Geometry of the Cramer-Rao bound

The Fisher information matrix determines how much information is given by a measurement about the parameters that index the underlying probability distribution. This paper assumes that the parameters structure the mean value vector in a multivariate normal distribution. The Fisher matrix is then a Gramian constructed from the sensitivity vectors that characterize the first-order variation in the mean with respect to the parameters. The inverse of the Fisher matrix has several geometrical properties that bring insight into the problem of identifying multiple parameters. The angle between a given sensitivity vector and the linear subspace spanned by all others determines the variance bound for identifying a given parameter. Similarly, the covariance for identifying the linear influence of two different subsets of parameters depends on the principal angles between the linear subspaces spanned by the sensitivity vectors for the respective subsets