The Bayesian Cram\'er-Rao bound (CRB) provides a lower bound on the error of any Bayesian estimator under mild regularity conditions. It can be used to benchmark the performance of estimators, and provides a principled design metric for guiding system design and optimization. However, the Bayesian CRB depends on the prior distribution, which is often unknown for many problems of interest. This work develops a new data-driven estimator for the Bayesian CRB using score matching, a statistical estimation technique, to model the prior distribution. The performance of the estimator is analyzed in both the classical parametric modeling regime and the neural network modeling regime. In both settings, we develop novel non-asymptotic bounds on the score matching error and our Bayesian CRB estimator. Our proofs build on results from empirical process theory, including classical bounds and recently introduced techniques for characterizing neural networks, to address the challenges of bounding the score matching error. The performance of the estimator is illustrated empirically on a denoising problem example with a Gaussian mixture prior.

翻译：贝叶斯克拉美-罗界（CRB）在温和的正则条件下为任何贝叶斯估计量的误差提供了下界。它可用于评估估计量的性能，并为指导系统设计与优化提供了理论上的设计度量。然而，贝叶斯CRB依赖于先验分布，而许多关注问题中的先验分布通常是未知的。本文利用评分匹配这一统计估计技术来建模先验分布，提出了一种新的数据驱动型贝叶斯CRB估计量。我们在经典参数化建模场景和神经网络建模场景下均分析了该估计量的性能。在这两种设定中，我们针对评分匹配误差及所提出的贝叶斯CRB估计量，推导了新的非渐近界。我们的证明建立在经验过程理论结果的基础上，包括经典界以及近期引入的用于刻画神经网络的技术，以解决评分匹配误差有界性证明中的挑战。最后，通过一个具有高斯混合先验的去噪问题实例，从经验上验证了该估计量的性能。