Variable selection is a classic problem in statistics. In this paper, we consider a Bayes variable selection problem based on spike-and-slab prior with mixed normal distribution proposed by Ro\v{c}kov\'a and George (2014). Motivated by Ormerod and You (2017, 2023), we use the variational inference and collapsed variational inference method to solve the Bayesian problem instead of MCMC. Like Ormerod and You (2017, 2023), we also explain how the sparsity estimator is induced, and under certain mild assumptions, we also prove the consistent and asymptotic results.

翻译：变量选择是统计学中的一个经典问题。本文基于Ro\v{c}kov\'a和George（2014）提出的混合正态分布尖峰-厚尾先验，考虑贝叶斯变量选择问题。受Ormerod和You（2017, 2023）的启发，我们采用变分推断和折叠变分推断方法代替MCMC来解决该贝叶斯问题。与Ormerod和You（2017, 2023）类似，我们同样解释了稀疏性估计量的导出过程，并在某些温和假设下证明了估计量的一致性和渐近性质。