We can use the same approach we're currently using for deriving Gibbs samplers to instead identify and use proximal envelopes for many non-standard and discrete prior and observed distributions (see here for an overview of the idea).

We can start by deriving the basic proximal gradient routine for models (see here) and expand to other approaches and distributions later (e.g. splitting approaches, divide and concur, proximal Langevin for posterior sampling, etc.)