  Prior information often takes the form of parameter constraints. Bayesian
methods include such information through prior distributions having constrained
support. By using posterior sampling algorithms, one can quantify uncertainty
without relying on asymptotic approximations. However, sharply constrained
priors are (a) not necessary in some settings; and (b) tend to limit modeling
scope to a narrow set of distributions that are tractable computationally.
Inspired by the vast literature that replaces the slab-and-spike prior with a
continuous approximation, we propose to replace the sharp indicator function of
the constraint with an exponential kernel, thereby creating a
close-to-constrained neighborhood within the Euclidean space in which the
constrained subspace is embedded. This kernel decays with distance from the
constrained space at a rate depending on a relaxation hyperparameter. By
avoiding the sharp constraint, we enable use of off-the-shelf posterior
sampling algorithms, such as Hamiltonian Monte Carlo, facilitating automatic
computation in broad models. We study the constrained and relaxed distributions
under multiple settings, and theoretically quantify their differences. We
illustrate the method through multiple novel modeling examples.
