Daniel Bernoulli, Leonhard Euler, and Maximum Likelihood

The history of statistical concepts usually hinges on subtle questions of definition, on what one sees as a crucial element in the concept. Is the simple statement of a goal crucial? Or do we require the investigation of the implications of pursuing that goal, perhaps including the discovery of anomalies that require specification of conditions under which a claimed property holds? Or the detailed successful exploration of those conditions? Such considerations certainly arise in the case of the method of maximum likelihood. If the object of study is the modern theory of maximum likelihood, of its efficiency in large samples in a parametric setting, then an argument could be made for beginning with Edgeworth (1908–1909) (see Pratt (1976)), or Fisher (1912 or1922or1935) (see Edwards, 1974), or even Wald (1949) or Le Cam (1953). It might be thought that the question would be easy to resolve if instead of worrying about mathematical rigor and the deeper questions of inference, including the interpretation of statistical information, we only asked about the introduction of the idea of choosing, as an estimate, that value which maximizes the likelihood function, but that is not the case. Even at that level difficulties of interpretation arise. Was Gauss employing maximum likelihood in 1809 when he arrived at the method of least squares, or, as some of his development would lead you to believe, was he maximizing a posterior density with a uniform prior?