We develop a post-selective Bayesian framework to jointly and consistently estimate parameters in group-sparse linear regression models. After selection with the Group LASSO (or generalized variants such as the overlapping, sparse, or standardized Group LASSO), uncertainty estimates for the selected parameters are unreliable in the absence of adjustments for selection bias. Existing post-selective approaches are limited to uncertainty estimation for (i) real-valued projections onto very specific selected subspaces for the group-sparse problem, (ii) selection events categorized broadly as polyhedral events that are expressible as linear inequalities in the data variables. Our Bayesian methods address these gaps by deriving a likelihood adjustment factor, and an approximation thereof, that eliminates bias from selection. Paying a very nominal price for this adjustment, experiments on simulated data, and data from the Human Connectome Project demonstrate the efficacy of our methods for a joint estimation of group-sparse parameters and their uncertainties post selection.
翻译:我们开发了一个后选择性贝叶斯框架,以共同和一致地估计群体分析线性回归模型中的参数。在与LASSO集团(或通用变体,如重叠、稀少或标准化的LASSO集团)选定之后,在没有对选择偏差进行调整的情况下,选定参数的不确定性估计数是不可靠的。现有的选择后办法仅限于以下的不确定性估计:(一) 对群体偏差问题的非常具体的选定子空间进行实际价值预测,(二) 被广泛归类为多面事件的选择事件,在数据变量中可表现为线性不平等。我们的巴伊西亚方法通过得出可能调整系数及其近似值来弥补这些差距,从而消除了选择中的偏差。为这一调整支付非常象征性的价格,对模拟数据进行实验,以及人类连接项目的数据,显示了我们联合估计群体偏差参数及其后选择不确定性的方法的有效性。