We present a novel Bayesian approach for high-dimensional grouped regression under sparsity. We leverage a sparse projection method that uses a sparsity-inducing map to derive an induced posterior on a lower-dimensional parameter space. Our method introduces three distinct projection maps based on popular penalty functions: the Group LASSO Projection Posterior, Group SCAD Projection Posterior, and Adaptive Group LASSO Projection Posterior. Each projection map is constructed to immerse dense posterior samples into a structured, sparse space, allowing for effective group selection and estimation in high-dimensional settings. We derive optimal posterior contraction rates for estimation and prediction, proving that the methods are model selection consistent. Additionally, we propose a Debiased Group LASSO Projection Map, which ensures exact coverage of credible sets. Our methodology is particularly suited for applications in nonparametric additive models, where we apply it with B-spline expansions to capture complex relationships between covariates and response. Extensive simulations validate our theoretical findings, demonstrating the robustness of our approach across different settings. Finally, we illustrate the practical utility of our method with an application to brain MRI volume data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), where our model identifies key brain regions associated with Alzheimer's progression.

翻译：本文提出了一种新颖的贝叶斯方法，用于稀疏性假设下的高维分组回归问题。我们利用一种稀疏投影方法，通过引入稀疏诱导映射，在低维参数空间上推导出诱导后验分布。本方法基于三种常用的惩罚函数构建了不同的投影映射：分组LASSO投影后验、分组SCAD投影后验以及自适应分组LASSO投影后验。每种投影映射的设计旨在将稠密后验样本嵌入到具有结构化的稀疏空间中，从而在高维场景下实现有效的分组选择和参数估计。我们推导了估计与预测的最优后验收敛速率，并证明了这些方法具有模型选择一致性。此外，我们提出了去偏分组LASSO投影映射，该映射能够确保可信集合的精确覆盖。我们的方法特别适用于非参数可加模型的应用场景，其中我们结合B样条展开来捕捉协变量与响应变量之间的复杂关系。大量仿真实验验证了我们的理论结果，证明了该方法在不同设置下的鲁棒性。最后，我们通过将其应用于阿尔茨海默病神经影像学倡议（ADNI）的脑部MRI体积数据来展示本方法的实际效用，该模型成功识别出与阿尔茨海默病进展相关的关键脑区。