Duan Leo L, Young Alexander L, Nishimura Akihiko, Dunson David B
Department of Statistics, University of Florida, 101C Griffin-Floyd Hall, P.O. Box 118545, Gainesville, Florida 32611, U.S.A.
Department of Statistical Science, Duke University, Box 90251, Durham, North Carolina 27708, U.S.A.
Biometrika. 2020 Mar;107(1):191-204. doi: 10.1093/biomet/asz069. Epub 2019 Dec 24.
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.
先验信息通常采用参数约束的形式。贝叶斯方法通过具有约束支持的先验分布纳入此类信息。通过使用后验抽样算法，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，严格的约束先验并非必要，而且往往会将建模范围限制在一组计算上易于处理的狭窄分布中。我们建议用指数核取代约束的尖锐指示函数，从而在嵌入了约束子空间的欧几里得空间内创建一个接近约束的邻域。该核随着与约束空间距离的增加而衰减，衰减速率取决于一个松弛超参数。通过避免严格的约束，我们能够使用现成的后验抽样算法，如哈密顿蒙特卡罗算法，便于在广泛的模型中进行自动计算。我们在多种情况下研究了约束分布和松弛分布，并从理论上量化了它们的差异。通过几个新颖的建模示例说明了该方法的应用。
