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Perhaps the strongest criticisms of inverse probability, although indirect, are implied by the fact that Laplace after 1811 and Gauss after 1816 based their theory of linear minimum variance estimation on direct probability. Nevertheless inverse probability continued to be used; only few of the critics went as far as to reject the theory completely.

It is a remarkable fact that Laplace simultaneously worked on statistical inference by inverse probability, 1774–1786, and by direct probability, 1776–1781. In 1776 he derived the distribution of the arithmetic mean for continuous rectangularly distributed variables by repeated applications of the convolution formula. In his comprehensive 1781 paper he derived the distribution of the mean for independent variables having an arbitrary, piecewise continuous density. As a special case he found the distribution of the mean for variables with a polynomial density, thus covering the rectangular, triangular, and parabolic cases. In principle he had solved the problem but his formula did not lead to manageable results because the densities then discussed resulted in complicated mathematical expressions and cumbersome numerical work even for small samples. He had thus reached a dead end and it was not until 1810 that he returned to the problem, this time looking for an approximative solution, which he found by means of the central limit theorem.

Gauss’s paper on the “Theory of the Combination of Observations Leading to Minimum Errors” was published in three parts, the first two in 1823 and a supplement in 1828.

Gauss did not take the trouble to rewrite his first proof of the method of least squares in terms of direct probability. This task was carried out by astronomers and geodesists writing elementary textbooks on the method of least squares. They found Gauss’s second proof too cumbersome for their readers and did not need the generalization involved because the measurement errors encountered in their fields were in most cases nearly normally distributed. As far as error theory is concerned, they realized that the principle of inverse probability was superfluous. The method of maximizing the posterior density could be replaced by the method of maximizing the density (|) of the observations, which would lead to the same estimates because (|) α (|). This method has an obvious intuitive appeal and goes back to Daniel Bernoulli and Lambert; see Todhunter [261], pp. 236–237) and Edwards [50]. Todhunter writes:

During the period from about 1830 to 1900 statistical methods gradually came to be used in fields other than the natural sciences. Three pioneers were Quetelet (anthropometry, social sciences), Fechner (psychophysics, factorial experiments), and Galton (genetics, biology, regression, correlation). Applications also occurred in demography, insurance, economics, and medicine. The normal distribution, originally introduced for describing the variation of errors of measurement, was used by Quetelet and Galton to describe the variation of characteristics of individuals. However, in many of the new applications skew distributions were encountered that led to the invention of systems of nonnormal distributions.

We employ the modern notation of the multivariate normal distribution. Let = (, . . ., ) be a vector of normally correlated random variables with density where is the vector of expectations and a positive definite × matrix.

In the present chapter it is assumed that , . . ., are independently and normally distributed ().

Ronald Aylmer Fisher (1890–1962) was born in London as the son of a prosperous auctioneer whose business collapsed in 1906, whereafter Ronald had to fend for himself. In 1909 he won a scholarship in mathematics to Cambridge University where he graduated in 1912 as a Wrangler in the Mathematical Tripos. He was awarded a studentship and spent another year in Cambridge studying statistical mechanics and quantum theory. In addition he used much of his time studying Darwin’s evolutionary theory, Galton’s eugenics, and Pearson’s biometrical work.

During the period 1912–1921 Fisher had, at least for himself, developed new concepts of fundamental importance for the theory of estimation. He had rejected inverse probability as arbitrary and leading to noninvariant estimates; instead he grounded his own theory firmly on the frequency interpretation of probability. He had proposed to use invariance and the method of maximum likelihood estimation as basic concepts and had introduced the concept of sufficiency by an important example. Thus prepared, he was ready to publish a general theory of estimation, which he did in the paper [67], “On the Mathematical Foundations of Theoretical Statistics.” For the first time in the history of statistics a framework for a frequency-based general theory of parametric statistical inference was clearly formulated.

Fisher never tired of emphasizing the importance of Gosset’s idea; likewise he often repeated his own derivation of . It was not until 1935 that he acknowledged Helmert’s [120] priority with respect to the distribution of [] = , he never mentioned that Helmert also derived .