The posterior Cramér–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.

中文翻译：

---

使用高斯混合模型近似求解非线性滤波问题的后Cramér-Rao界

后Cramér-Rao界（PCRB）是评估贝叶斯估计问题的准确性极限的基本工具。在本文中，我们提出了一个新颖的框架来计算具有加性高斯白噪声的一般非线性滤波问题的PCRB。它使用高斯混合模型来表示和传播状态向量中包含的不确定性，并使用高斯-赫尔姆特正交规则来计算状态变量的向量值非线性函数的数学期望。还介绍了针对小分量协方差情况和大分量协方差情况的详细伪代码。进行了三个数值实验。所有结果表明，在小分量协方差情况下，该方法具有较高的准确性，并且比普通的蒙特卡洛积分方法更有效。