Statistical Science

On the history of maximum likelihood in relation to inverse probability and least squares

Anders Hald

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It is shown that the method of maximum likelihood occurs in rudimentary forms before Fisher [Messenger of Mathematics 41 (1912) 155–160], but not under this name. Some of the estimates called “most probable” would today have been called “most likely.” Gauss [Z. Astronom. Verwandte Wiss. 1 (1816) 185–196] used invariance under parameter transformation when deriving his estimate of the standard deviation in the normal case. Hagen [Grundzüge der Wahrschein­lichkeits­Rechnung, Dümmler, Berlin (1837)] used the maximum likelihood argument for deriving the frequentist version of the method of least squares for the linear normal model. Edgeworth [J. Roy. Statist. Soc. 72 (1909) 81–90] proved the asymptotic normality and optimality of the maximum likelihood estimate for a restricted class of distributions. Fisher had two aversions: noninvariance and unbiasedness. Replacing the posterior mode by the maximum likelihood estimate he achieved invariance, and using a two­stage method of maximum likelihood he avoided appealing to unbiasedness for the linear normal model.

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Statist. Sci. Volume 14, Number 2 (1999), 214-222.

First available in Project Euclid: 24 December 2001

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Chauvenet confidence limits credible limits Edgeworth Encke Fisher Gauss Gosset Hagen invariance inverse probability Laplace least squares likelihood limits linear normal model maximum likelihood Merriman posterior mode reparameterization t­distribution two­stage maximum likelihood method unbiasedness


Hald, Anders. On the history of maximum likelihood in relation to inverse probability and least squares. Statist. Sci. 14 (1999), no. 2, 214--222. doi:10.1214/ss/1009212248.

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