Solving most image processing tasks, such as denoising, deblurring, inpainting, etc, require explicitly or implicitly a prior probability model of natural images. Classically, both learning (by maximizing likelihood) and using (via Bayes’ rule) such prior models are intractable due to the curse of dimensionality. Diffusion models take a different approach, where the prior density is replaced by a family of score vector fields across noise levels. They have led to impressive success in generative modeling, but the learned density is not explicit nor is it readily usable as a prior for solving inverse problems. In this talk, I will show how to bridge this gap by adopting a scale-space representation of the prior density itself. This leads to a procedure for learning normalized density functions from data.  The resulting prior can be used in inverse problems to efficiently access the normalized posterior density, its mean, and draw posterior samples. Further, the scale space gives access to geometric properties of the learned probability distribution that have both information- and estimation-theoretic interpretations. I will show how this can be used to test the common hypothesis that natural images lie on a low-dimensional manifold and define a perceptual distance function between images that predicts human judgments.