Bayesian Regression Using a Prior on the Model Fit: The R2-D2 Shrinkage Prior
基于模型拟合度先验的贝叶斯回归：R2-D2 收缩先验AI 解读 ↓
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基于模型拟合度先验的贝叶斯回归：R2-D2 收缩先验
- 研究问题
- 高维回归系数的联合先验难以直接设定时，能否从模型解释度出发构造收缩先验，同时兼顾零点集中与尾部行为？
- 方法与思路
- 作者先对决定系数设定先验，再将其分配到回归系数，考察所得先验在零点和尾部的性质及后验收缩表现。
- 材料陈述的贡献
- 摘要称新先验兼具零点集中和尾部性质，并达到与尖峰与平板先验相同的近极小极大后验收缩率，另报告经验表现。
- 阅读关注点
- 应核对决定系数先验及系数分配机制、先验适当性与极限表述，并检查后验收缩所需的设计、稀疏度和超参数条件。
- 阅读边界
- 仅据稿件版本未注明的 OpenAlex 索引摘要；摘要未说明极限行为的具体参数化与定理条件，所述最优性不等于普遍估计性能最优。
原文摘要
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.