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In this paper, we propose to develop a new Cram\'er-Rao Bound (CRB) when theparameter to estimate lies in a manifold and follows a prior distribution. Thisderivation leads to a natural inequality between an error criteria based ongeometrical properties and this new bound. This main contribution isillustrated in the problem of covariance estimation when the data follow aGaussian distribution and the prior distribution is an inverse Wishart.Numerical simulation shows new results where the proposed CRB allows to exhibitinteresting properties of the MAP estimator which are not observed with theclassical Bayesian CRB.

Natural Bayesian Cram\'er-Rao Bound with an Application to Covariance Estimation