We use a Bayesian version of the Cramér-Rao lower bound due to van Trees to give an elementary proof that the limiting distribution of any regular estimator cannot have a variance less than the classical information bound, under minimal regularity conditions. We also show how minimax convergence rates can be derived in various non- and semi-parametric problems from the van Trees inequality. Finally we develop multivariate versions of the inequality and give applications.
"Applications of the van Trees inequality: a Bayesian Cramér-Rao bound." Bernoulli 1 (1-2) 59 - 79, March 1995.