# Learned Bayesian Cram'er-Rao Bound for Unknown Measurement Models Using Score Neural Networks

arXiv: https://arxiv.org/abs/2502.00724 · HTML: https://mlsanity.com/paper/2502.00724/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 66 references that are in the map.
- First posted on arXiv 2 Feb 2025.

## Builds on (most cited first, 23 of 23)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Model-Based Deep Learning](https://mlsanity.com/paper/2012.08405/) (2020)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization](https://mlsanity.com/paper/2205.02640/) (2022)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Deep Network Approximation: Beyond ReLU to Diverse Activation Functions](https://mlsanity.com/paper/2307.06555/) (2023)
- [Smooth function approximation by deep neural networks with general activation functions](https://mlsanity.com/paper/1906.06903/) (2019)
- [Physics-Informed Computer Vision: A Review and Perspectives](https://mlsanity.com/paper/2305.18035/) (2023)
- [Estimating Density Models with Truncation Boundaries using Score Matching](https://mlsanity.com/paper/1910.03834/) (2019)
- [Conditional Adversarial Generative Flow for Controllable Image Synthesis](https://mlsanity.com/paper/1904.01782/) (2019)
- [Neural Network Architecture Beyond Width and Depth](https://mlsanity.com/paper/2205.09459/) (2022)
- [SepIt: Approaching a Single Channel Speech Separation Bound](https://mlsanity.com/paper/2205.11801/) (2022)
- [Generalized Score Matching for General Domains](https://mlsanity.com/paper/2009.11428/) (2020)
- [Empirical non-parametric estimation of the Fisher Information](https://mlsanity.com/paper/1408.1182/) (2014)
- [The Cramer-Rao Bound for Signal Parameter Estimation from Quantized Data](https://mlsanity.com/paper/2209.13182/) (2022)
- [Learning to Bound: A Generative Cram'er-Rao Bound](https://mlsanity.com/paper/2203.03695/) (2022)
- [Towards Robust Data-Driven Underwater Acoustic Localization: A Deep CNN Solution with Performance Guarantees for Model Mismatch](https://mlsanity.com/paper/2305.17920/) (2023)

## Built on by (most cited first, 2 of 2)

- [Weighted Bayesian Cramacuteer-Rao Bound for Mixed-Resolution Parameter Estimation](https://mlsanity.com/paper/2508.20761/) (2025)
- [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 (BCRB) is a crucial tool in signal processing for assessing the fundamental limitations of any estimation problem as well as benchmarking within a Bayesian frameworks. However, the BCRB cannot be computed without full knowledge of the prior and the measurement distributions. In this work, we propose a fully learned Bayesian Cram'er-Rao bound (LBCRB) that learns both the prior and the measurement distributions. Specifically, we suggest two approaches to obtain the LBCRB: the Posterior Approach and the Measurement-Prior Approach. The Posterior Approach provides a simple method to obtain the LBCRB, whereas the Measurement-Prior Approach enables us to incorporate domain knowledge to improve the sample complexity and interpretability. To achieve this, we introduce a Physics-encoded score neural network which enables us to easily incorporate such domain knowledge into a neural network. We study the learning errors of the two suggested approaches theoretically, and validate them numerically. We demonstrate the two approaches on several signal processing examples, including a linear measurement problem with unknown mixing and Gaussian noise covariance matrices, frequency estimation, and quantized measurement. In addition, we test our approach on a nonlinear signal processing problem of frequency estimation with real-world underwater ambient noise.