锥稀疏性：闭凸多面体锥中回归参数的估计
Conic Sparsity: Estimation of Regression Parameters in Closed Convex Polyhedral Cones
AI总结：
本文提出一种基于对偶形式参数化的“锥稀疏性”概念，用于刻画闭凸多面体锥低维面上的回归参数，并推广spike-and-slab先验以高效计算后验，有效解决带线性等式与不等式约束的参数估计问题。
摘要：
统计问题通常涉及对模型参数的线性等式和不等式约束。直接估计受限于一般多面体锥的参数，特别是当人们感兴趣于估计低维特征时，可能具有挑战性。我们使用对偶形式参数化来刻画受限于多面体锥低维面的参数向量，并利用该刻画在此类锥上定义“稀疏性”的概念。我们证明所提出的概念在无约束情况下与通常的稀疏性概念一致，并证明了所提出的定义作为稀疏性度量的有效性。低维面的可识别参数化允许将流行的spike-and-slab先验推广到闭凸多面体锥。该先验测度通过在锥的极射线的邻接图上定义马尔可夫随机场来利用锥的几何结构。我们描述了一种在受约束情况下计算参数后验的有效方法。我们通过使用来自国家健康与营养检查调查（NHANES）actigraph研究的可穿戴设备数据，说明了所提出的方法论在施加线性等式和不等式约束方面的有用性，其中参与者的每日平均活动曲线表现出似乎遵守此类约束的模式。
英文摘要：
Statistical problems often involve linear equality and inequality constraints on model parameters. Direct estimation of parameters restricted to general polyhedral cones, particularly when one is interested in estimating low dimensional features, may be challenging. We use a dual form parameterization to characterize parameter vectors restricted to lower dimensional faces of polyhedral cones and use the characterization to define a notion of 'sparsity' on such cones. We show that the proposed notion agrees with the usual notion of sparsity in the unrestricted case and prove the validity of the proposed definition as a measure of sparsity. The identifiable parameterization of the lower dimensional faces allows a generalization of popular spike-and-slab priors to a closed convex polyhedral cone. The prior measure utilizes the geometry of the cone by defining a Markov random field over the adjacency graph of the extreme rays of the cone. We describe an efficient way of computing the posterior of the parameters in the restricted case. We illustrate the usefulness of the proposed methodology for imposing linear equality and inequality constraints by using wearables data from the National Health and Nutrition Examination Survey (NHANES) actigraph study where the daily average activity profiles of participants exhibit patterns that seem to obey such constraints.
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