Likelihood function

In statistical inference, the likelihood function quantifies the plausibility of parameter values characterizing a statistical model in light of observed data. Its most typical usage is to compare possible parameter values (under a fixed set of observations and a particular model), where higher values of likelihood are preferred because they correspond to more probable parameter values. While it is derived from the joint probability distribution on the observed data, it is not necessarily a measure of probability because it can return values larger than 1, especially when considering statistical models involving continuous random variables. Since it can be used to choose parameter values, it is a common utility function in situations that consider randomness. In maximum likelihood estimation, the arg max (over the parameter ) of the likelihood function serves as a point estimate for , while the Fisher information (often approximated by the likelihood's Hessian matrix) indicates the estimate's precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' rule. The likelihood function, parameterized by a (possibly multivariate) parameter , is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function where is a realization of the random variable , the likelihood function is often written In other words, when is viewed as a function of with fixed, it is a probability density function, and when viewed as a function of with fixed, it is a likelihood function. The likelihood function does not specify the probability that is the truth, given the observed sample . Such an interpretation is a common error, with potentially disastrous consequences (see prosecutor's fallacy). Let be a discrete random variable with probability mass function depending on a parameter .