We consider the problem of model selection when grouping structure is inherent within the regressors. Using a Bayesian approach, we model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. In the context of variable selection, this class of priors was studied by Tang et al. (2018). A modified form of the usual class of global-local shrinkage priors with polynomial tail on the group regression coefficients is proposed. The resulting threshold rule selects the active group if within a group, the ratio of the $L_2$ norm of the posterior mean of its group coefficient to that of the corresponding ordinary least square group estimate is greater than a half. In the theoretical part of this article, we have used the global shrinkage parameter either as a tuning one or an empirical Bayes estimate of it depending on the knowledge regarding the underlying sparsity of the model. When the proportion of active groups is known, using $\tau$ as a tuning parameter, we have proved that our method is oracle. In case this proportion is unknown, we propose an empirical Bayes estimate of $\tau$. Even if this empirical Bayes estimate is used, then also our half-thresholding rule captures the truly important groups and obtains optimal estimation rate of the group coefficients simultaneously. Though our theoretical works rely on a special form of the design matrix, for general design matrices also, our simulation results show that the half-thresholding rule yields results similar to that of Yang and Narisetty (2020). As a consequence of this, in a high dimensional sparse group selection problem, instead of using the so-called `gold standard' spike and slab prior, one can use the one-group global-local shrinkage priors with polynomial tail to obtain similar results.

翻译：我们考虑当回归变量内存在固有分组结构时的模型选择问题。采用贝叶斯方法，通过属于广泛一类全局-局部收缩先验（包括马蹄铁先验）的单组先验对均值向量进行建模。在变量选择情境下，Tang等人（2018）研究了这类先验。本文提出了一种改进形式的常用全局-局部收缩先验，其组回归系数具有多项式尾部。若某分组内，该组系数后验均值的$L_2$范数与相应普通最小二乘组估计的$L_2$范数之比大于二分之一，则所得阈值规则将该组判定为活跃组。在理论部分，我们根据模型潜在稀疏性的已知程度，将全局收缩参数作为调谐参数或对其进行经验贝叶斯估计。当活跃组比例已知时，使用$\tau$作为调谐参数，我们证明了该方法具有oracle性质。若该比例未知，则提出$\tau$的经验贝叶斯估计。即使采用该经验贝叶斯估计，我们的半阈值规则仍能捕获真正重要的分组，并同时获得组系数的最优估计速率。尽管理论工作依赖于设计矩阵的特殊形式，但模拟结果表明，对于一般设计矩阵，半阈值规则也能获得与Yang和Narisetty（2020）相似的结果。据此，在高维稀疏组选择问题中，我们可使用具有多项式尾部的单组全局-局部收缩先验替代所谓"黄金标准"的spike-and-slab先验，以获得类似结果。