In this paper we introduce a natural image prior that directly represents a Gaussian-smoothed version of the natural image distribution. We include our prior in a formulation of image restoration as a Bayes estimator that also allows us to solve noise-blind image restoration problems. The gradient of a bound of our estimator involves the gradient of the logarithm of our prior. This gradient corresponds to the mean-shift vector on the natural image distribution, and we learn the mean-shift vector field using denoising autoencoders. We demonstrate competitive results for noise-blind deblurring, super-resolution, and demosaicing.