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，用于稀疏假设下的高维分组回归。我们利用稀疏投影方法，通过引入稀疏诱导映射在高维参数空间的低维子空间上推导出诱导后验分布。基于三种常见罚函数，我们的方法构建了三种不同的投影映射：Group LASSO投影后验、Group SCAD投影后验和自适应Group LASSO投影后验。每个投影映射旨在将稠密后验样本嵌入结构化的稀疏空间，从而在高维场景中实现有效的组变量选择和估计。我们推导了估计与预测的最优后验收缩速率，证明该方法具有模型选择一致性。此外，我们提出了去偏Group LASSO投影映射，确保可信集的精确覆盖。该方法特别适用于非参数加性模型，我们结合B样条展开以捕捉协变量与响应变量间的复杂关系。大量仿真实验验证了我们的理论结果，在不同场景下展现了方法的鲁棒性。最后，我们将该方法应用于阿尔茨海默病神经影像学倡议（ADNI）的大脑MRI体积数据，模型成功识别出与阿尔茨海默病进展相关的关键脑区。