Maximum Likelihood

Maximum likelihood estimation (MLE) is a popular statistical method used to calculate the best way of fitting a mathematical model to some data. Modeling real world data by estimating maximum likelihood offers a way of tuning the free parameters of the model to provide an optimum fit.The method was pioneered by geneticist and statistician Sir R. A. Fisher between 1912 and 1922. It has widespread applications in various fields, including:linear models and generalized linear models are commonly fit by maximum likelihood.econometrics and hypothesis testing in medical research.time-delay of arrival (TDOA) in acoustic detection.data modeling in nuclear and particle physics.origin/destination and path-choice modeling in transport networks. The method of maximum likelihood corresponds to many well-known estimation methods in statistics. For example, suppose you are interested in the heights of Americans. You have a sample of some number of Americans, but not the entire population, and record their heights. Further, you are willing to assume that heights are normally distributed with some unknown mean and variance. The sample mean is then the maximum likelihood estimator of the population mean, and the sample variance is a close approximation to the maximum likelihood estimator of the population variance (see examples below).
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An common alternative to the least squares loss function is to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function; the term maximum likelihood was first used by Fisher, 1922a). These functions are typically used when fitting non-linear models. In most general terms, the likelihood function is defined as:

The method of maximum likelihood (the term first used by Fisher, 1922a) is a general method of estimating parameters of a population by values that maximize the likelihood (L) of a sample. The likelihood L of a sample of n observations x1, x2, ..., xn, is the joint probability function p(x1, x2, ..., xn) when x1, x2, ..., xn are discrete random variables. If x1, x2, ..., xn are continuous random variables, then the likelihood L of a sample of n observations, x1, x2, ..., xn, is the joint density function f(x1, x2, ..., xn).
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Maximum likelihood
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