Sampling from multi-dimensional and complex distributions is still a challenging issue for the signal processing community. In this research area, Hamiltonian Monte Carlo (HMC) schemes have been proposed several years ago, using the target distribution geometry to perform efficient sampling. More recently, a non-smooth HMC (ns-HMC) scheme has been proposed to generalize HMC for distributions having non-smooth energy functions. This new scheme relies on the use of a proximity operator, which cannot be explicitly calculated for a large class of energy functions. We propose in this paper a fast and more general ns-HMC scheme that can be applied to any energy function by using a Bayesian calculation of the proximity operator, which makes the proposed scheme applicable to any energy function. Moreover, the proposed scheme relies on an interesting property of the proximity operator avoiding heavy calculations at each sampling step. The proposed scheme is tested on different sampling examples involving ℓp and total variation energy functions.