Abstract

Prior distributions for high-dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients, which is inherently difficult. We instead propose a new class of shrinkage priors for linear regression via specifying a prior first on the model fit, in particular, the coefficient of determination, and then distributing through to the coefficients in a novel way. The proposed method compares favorably to previous approaches in terms of both concentration around the origin and tail behavior, which leads to improved performance both in posterior contraction and in empirical performance. The limiting behavior of the proposed prior is $1/x$ , both around the origin and in the tails. This behavior is optimal in the sense that it simultaneously lies on the boundary of being an improper prior both in the tails and around the origin. None of the existing shrinkage priors obtain this behavior in both regions simultaneously. We also demonstrate that our proposed prior leads to the same near-minimax posterior contraction rate as the spike-and-slab prior. Supplementary materials for this article are available online.

摘要

高维线性回归的先验分布需要为未观察到的回归系数指定联合分布，这本质上是困难的。相反，我们提出了一类新的线性回归收缩先验，方法是先在模型拟合上指定先验，特别是确定系数，然后以一种新颖的方式分配给系数。所提出的方法在围绕起源和尾部行为的集中度方面优于以前的方法，这导致在后收缩和经验性能方面的性能都有所提高。所提出的先验的限制行为是 $1/X$ ，围绕原点和尾部。这种行为是最优的，因为它同时位于尾部和原点周围的不适当先验的边界上。现有的收缩先验都没有同时在两个区域中获得这种行为。我们还证明了我们提出的先验导致与尖峰和平板先验相同的近极小后收缩率。本文的补充材料可在线获取。