Learning to Bound: A Generative Cramér-Rao Bound

Hai Victor Habi, Hagit Messer, Yoram Bresler

Research output: Contribution to journalArticlepeer-review


The Cramér-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 Cramér-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.

Original languageEnglish
Pages (from-to)1216-1231
Number of pages16
JournalIEEE Transactions on Signal Processing
StatePublished - 2023
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 1991-2012 IEEE.


  • CRB
  • Generative models
  • normalizing flows
  • parameter estimation


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