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We use a Bayesian version of the Cramer-Rao lower bound due to van Trees to give an elementary proof that the limiting distibution 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