Bayesian Sparse Group Selection

This article proposes a Bayesian approach for the sparse group selection problem in the regression model. In this problem, the variables are partitioned into different groups. It is assumed that only a small number of groups are active for explaining the response variable, and it is further assumed that within each active group only a small number of variables are active. We adopt a Bayesian hierarchical formulation, where each candidate group is associated with a binary variable indicating whether the group is active or not. Within each group, each candidate variable is also associated with a binary indicator, too. Thus, the sparse group selection problem can be solved by sampling from the posterior distribution of the two layers of indicator variables. We adopt a group-wise Gibbs sampler for posterior sampling. We demonstrate the proposed method by simulation studies as well as real examples. The simulation results show that the proposed method performs better than the sparse group Lasso in terms of selecting the active groups as well as identifying the active variables within the selected groups. Supplementary materials for this article are available online.

本文针对回归模型中的稀疏组选择问题，提出一种贝叶斯（Bayesian）方法。该问题中将变量划分为若干不同组别，我们假设仅有少量组别可用于解释响应变量，且进一步假定：在每个活跃组内，同样仅有少量变量处于活跃状态。我们采用贝叶斯分层建模框架，为每个候选组分配一个二元变量，以标识该组是否处于活跃状态；在每个组内，每个候选变量也配有一个二元标识变量。据此，稀疏组选择问题可通过对两层标识变量的后验分布进行采样求解。我们采用组级吉布斯采样器（Gibbs Sampler）完成后验采样。我们通过仿真实验与真实案例对所提方法进行验证。仿真结果表明，在活跃组选择以及识别所选组内活跃变量两方面，本文所提方法的性能均优于稀疏组套索（sparse group Lasso）。本文的补充材料可在线获取。