Bayesian Inference with Projected Densities

Constraints are a natural choice for prior information in Bayesian inference. In various applications, the parameters of interest lie on the boundary of the constraint set. In this paper, we use a method that implicitly defines a constrained prior such that the posterior assigns positive probability to the boundary of the constraint set. We show that by projecting posterior mass onto a polyhedral constraint set, we obtain a new posterior with a rich probabilistic structure on the boundary of that set. If the original posterior is a Gaussian, then such a projection can be done efficiently. We apply the method to Bayesian linear inverse problems, in which case samples can be obtained by repeatedly solving constrained least squares problems, similar to an MAP estimate but with perturbations in the data. When combined into a Bayesian hierarchical model and the constraint set is a polyhedral cone, we can derive a Gibbs sampler to efficiently sample from the hierarchical model. To show the effect of projecting the posterior, we applied the method to deblurring and CT examples.