In the context of a high-dimensional linear regression model, we propose the use of an empirical correlation-adaptive prior that makes use of information in the observed predictor variable matrix to adaptively address high collinearity, determining if parameters associated with correlated predictors should be shrunk together or kept apart. Under suitable conditions, we prove that this empirical Bayes posterior concentrates around the true sparse parameter at the optimal rate asymptotically. A simplified version of a shotgun stochastic search algorithm is employed to implement the variable selection procedure, and we show, via simulation experiments across different settings and a real-data application, the favorable performance of the proposed method compared to existing methods.

中文翻译：

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使用经验相关自适应先验的高维线性模型中的贝叶斯推理

在高维线性回归模型的背景下，我们建议使用经验相关自适应先验，利用观察到的预测变量矩阵中的信息来自适应地解决高共线性问题，确定是否应缩小与相关预测变量相关的参数在一起或分开。在合适的条件下，我们证明了这种经验贝叶斯后验以最佳速率渐近地集中在真正的稀疏参数周围。使用霰弹枪随机搜索算法的简化版本来实现变量选择程序，我们通过不同设置的模拟实验和实际数据应用展示了所提出的方法与现有方法相比的良好性能。