  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
the 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 a 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 computed
tomography examples.
