Approximating Posterior Cramer-Rao Bounds for Nonlinear Filtering Problems Using Gaussian Mixture Models

The posterior Cramer-Rao bound (PCRB) is a fundamental tool to assess the accuracy limit of the Bayesian estimation problem. In this article, we propose a novel framework to compute the PCRB for the general nonlinear filtering problem with additive white Gaussian noise. It uses the Gaussian mixture model to represent and propagate the uncertainty contained in the state vector and uses the Gauss-Hermite quadrature rule to compute mathematical expectations of vector-valued nonlinear functions of the state variable. The detailed pseudocodes for both the small and large component covariance cases are also presented. Three numerical experiments are conducted. All of the results show that the proposed method has high accuracy and it is more efficient than the plain Monte Carlo integration approach in the small component covariance case.