Bayesian <i>L</i>_1/2 Regression

It is well known that Bridge regression enjoys superior theoretical properties when compared to traditional LASSO. However, the current latent variable representation of its Bayesian counterpart, based on the exponential power prior, is computationally expensive in higher dimensions. In this article, we show that the exponential power prior has a closed form scale mixture of normal decomposition for α=(12)γ,γ∈{1,2,…}. We call these types of priors L12 prior for short. We develop an efficient partially collapsed Gibbs sampling scheme for computation using the L12 prior and study theoretical properties when p&gt;n. In addition, we introduce a non-separable Bridge penalty function inspired by the fully Bayesian formulation and a novel, efficient coordinate descent algorithm. We prove the algorithm’s convergence and show that the local minimizer from our optimization algorithm has an oracle property. Finally, simulation studies were carried out to illustrate the performance of the new algorithms. Supplementary materials for this article are available online.众所周知，桥回归（Bridge regression）相较于传统套索回归（LASSO）具备更优异的理论特性。然而，当前基于指数幂先验（exponential power prior）的桥回归贝叶斯模型的隐变量表示方法，在高维场景下计算开销巨大。本文证明，当α=(1/2)γ且γ∈{1,2,…}时，指数幂先验可表示为闭合形式的正态尺度混合分解，我们将这类先验简称为L12先验（L12 prior）。基于L12先验，我们提出了一种高效的部分折叠吉布斯采样方案用于模型推断，并探究了变量维度p大于样本量n时的理论特性。此外，我们受全贝叶斯建模框架启发，提出了一种不可分桥惩罚函数，以及一种新颖高效的坐标下降算法。我们证明了该算法的收敛性，并证实我们的优化算法所得的局部极小值点具备神谕性质（oracle property）。最后，我们通过模拟实验验证了新算法的性能表现。本文的配套补充材料可在线获取。