The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either restrict to specific sub-classes of sparsity patterns and/or are not scalable to settings with hundreds of responses and predictors. Bayesian approaches which only focus on estimating the posterior mode are scalable, but do not generate samples from the posterior distribution for uncertainty quantification. Using a bi-convex regression based generalized likelihood and spike-and-slab priors, we develop an algorithm called Joint Regression Network Selector (JRNS) for joint regression and covariance selection which (a) can accommodate general sparsity patterns, (b) provides posterior samples for uncertainty quantification, and (c) is scalable and orders of magnitude faster than the state-of-the-art Bayesian approaches providing uncertainty quantification. We demonstrate the statistical and computational efficacy of the proposed approach on synthetic data and through the analysis of selected cancer data sets. We also establish high-dimensional posterior consistency for one of the developed algorithms.
翻译:本文涉及回归系数矩阵和贝叶西亚模式中高维多变量回归模型的误差精确度(反共差)矩阵(反共差)矩阵中的联合宽度选择。 选定的宽度模式对于帮助理解预测变量和响应变量之间的关系网络以及后者之间的有条件关系至关重要。 虽然巴伊西亚方法的优势在于通过后置包容概率和可信的间隔提供自然不确定性的量化,但当前的巴伊西亚方法要么局限于特定的次类宽度模式和(或)无法与数百个反应和预测器相适应的设置相适应。 仅侧重于估算后端模式的贝伊斯方法是可伸缩的,但并不产生从后端分布变量变量和响应变量之间的样本,以及后者之间的有条件关系。 使用基于普遍可能性和峰值和悬浮点前的双convex回归法,我们开发了一种算法,称为联合回归网络选择器,用于联合回归和变量选择,这(a) 能够适应一般宽度模式, (b) 仅提供后端样本,用于不确定性的后端模式,而不能产生后端分配的后序值样本, 以更快的方式,(c) 提供数据排序的升级。