We study the relationships between diffusivity functions in a nonlinear diffusion scheme and probabilities of edge presence under a marginal prior on ideal, noise-free image gradient. In particular we impose a Laplacian-shaped prior for the ideal gradient and we define the diffusivity function explicitly in terms of edge probabilities under this prior. The resulting diffusivity function has no free parameters to optimize. Our results demonstrate that the new diffusivity function, automatically, i.e., without any parameter adjustments,satisfies the well accepted criteria for the goodness of edge-stopping functions. Our results also offer a new and interesting interpretation of some widely used diffusivity functions, which are now compared to edge-stopping functions under a marginal prior for the ideal image gradient.