高维图模型的贝叶斯变量选择与稀疏估计
Bayesian Variable Selection and Sparse Estimation for High-Dimensional Graphical Models
AI总结：
提出一种新颖的贝叶斯方法，利用分层spike and slab先验的贝叶斯条件随机场模型，实现高维高斯图模型中回归系数矩阵、响应间及响应与协变量间三种稀疏结构的同时估计，并证明了统计准确性。
摘要：
我们提出了一种新颖的贝叶斯方法，用于在涉及多响应的高维高斯图模型中进行协变量选择和稀疏精度矩阵估计。我们的方法提供了对三种不同稀疏结构的稀疏估计：回归系数矩阵、响应之间的条件依赖结构，以及响应与协变量之间的结构。这与现有方法形成对比，现有方法通常关注这些结构中的任意两个，但很少实现对所有三个结构的同时稀疏估计。我们方法的一个关键方面是，它利用从数据集中存在的无关协变量获得的结构稀疏性信息，在精度矩阵和回归系数矩阵中引入协变量级别的稀疏性。这是通过使用分层spike and slab先验设置的贝叶斯条件随机场模型实现的。尽管问题具有非凸性质，我们为高后验密度区域中的点（包括最大后验概率（MAP）估计器）建立了统计准确性。我们还提出了一种高效的期望最大化（EM）算法来计算估计器。通过模拟实验，我们展示了我们方法的竞争性能，特别是在精度矩阵中信号强度较弱的情况下。最后，我们将该方法应用于一个共享单车数据集，展示了其预测性能。
英文摘要：
We introduce a novel Bayesian approach for both covariate selection and sparse precision matrix estimation in the context of high-dimensional Gaussian graphical models involving multiple responses. Our approach provides a sparse estimation of the three distinct sparsity structures: the regression coefficient matrix, the conditional dependency structure among responses, and between responses and covariates. This contrasts with existing methods, which typically focus on any two of these structures but seldom achieve simultaneous sparse estimation for all three. A key aspect of our method is that it leverages the structural sparsity information gained from the presence of irrelevant covariates in the dataset to introduce covariate-level sparsity in the precision and regression coefficient matrices. This is achieved through a Bayesian conditional random field model using a hierarchical spike and slab prior setup. Despite the non-convex nature of the problem, we establish statistical accuracy for points in the high posterior density region, including the maximum-a-posteriori (MAP) estimator. We also present an efficient Expectation-Maximization (EM) algorithm for computing the estimators. Through simulation experiments, we demonstrate the competitive performance of our method, particularly in scenarios with weak signal strength in the precision matrices. Finally, we apply our method to a bike-share dataset, showcasing its predictive performance.
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