We propose a novel, Bayesian formulation of the edge-stopping (diffusivity) function in a nonlinear diffusion scheme in terms of edge probability under a marginal prior on noise-free gradient. This formulation differs from the existing probabilistic diffusion approaches that give stochastic formulations for the conductivity but not for the diffusivity function of the gradient. In particular, we impose a Laplacian prior for the ideal gradient, but the proposed formulation is general and can be used with other marginal distributions. We also make links to related works that treat correspondences between nonlinear diffusion and wavelet shrinkage.