Natural Bayesian Cramér-Rao Bound with an Application to Covariance
  Estimation
Florent Bouchard Alexandre Renaux Guillaume Ginolhac Arnaud Breloy
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.