Many recent algorithms for sparse signal recovery can be interpreted as maximum-a-posteriori (MAP) estimators relying on some specific priors. From this Bayesian perspective, state-of-the-art methods based on discrete-gradient regularizers, such as total-variation (TV) minimization, implicitly assume the signals to be sampled instances of Levy processes with independent Laplace-distributed increments. By extending the concept to more general Levy processes, we propose an efficient minimum-mean-squared error (MMSE) estimation method based on message-passing algorithms on factor graphs. The resulting algorithm can be used to benchmark the performance of the existing or design new algorithms for the recovery of sparse signals.