Bayesian constraint relaxation

Leo L. Duan, Alexander L. Young, Akihiko Nishimura, David B. Dunson

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 not necessary in some settings and tend to limit modelling scope to a narrow set of distributions that are tractable computationally. We propose to replace the sharp indicator function of the constraint with an exponential kernel, thereby creating a closeto-constrained neighbourhood 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-Theshelf posterior sampling algorithms, such as Hamiltonian Monte Carlo, facilitating automatic computation in a broad range of models. We study the constrained and relaxed distributions under multiple settings and theoretically quantify their differences. Application of the method is illustrated through several novel modelling examples.

Original languageEnglish (US)
Pages (from-to)191-204
Number of pages14
Issue number1
StatePublished - Mar 1 2020
Externally publishedYes


  • Constrained Bayes
  • Constraint function
  • Factor model
  • Manifold constraint
  • Ordered simplex
  • Orthonormality
  • Parameter restriction
  • Shrinkage

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Agricultural and Biological Sciences (miscellaneous)
  • Agricultural and Biological Sciences(all)
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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