  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.
