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

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