Bayesian Constraint Relaxation

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) unrealistic in many settings; and (b) tend to limit modeling scope to a narrow set of distributions that are tractable computationally. We propose to solve both of these problems via a general class of Bayesian constraint relaxation methods. The key idea is to replace the sharp indicator function of the constraint holding with an exponential kernel. This kernel decays with distance from the constrained space at a rate depending on a relaxation hyperparameter. By avoiding the sharp constraint, we enable the 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.