Maximum likelihood estimation
A method maximizing likelihood to estimate distribution parameters.
Maximum likelihood estimation (MLE) is a method in statistics for estimating the parameters of an assumed probability distribution given observed data. It works by maximizing a likelihood function, making the observed data most probable under the chosen statistical model. The resulting point in the parameter space is called the maximum likelihood estimate. The method has become a dominant means of statistical inference due to its intuitive logic and flexibility.
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- Maximum likelihood estimation (MLE) method for parameter estimation
Lore & Background
Maximum likelihood estimation models a set of observations as a random sample from an unknown joint probability distribution expressed in terms of parameters. The goal is to determine the parameters for which the observed data have the highest joint probability. The likelihood function is defined as the joint density evaluated at the observed data sample, and for independent random variables it becomes the product of univariate density functions. The maximum likelihood estimate is the value in the parameter space that maximizes this function.
Reader's Guide
Maximum likelihood estimation is a foundational technique in statistical inference. Its significance lies in its ability to provide a principled and flexible way to estimate parameters from data. The method is intuitive: it selects the parameter values that make the observed data most probable. In practice, the natural logarithm of the likelihood function is often used because it is monotonic and easier to work with. The method has connections to other inferential frameworks: from a Bayesian perspective, MLE is generally equivalent to maximum a posteriori estimation with a uniform prior; in frequentist inference, it is a special case of an extremum estimator. The ordinary least squares estimator for linear regression, under normal errors with equal variance, is an example of an MLE that can be solved analytically. The method's flexibility and intuitive appeal have made it a dominant tool across many fields of science and engineering.
Did You Know?
- Maximum likelihood estimation maximizes a likelihood function so that, under the assumed statistical model, the observed data is most probable.
- If the likelihood function is differentiable, the derivative test for finding maxima can be applied.
- From the perspective of Bayesian inference, MLE is generally equivalent to maximum a posteriori estimation with a prior distribution that is uniform in the region of interest.
- In frequentist inference, MLE is a special case of an extremum estimator, with the objective function being the likelihood.
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