# Bayesian Cram'er-Rao Bound Estimation with Score-Based Models

arXiv: https://arxiv.org/abs/2309.16076 · HTML: https://mlsanity.com/paper/2309.16076/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 13 of its 54 references that are in the map.
- First posted on arXiv 28 Sep 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Learning to Estimate Without Bias](https://mlsanity.com/paper/2110.12403/) (2021)
- [Empirical non-parametric estimation of the Fisher Information](https://mlsanity.com/paper/1408.1182/) (2014)
- [Fundamental Limits in Multi-image Alignment](https://mlsanity.com/paper/1602.01541/) (2016)
- [Rapid and deterministic estimation of probability densities using scale-free field theories](https://mlsanity.com/paper/1312.6661/) (2013)
- [Optimal Experiment Design for Magnetic Resonance Fingerprinting: Cram'er-Rao Bound Meets Spin Dynamics](https://mlsanity.com/paper/1710.08062/) (2017)
- [Learning to Bound: A Generative Cram'er-Rao Bound](https://mlsanity.com/paper/2203.03695/) (2022)

## Built on by (most cited first, 3 of 3)

- [Optimal Defenses Against Gradient Reconstruction Attacks](https://mlsanity.com/paper/2411.03746/) (2024)
- [Fundamental Recovery Bounds for SPAD Signals under Stationary Flux](https://mlsanity.com/paper/2601.07599/) (2026)
- [Latent Score-Based Bayesian Cram'er-Rao Bound Estimation for High-Dimensional Imaging Systems](https://mlsanity.com/paper/2610.03956/) (2026)

## Abstract (arXiv)

The Bayesian Cram'er-Rao bound (CRB) provides a lower bound on the mean square error of any Bayesian estimator under mild regularity conditions. It can be used to benchmark the performance of statistical estimators, and provides a principled metric for system design and optimization. However, the Bayesian CRB depends on the underlying prior distribution, which is often unknown for many problems of interest. This work introduces a new data-driven estimator for the Bayesian CRB using score matching, i.e., a statistical estimation technique that models the gradient of a probability distribution from a given set of training data. The performance of the proposed 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 based on the results from empirical process theory, including classical bounds and recently introduced techniques for characterizing neural networks. We illustrate the performance of the proposed estimator with two application examples: a signal denoising problem and a dynamic phase offset estimation problem in communication systems.