Encyclopedia > Maximum likelihood

  Article Content

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
<math>L(p)=P(X=x\mid p).</math>
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
<math>L(p)={n \choose x}p^x(1-p)^{n-x}.</math>
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:
<math>\frac{x}{p}-\frac{n-x}{1-p}=0</math>
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 typically better than unbiased estimators. 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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Europium

... in the spectrums of the sun and certain stars. Compounds Europium compounds include: Fluorides EuF2[?] EuF3[?] Chlorides EuCl2[?] EuCl3[?] ...