Maximum likelihoodIn statistics, the method of maximum likelihood, pioneered by geneticis/statistician Sir Ronald A. Fisher, is a method of point estimation, that uses as an estimate of an unobservable population parameter the member of the parameter space that maximizes the likelihood function. For the moment let p denote the unobservable population parameter to be estimated. Let X denote the random variable observed (which in general will not be scalar-valued, but often will be a vector of probabilistically independent scalar-valued random variables. The probability of an observed outcome X=x (this is case-sensitive notation!), or the value at (lower-case) x of the probability density function of the random variable (Capital) X, as a function of p with x held fixed is the likelihood function
If we replace the lower-case x with capital X then we have, not the observed value in a particular case, but rather a random variable, which, like all random variables, has a probability distribution. The value (lower-case) observed in a particular case is an estimate is an estimator. The statistician may take the nature of the probability distribution of the estimator to indicate how good the estimator is; in particular it is desirable that the probability that the estimator is far from the parameter p be small. Maximum-likelihood estimators are typically better than unbiased estimatorss. They also have a property called "functional invariance" that unbiased estimators lack: for any function f, the maximum-likelihood estimator of f(p) is f(T), where T is the maximum-likelihood estimator of p.