Learning to Bound: A Generative Cramer-Rao Bound

The Cramer-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 Cramer-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.

Automated summary

The Cramer-Rao bound is a well-known lower bound that evaluates the performance of unbiased parameter estimators in various problems. However, obtaining the Cramer-Rao bound requires an analytical expression for the likelihood of measurements or a statistical model for the data, which may not be available in many applications. This study introduces a data-driven approach, based on deep generative models, to approximate the Cramer-Rao bound without the need for an analytical statistical model. The proposed Generative Cramer-Rao Bound (GCRB) utilizes a learned normalizing flow model to estimate the bound and demonstrates its efficacy in image denoising and edge detection tasks.