Anantharam, Jog, and Nair recently unified the Shannon–Stam inequality and the entropic form of the Brascamp–Lieb inequalities under a common inequality. They left open the problems of extremizability and characterization of extremizers. Both questions are resolved in the present paper. Separately, we show that the Anantharam–Jog–Nair inequality may be derived as a corollary of the Brascamp–Lieb inequalities, establishing a formal equivalence between the two.

We establish a family of sharp entropy inequalities with Gaussian extremizers. These inequalities hold for certain dependent random variables, namely entropy-maximizing couplings subject to information constraints. Several well-known results, such as the Zamir-Feder and Brunn-Minkowski inequalities, follow as special cases.

Under minimal regularity assumptions, we establish a family of information-theoretic Bayesian Cramér-Rao bounds, indexed by probability measures that satisfy a logarithmic Sobolev inequality. This family includes as a special case the known Bayesian Cramér-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 Cramér-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.