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A general class of Bayesian lower bounds when the underlying loss function isa Bregman divergence is demonstrated. This class can be considered as anextension of the Weinstein--Weiss family of bounds for the mean squared errorand relies on finding a variational characterization of Bayesian risk. Theapproach allows for the derivation of a version of the Cram\'er--Rao bound thatis specific to a given Bregman divergence. The new generalization of theCram\'er--Rao bound reduces to the classical one when the loss function istaken to be the Euclidean norm. The effectiveness of the new bound is evaluatedin the Poisson noise setting and the Binomial noise setting.

A Cram\'er-Rao Type Bound for Bayesian Risk with Bregman Loss