  Constrained learning is prevalent in many statistical tasks. Recent work
proposes distance-to-set penalties to derive estimators under general
constraints that can be specified as sets, but focuses on obtaining point
estimates that do not come with corresponding measures of uncertainty. To
remedy this, we approach distance-to-set regularization from a Bayesian lens.
We consider a class of smooth distance-to-set priors, showing that they yield
well-defined posteriors toward quantifying uncertainty for constrained learning
problems. We discuss relationships and advantages over prior work on Bayesian
constraint relaxation. Moreover, we prove that our approach is optimal in an
information geometric-sense for finite penalty parameters $\rho$, and enjoys
favorable statistical properties when $\rho\to\infty$. The method is designed
to perform effectively within gradient-based MCMC samplers, as illustrated on a
suite of simulated and real data applications.
