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 .
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In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.
In statistics, the likelihood principle is the proposition that, given a statistical model, all the evidence in a sample relevant to model parameters is contained in the likelihood function. A likelihood function arises from a probability density function considered as a function of its distributional parameterization argument.
Bayesian inference (ˈbeɪziən or ˈbeɪʒən ) is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law.
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