Studies in the history of probability and statistics: XI. Daniel Bernoulli on maximum likelihood

1. Almost as soon as the calculus of probabilities began to take a definite shape mathematicians were concerned with the use of probabilistic ideas in reconciling discrepant observations. James Bernoulli's Ars Coniectandi was published in 1713. Within 9 years we find Roger Cotes (1722), in a work on the estimation of errors in trigonometrical mensuration, discussing what would nowadays be described as an estimation problem in a plane. Let p, q, r, s be four different determinations of a point o, with weights P, Q, R, S which are inversely proportional to distance from o (pondera reciproce proportionalia spatiis evagationum). Put weights P t secondly, they must be finite in range. Lagrange reproduced Simpson's work without acknowledgement in a memoir published between 1770 and 1773, but Lagrange's contributions are more of analytical than of probabilistic interest. 4. Daniel Bernoulli was born in 1700 and lived to be 82. Throughout his productive life he made contributions to the theory of probability and although his mathematical methods are not now of much importance, the originality of his thinking on such matters as moral expectation entitles him to a permanent place among the founders of the subject. In particular, the memoir on maximum likelihood reproduced in the following pages is astonishingly in advance of its time. The author was 78 when it was published and it appears that he excogitated the basic ideas for himself without reference to previous writings. The memoir may, in actual fact, have been written rather earlier. Laplace's