  The use of non-differentiable priors in Bayesian statistics has become
increasingly popular, in particular in Bayesian imaging analysis. Current state
of the art methods are approximate in the sense that they replace the posterior
with a smooth approximation via Moreau-Yosida envelopes, and apply
gradient-based discretized diffusions to sample from the resulting
distribution. We characterize the error of the Moreau-Yosida approximation and
propose a novel implementation using underdamped Langevin dynamics. In
misson-critical cases, however, replacing the posterior with an approximation
may not be a viable option. Instead, we show that Piecewise-Deterministic
Markov Processes (PDMP) can be utilized for exact posterior inference from
distributions satisfying almost everywhere differentiability. Furthermore, in
contrast with diffusion-based methods, the suggested PDMP-based samplers place
no assumptions on the prior shape, nor require access to a computationally
cheap proximal operator, and consequently have a much broader scope of
application. Through detailed numerical examples, including a
non-differentiable circular distribution and a non-convex genomics model, we
elucidate the relative strengths of these sampling methods on problems of
moderate to high dimensions, underlining the benefits of PDMP-based methods
when accurate sampling is decisive.
