  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.
