The Epic Story of Maximum Likelihood

摘要

At a superficial level, the idea of maximum likelihood must be prehistoric:early hunters and gatherers may not have used the words ``method of maximumlikelihood'' to describe their choice of where and how to hunt and gather, butit is hard to believe they would have been surprised if their method had beendescribed in those terms. It seems a simple, even unassailable idea: Who wouldrise to argue in favor of a method of minimum likelihood, or even mediocrelikelihood? And yet the mathematical history of the topic shows this ``simpleidea'' is really anything but simple. Joseph Louis Lagrange, Daniel Bernoulli,Leonard Euler, Pierre Simon Laplace and Carl Friedrich Gauss are only some ofthose who explored the topic, not always in ways we would sanction today. Inthis article, that history is reviewed from back well before Fisher to the timeof Lucien Le Cam's dissertation. In the process Fisher's unpublished 1930characterization of conditions for the consistency and efficiency of maximumlikelihood estimates is presented, and the mathematical basis of his threeproofs discussed. In particular, Fisher's derivation of the informationinequality is seen to be derived from his work on the analysis of variance, andhis later approach via estimating functions was derived from Euler's Relationfor homogeneous functions. The reaction to Fisher's work is reviewed, and somelessons drawn.