In Bayesian inference, the Bernstein–von Mises theorem provides the basis for the important result that the posterior distribution for unknown quantities in any problem is effectively independent of the prior distribution (assuming it obeys Cromwell's rule) once the amount of information supplied by a sample of data is large enough.
The theorem is named after Richard von Mises and S. N. Bernstein even though the first proper proof was given by Joseph Leo Doob in 1949 for random variables with finite probability space. Later Lucien Le Cam, his PhD student Lorraine Schwarz, David A. Freedman and Persi Diaconis extended the proof under more general assumptions. A remarkable result was found by Freedman in 1965: the Bernstein-von Mises theorem does not hold almost surely if the random variable has an infinite countable probability space.
The statistician A. W. F. Edwards has remarked, "It is sometimes said, in defence of the Bayesian concept, that the choice of prior distribution is unimportant in practice, because it hardly influences the posterior distribution at all when there are moderate amounts of data. The less said about this 'defence' the better."[1]