# Learning to Bound: A Generative Cram'er-Rao Bound

arXiv: https://arxiv.org/abs/2203.03695 · HTML: https://mlsanity.com/paper/2203.03695/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 52 references that are in the map.
- First posted on arXiv 7 Mar 2022.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Parallel WaveNet: Fast High-Fidelity Speech Synthesis](https://mlsanity.com/paper/1711.10433/) (2017)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [Noise Flow: Noise Modeling with Conditional Normalizing Flows](https://mlsanity.com/paper/1908.08453/) (2019)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [Learning Discrete Distributions by Dequantization](https://mlsanity.com/paper/2001.11235/) (2020)
- [On the expressivity of bi-Lipschitz normalizing flows](https://mlsanity.com/paper/2107.07232/) (2021)
- [Closing the Dequantization Gap: PixelCNN as a Single-Layer Flow](https://mlsanity.com/paper/2002.02547/) (2020)
- [Universal Approximation of Residual Flows in Maximum Mean Discrepancy](https://mlsanity.com/paper/2103.05793/) (2021)
- [Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications](https://mlsanity.com/paper/1709.08210/) (2017)
- [Estimating covariance and precision matrices along subspaces](https://mlsanity.com/paper/1909.12218/) (2019)

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

- [Bayesian Cram'er-Rao Bound Estimation with Score-Based Models](https://mlsanity.com/paper/2309.16076/) (2023)
- [Learned Bayesian Cram'er-Rao Bound for Unknown Measurement Models Using Score Neural Networks](https://mlsanity.com/paper/2502.00724/) (2025)

## Abstract (arXiv)

The Cram'er-Rao bound (CRB), a well-known lower bound on the performance of any unbiased parameter estimator, has been used to study a wide variety of problems. However, to obtain the CRB, requires an analytical expression for the likelihood of the measurements given the parameters, or equivalently a precise and explicit statistical model for the data. In many applications, such a model is not available. Instead, this work introduces a novel approach to approximate the CRB using data-driven methods, which removes the requirement for an analytical statistical model. This approach is based on the recent success of deep generative models in modeling complex, high-dimensional distributions. Using a learned normalizing flow model, we model the distribution of the measurements and obtain an approximation of the CRB, which we call Generative Cram'er-Rao Bound (GCRB). Numerical experiments on simple problems validate this approach, and experiments on two image processing tasks of image denoising and edge detection with a learned camera noise model demonstrate its power and benefits.