We propose a scalable variational Bayes method for statistical inference for a single or low-dimensional subset of the coordinates of a high-dimensional parameter in sparse linear regression. Our approach relies on assigning a mean-field approximation to the nuisance coordinates and carefully modelling the conditional distribution of the target given the nuisance. This requires only a preprocessing step and preserves the computational advantages of mean-field variational Bayes, while ensuring accurate and reliable inference for the target parameter, including for uncertainty quantification. We investigate the numerical performance of our algorithm, showing that it performs competitively with existing methods. We further establish accompanying theoretical guarantees for estimation and uncertainty quantification in the form of a Bernstein--von Mises theorem.

翻译：本文提出了一种可扩展的变分贝叶斯方法，用于稀疏线性回归中高维参数的单个或低维子坐标的统计推断。我们的方法通过对冗余坐标采用平均场近似，并精细建模目标参数在给定冗余参数条件下的分布来实现。该方法仅需一个预处理步骤，在保持平均场变分贝叶斯计算优势的同时，确保了对目标参数（包括不确定性量化）的准确可靠推断。我们通过数值实验验证了算法的性能，结果表明其与现有方法相比具有竞争优势。进一步地，我们以伯恩斯坦-冯·米塞斯定理的形式建立了估计与不确定性量化的理论保证。