Bayesian Cramér-Rao Bound Estimation with Score-Based Models

The Bayesian Cramér-Rao bound (CRB) provides a lower bound on the mean square error of any estimator in Bayesian inference under mild technical conditions. It benchmarks the performance of statistical estimation and can also serve as a principled metric for system design and optimization. However, it is difficult to calculate the Bayesian CRB without explicit knowledge of the prior distribution. In this paper, we introduce a novel data-driven method for Bayesian CRB estimation, leveraging state-of-the-art score estimation and deep generative modeling techniques. We show that the proposed estimator is asymptotically consistent and illustrate its performance in a denoising problem.