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The use of non-differentiable priors in Bayesian statistics has becomeincreasingly popular, in particular in Bayesian imaging analysis. Current stateof the art methods are approximate in the sense that they replace the posteriorwith a smooth approximation via Moreau-Yosida envelopes, and applygradient-based discretized diffusions to sample from the resultingdistribution. We characterize the error of the Moreau-Yosida approximation andpropose a novel implementation using underdamped Langevin dynamics. Inmisson-critical cases, however, replacing the posterior with an approximationmay not be a viable option. Instead, we show that Piecewise-DeterministicMarkov Processes (PDMP) can be utilized for exact posterior inference fromdistributions satisfying almost everywhere differentiability. Furthermore, incontrast with diffusion-based methods, the suggested PDMP-based samplers placeno assumptions on the prior shape, nor require access to a computationallycheap proximal operator, and consequently have a much broader scope ofapplication. Through detailed numerical examples, including anon-differentiable circular distribution and a non-convex genomics model, weelucidate the relative strengths of these sampling methods on problems ofmoderate to high dimensions, underlining the benefits of PDMP-based methodswhen accurate sampling is decisive.

Gradient-Based Markov Chain Monte Carlo for Bayesian Inference With Non-Differentiable Priors