This paper addresses the problem of shadow extraction from a single image of a complex natural scene. No simplifying assumption on the camera and the light source other than the Lambertian assumption is used. Our method is unique because it is capable of translating very rough user-supplied hints into the effective likelihood and prior functions for our Bayesian optimization. The likelihood function requires a decent estimation of the shadowless image, which is obtained by solving the associated Poisson equation. Our Bayesian framework allows for the optimal extraction of smooth shadows while preserving texture appearance under the extracted shadow. Thus our technique can be applied to shadow removal, producing some best results to date compared with the current state-of-the-art techniques using a single input image. We propose related applications in shadow compositing and image repair using our Bayesian technique.