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.

翻译：统计问题常涉及模型参数的线性等式与不等式约束。当研究者关注低维特征的估计时，直接估计受限在一般多面锥内的参数可能面临挑战。我们通过一种对偶形式的参数化方法，刻画了受限在多面锥低维面上的参数向量，并以此定义了该类锥上的"稀疏性"概念。研究表明，所提出的概念在无约束情形下与常规稀疏性定义一致，且验证了该定义作为稀疏性度量的有效性。低维面的可辨识参数化使得经典的spike-and-slab先验能够推广至闭凸多面锥。该先验测度利用锥的极射线邻接图构建马尔可夫随机场，从而融合了锥的几何结构。我们描述了受限情形下参数后验计算的高效方法。通过美国国家健康与营养调查（NHANES）加速度计研究中可穿戴设备数据的应用，展示了所提方法在施加线性等式与不等式约束上的有效性——在该数据中，参与者日均活动轮廓曲线呈现出符合此类约束的规律性模式。