In many regression settings the unknown coefficients may have some known structure, for instance they may be ordered in space or correspond to a vectorized matrix or tensor. At the same time, the unknown coefficients may be sparse, with many nearly or exactly equal to zero. However, many commonly used priors and corresponding penalties for coefficients do not encourage simultaneously structured and sparse estimates. In this paper we develop structured shrinkage priors that generalize multivariate normal, Laplace, exponential power and normal-gamma priors. These priors allow the regression coefficients to be correlated a priori without sacrificing elementwise sparsity or shrinkage. The primary challenges in working with these structured shrinkage priors are computational, as the corresponding penalties are intractable integrals and the full conditional distributions that are needed to approximate the posterior mode or simulate from the posterior distribution may be non-standard. We overcome these issues using a flexible elliptical slice sampling procedure, and demonstrate that these priors can be used to introduce structure while preserving sparsity.

翻译：在许多回归场景中，未知系数可能具有某种已知结构，例如它们在空间上具有顺序排列，或对应于向量化矩阵或张量。同时，未知系数可能具有稀疏性，即大量系数接近或等于零。然而，许多常用的先验及相应的惩罚函数并不能同时鼓励结构化与稀疏性估计。本文提出了结构化收缩先验，该先验推广了多元正态、拉普拉斯、指数幂及正态-伽马先验。这些先验允许回归系数先验相关而不牺牲元素层面的稀疏性或收缩特性。处理这些结构化收缩先验的主要挑战在于计算层面——相应的惩罚函数涉及不可处理的积分，且用于近似后验众数或从后验分布取样的完全条件分布可能非常规。我们通过灵活的椭圆切片采样方法解决了这些问题，并论证了这些先验能够在保持稀疏性的同时引入结构。