The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood function indicates which parameter values are more likely than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as instead of , to emphasize that it is to be understood as a function of the parameters instead of the random variable . In maximum likelihood estimation, the arg max 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. Definition 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). Discrete probability distribution Let be a discrete random variable with probability mass function depending on a parameter . Then the function considered as a function of , is the likelihood function, given the outcome of the random variable . Sometimes the probability of "the value of for the parameter value " is written as or . The likelihood is the probability that a particular outcome is observed when the true value of the parameter is , equivalent to the probability mass on ; it is not a probability density over the parameter . The likelihood, , should not be confused with , which is the posterior probability of given the data . Given no event (no data), the likelihood is 1; any non-trivial event will have a lower likelihood. Example Consider a simple statistical model of a coin flip: a single parameter that expresses the