Intrinsic Cramér–Rao bounds for distributed Bayesian estimator

Abstract

This paper addresses the intrinsic Cramér–Rao bounds (CRBs) for a distributed Bayesian estimator whose states and measurements are on Riemannian manifolds. As Euclidean-based CRBs for recursive Bayesian estimator are no longer applicable to general Riemannian manifolds, the bounds need redesigning to accommodate the non-zero Riemannian curvature. To derive the intrinsic CRBs, we append a coordination step to the recursive Bayesian procedure, where the proposed sequential steps are prediction, measurement and coordination updates. In the coordination step, the estimator minimises the Kullback–Liebler divergence to obtain the consensus of multiple probability density functions (PDFs). Employing the PDFs from those steps together with the affine connection on manifolds the Fisher Information Matrix (FIM) and the curvature terms of the corresponding intrinsic bounds are derived. Subsequently, the design of a distributed estimator for Riemannian information manifold with Gaussian distribution – referred to as distributed Riemannian Kalman filter – is also presented to exemplify the application of the proposed intrinsic bounds. Finally, simulations utilising the designed filter for a distributed quaternionic estimation problem verifies that the covariance matrices of the filter are never below the formulated intrinsic CRBs.