A Family of Bayesian Cram\'er-Rao Bounds, and Consequences for Log-Concave Priors

Under minimal regularity assumptions, we establish a family of information-theoretic Bayesian Cram\'er-Rao bounds, indexed by probability measures that satisfy a logarithmic Sobolev inequality. This family includes as a special case the known Bayesian Cram\'er-Rao bound (or van Trees inequality), and its less widely known entropic improvement due to Efroimovich. For the setting of a log-concave prior, we obtain a Bayesian Cram\'er-Rao bound which holds for any (possibly biased) estimator and, unlike the van Trees inequality, does not depend on the Fisher information of the prior.