This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which claimed that the half-Cauchy prior is a sensible default prior for a scale parameter in hierarchical models. For estimation of a normal mean vector under the quadratic loss, they showed that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In terms of the shrinkage coefficient, the half-Cauchy prior has a U-shape and can be interpreted as a continuous spike and slab prior. In this paper, we consider a general class of priors with U-shapes and theoretically establish sufficient conditions for the minimaxity of the corresponding (generalized) Bayes estimators. We also develop an algorithm for posterior sampling and present numerical results.

翻译：这是Polson和Scott（2012,《贝叶斯分析》）的后续论文，该文声称半柯西先验是层次模型中尺度参数的一个合理默认先验。针对二次损失下正态均值向量的估计问题，他们通过数值实验表明，采用半柯西先验的贝叶斯估计量似乎具有极小极大性。就收缩系数而言，半柯西先验呈U形，可解释为连续尖峰与板岩先验。本文考虑一类具有U形的一般先验，从理论上建立了相应（广义）贝叶斯估计量满足极小极大性的充分条件。我们还开发了一种后验采样算法，并给出了数值结果。