# Bayesian Constraint Relaxation

arXiv: https://arxiv.org/abs/1801.01525 · HTML: https://mlsanity.com/paper/1801.01525/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2018.

## Built on by (most cited first, 4 of 4)

- [Bayesian Joint Additive Factor Models for Multiview Learning](https://mlsanity.com/paper/2406.00778/) (2024)
- [Bayesian Transfer Learning](https://mlsanity.com/paper/2312.13484/) (2023)
- [Sequential Adaptive Priors for Orthogonal Functions](https://mlsanity.com/paper/2508.15552/) (2025)
- [Molecular Design beyond Training Data with Novel Extended Objective Functionals of Generative AI Models Driven by Quantum Annealing Computer](https://mlsanity.com/paper/2602.15451/) (2026)

## Abstract (arXiv)

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.