The Riemannian geometry of certain parameter estimation problems with singular Fisher information matrices

Many parametric statistical models suffer from "intrinsic ambiguities" in the sense that the distribution of the observation vector is invariant to smooth, structured changes in the model's parameters. The fact that certain members of the parametric statistical family are locally indistinguishable makes the Fisher information matrix (FIM) associated with the given statistical model singular. We examine such degenerate deterministic parameter estimation problems from a Riemannian geometric perspective. We start by replacing the original (ambiguous) parameter set by a lower-dimensional Riemannian (non-ambiguous) parameter set. The new parameter set comes in the form of a quotient space and is obtained by identifying equivalent family members in the initial parameterization. We specialize recently developed extensions of the Cramer-Rao bound (CRB) for the Riemannian setup to this particular setting. This offers a re-interpretation of the CRB inequality involving the pseudo-inverse of the FIM. Also, we present a lower bound on the variance (computed with respect to the geodesic distance) of unbiased estimators taking values in the quotient space. Geometrically, this corresponds to a fundamental limit on the capability of these estimators in discriminating adjacent parameter equivalence classes in the original problem parameterization. A numerical example involving the blind identification of single-input multiple-output (SIMO) channels driven by a Gaussian source is worked out.