The Bernstein-von Mises theorem is a result that links Bayesian inference with Frequentist inference. It assumes there is some true probabilistic process that generates the observations, as in frequentism, and then studies the quality of Bayesian methods of recovering that process, and making uncertainty statements about that process. In particular, it states that Bayesian credible sets of a certain credibility level will asymptotically be confidence sets of confidence level, which allows for the interpretation of Bayesian credible sets.
In case the maximum likelihood estimator is an efficient estimator, we can plug this in, and we recover a common, more specific, version of the Bernstein-von Mises theorem.
Implications
The most important implication of the Bernstein-von Mises theorem is that the Bayesian inference is asymptotically correct from a frequentist point of view. This means that for large amounts of data, one can use the posterior distribution to make, from a frequentist point of view, valid statements about estimation and uncertainty.
In case of a misspecified model, the posterior distribution will also become asymptotically Gaussian with a correct mean, but not necessarily with the Fisher information as the variance. This implies that Bayesian credible sets of level cannot be interpreted as confidence sets of level. In the case of nonparametric statistics, the Bernstein-von Mises theorem usually fails to hold with a notable exception of the Dirichlet process. 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; however, this depends on allowing a very broad range of possible priors. In practice, the priors used typically in research do have the desirable property even with an infinite countable probability space. Different summary statistics such as the mode and mean may behave differently in the posterior distribution. In Freedman's examples, the posterior density and its mean can converge on the wrong result, but the posterior mode is consistent and will converge on the correct result.
Quotations
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."