# Maximum likelihood

 related topics {math, number, function} {rate, high, increase} {household, population, female}

Maximum likelihood estimation (MLE) is a popular statistical method used for fitting a statistical model to data, and providing estimates for the model's parameters.

The method of maximum likelihood corresponds to many well-known estimation methods in statistics. For example, one may be interested in the heights of adult female giraffes, but be unable due to cost or time constraints, to measure the height of every single giraffe in a population. Assuming that the heights are normally (Gaussian) distributed with some unknown mean and variance, the mean and variance can be estimated with MLE while only knowing the heights of some sample of the overall population. MLE would accomplish this by taking the mean and variance as parameters and finding the specific values for these parameters that produces the distribution most likely to have produced the observed results (see examples below).

In general, for a fixed set of data and underlying probability model, the method of maximum likelihood selects values of the model parameters that produce the distribution most likely to have resulted in the observed data (i.e. the parameters that maximize the likelihood function). Maximum likelihood estimation gives a unified approach to estimation, which is well-defined in the case of the normal distribution and many other problems. However, in some complex problems, difficulties do occur: in such problems the maximum-likelihood estimators may be unsuitable or may not even exist.