Monte Carlo with Soft Constraints: the Surface Augmented Sampler

We describe an MCMC method for sampling distributions with soft constraints, which are constraints that are almost but not exactly satisfied. We sample a total distribution that is a convex combination of the target soft distribution with the nearby hard distribution supported on the constraint surface. Hard distribution moves lead to performance that is uniform in the softness parameter. On and Off moves related to the Holmes-Cerfon Stratification Sampler enable sampling the target soft distribution. Computational experiments verify that performance is uniform in the soft constraints limit.