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

翻译：本文提出了一种新的克拉美-罗下界（CRB），适用于估计参数位于流形上且服从先验分布的情形。该推导在基于几何性质的误差准则与这一新下界之间建立了自然不等式。这一主要贡献通过以下问题得以阐明：当数据服从高斯分布且先验分布为逆威沙特分布时的协方差估计。数值仿真展示了新结果，其中所提出的CRB能够揭示MAP估计器在经典贝叶斯CRB下未观察到的有趣性质。