihood" is often used as a synonym for "probability." But in statistical usage, a distinction is made depending on the roles of the outcome or parameter. Probability is used when describing a function of the outcome given a fixed parameter value. Likelihood is used when describing a function of a parameter given an outcome.
The likelihood of a set of parameter values, θ, given outcomes x, is equal to the probability of those observed outcomes given those parameter values, that is
The likelihood function is defined differently for discrete and continuous probability distributions.
Let X be a random variable with a discrete probability distribution p depending on a parameter θ. Then the function