标题： 高维贝叶斯回归模型中具有收缩先验的快速马尔可夫链蒙特卡洛方法 显示英文标题

标题： Fast Markov chain Monte Carlo for high dimensional Bayesian regression models with shrinkage priors

摘要： 在过去十年中，针对协变量数量 $p$ 较大的线性回归问题，开发了许多贝叶斯收缩模型。计算难以处理的后验分布通常通过基于表示收缩先验为正态分布尺度混合的三块Gibbs采样器（3BG）完成。另一种计算工具是先进的哈密顿蒙特卡洛（HMC）方法，可以在Stan软件中轻松实现。然而，我们发现这两种现有方法对于较大的 $p$ 问题效率低下且往往不切实际。受Rajaratnam等人（2018）的一般思想启发，我们提出了两块Gibbs采样器（2BG），用于三种常用的收缩模型，即贝叶斯组Lasso、贝叶斯稀疏组Lasso和贝叶斯融合Lasso模型。通过模拟和真实数据实例，我们展示了2BG背后的马尔可夫链比3BG收敛得快得多，并且不逊于HMC。同时，2BG每轮迭代的计算成本与3BG一样低，并且可能比HMC低几个数量级。因此，新提出的2BG是处理具有大 $p$ 数据集的贝叶斯收缩分析的唯一实用计算解决方案。此外，我们为2BG的优越性能提供了理论依据。我们建立了与这三个贝叶斯收缩模型相关的2BG马尔可夫链的几何遍历性（GE）。我们也证明了对于大多数贝叶斯组Lasso和贝叶斯稀疏组Lasso模型情况，2BG链的马尔可夫算子是迹类的。而对于所有三种贝叶斯收缩模型的所有情况，3BG链的马尔可夫算子甚至都不是希尔伯特-施密特算子。 显示英文摘要

摘要： In the past decade, many Bayesian shrinkage models have been developed for linear regression problems where the number of covariates, $p$, is large. Computing the intractable posterior are often done with three-block Gibbs samplers (3BG), based on representing the shrinkage priors as scale mixtures of Normal distributions. An alternative computing tool is a state of the art Hamiltonian Monte Carlo (HMC) method, which can be easily implemented in the Stan software. However, we found both existing methods to be inefficient and often impractical for large $p$ problems. Following the general idea of Rajaratnam et al. (2018), we propose two-block Gibbs samplers (2BG) for three commonly used shrinkage models, namely, the Bayesian group lasso, the Bayesian sparse group lasso and the Bayesian fused lasso models. We demonstrate with simulated and real data examples that the Markov chains underlying 2BG's converge much faster than that of 3BG's, and no worse than that of HMC. At the same time, the computing costs of 2BG's per iteration are as low as that of 3BG's, and can be several orders of magnitude lower than that of HMC. As a result, the newly proposed 2BG is the only practical computing solution to do Bayesian shrinkage analysis for datasets with large $p$. Further, we provide theoretical justifications for the superior performance of 2BG's. We establish geometric ergodicity (GE) of Markov chains associated with the 2BG for each of the three Bayesian shrinkage models. We also prove, for most cases of the Bayesian group lasso and the Bayesian sparse group lasso model, the Markov operators for the 2BG chains are trace-class. Whereas for all cases of all three Bayesian shrinkage models, the Markov operator for the 3BG chains are not even Hilbert-Schmidt.