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 close-to-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-the-shelf 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.

中文翻译：

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贝叶斯约束松弛

先验信息通常采用参数约束的形式。贝叶斯方法包括通过具有约束支持的先验分布的此类信息。通过使用后验采样算法，人们可以在不依赖渐近近似的情况下量化不确定性。然而，在某些情况下，严格约束的先验是不必要的，并且往往会将建模范围限制在一组易于计算的狭窄分布范围内。我们建议用指数核替换约束的锐指示函数，从而在嵌入约束子空间的欧几里得空间内创建接近约束的邻域。该内核随着距约束空间的距离而以取决于松弛超参数的速率衰减。通过避免尖锐的约束，我们可以使用现成的后验采样算法，例如哈密顿蒙特卡罗，促进各种模型的自动计算。我们研究多种设置下的约束分布和松弛分布，并从理论上量化它们的差异。通过几个新颖的建模示例说明了该方法的应用。