Conference Agenda

We review the theoretical progress and the history of the reception of Brownian motion in the first quarter of the 20th century.

The so-called "Bernstein-von-Mises Theorem" is a statement on the asymptotic normality of the posterior of a distribution parameter under relatively general conditions. It provides a pivotal link between "frequentist" and "Bayesian" statistics.

The history of this theorem starts with Laplace, who published on the binomial case under the assumption of a uniform prior in 1774. Later on, Laplace generalized his discussion to multinomial distributions and even with respect to means of continuous distributions.

We find several attempts to generalize Laplace's considerations by introducing non-uniform priors during the 19th century.

The first more or less rigorous proof (from the point of view of modern analysis) in the multinomial case is due to von Mises (1919). Interestingly, Neyman, one of the leading propontents of "frequentist" versus "Bayesian" statistics, proved a very similar assertion in 1929, initially without knowledge of von Mises's work, as he admitted.

The name "Bernstein-von-Mises Theorem" seems to be essentially due to LeCam, who gave a fairly general version of this theorem in 1953, and who repeatedly cited Bernstein's and von Mises's contributions in that paper and in later works. With respect to Bernstein, the reason for this naming remains unclear, however. According to all we know, Bernstein only treated the binomial case, and did not go much beyond Laplace in this respect.