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

Monte Carlo with Soft Constraints: the Surface Augmented Sampler