maximum likelihood: a standalone construct

R. A. Fisher named and formalised the method in his 1922 paper On the mathematical foundations of theoretical statistics and developed its properties further in 1925, coining the term likelihood to keep it separate from probability and proposing that the best estimate of an unknown quantity is the value that makes the observed data most probable. The idea had precursors in the least-squares work of Gauss and in Edgeworth's writing of the 1880s, but Fisher gave it a general and self-contained justification. In plain terms, maximum likelihood estimation asks: given a statistical model with unknown quantities, for example the average and spread of a population, which values of those quantities would have made the data we actually collected the most likely to occur, and it takes those values as the estimate. The number being maximised is the likelihood function, the probability of the observed data read as a function of the unknown parameters, and in practice one usually maximises its logarithm because sums are easier to handle than products. What makes maximum likelihood a landmark is its generality together with its large-sample behaviour: under broad conditions the estimates are consistent, meaning they converge on the true value as data accumulate, and asymptotically efficient, meaning no competing method does reliably better in large samples, which is why it underlies parameter estimation across statistics, econometrics, genetics, and machine learning.[1][2][3]

Unit

A point estimate of a model parameter or parameter vector, expressed in the units of that parameter

Recovery means

Not applicable

Measured by

Obtained by maximising the log-likelihood of the observed data over the parameter space, analytically where a closed form exists or by numerical optimisation such as Newton-Raphson or the EM algorithm