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Maximum likelihood

In statistics, the method of maximum likelihood, pioneered by geneticist/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
For example, in a large population of voters, the proportion p who will vote "yes" is unobservable, and is to be estimated based on a political opinion poll. A sample of n voters is chosen randomly, and it is observed that x of those n voters will vote "yes". Then the likelihood function is
The value of p that maximizes L(p) is the maximum-likelihood estimate of p. By finding the root of the first derivative one will obtain x/n as the maximum-likelihood estimate. In this case, as in many other cases, it is much easier to take the logarithm of the likelihood function before finding the root of the derivative:
Taking the logarithm of the likelihood is so common that the term log-likelihood is commonplace among statisticians. The log-likelihood is closely related to information entropy.

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) x/n observed in a particular case is an estimate; the random variable (Capital) X/n 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 sometimes 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.

However, the bias of maximum-likelihood estimators can be substantial. Consider a case where n tickets numbered from 1 through to n are placed in a box and one is selected at random, giving a value X. If n is unknown, then the maximum-likelihood estimator of n is X, even though the expectation of X is only n/2; we can only be certain that n is at least X and is probably more.