Graphs have been commonly used to represent complex data structures. In models dealing with graph-structured data, multivariate parameters may not only exhibit sparse patterns but have structured sparsity and smoothness in the sense that both zero and non-zero parameters tend to cluster together. We propose a new prior for high-dimensional parameters with graphical relations, referred to as the Tree-based Low-rank Horseshoe (T-LoHo) model, that generalizes the popular univariate Bayesian horseshoe shrinkage prior to the multivariate setting to detect structured sparsity and smoothness simultaneously. The T-LoHo prior can be embedded in many high-dimensional hierarchical models. To illustrate its utility, we apply it to regularize a Bayesian high-dimensional regression problem where the regression coefficients are linked by a graph, so that the resulting clusters have flexible shapes and satisfy the cluster contiguity constraint with respect to the graph. We design an efficient Markov chain Monte Carlo algorithm that delivers full Bayesian inference with uncertainty measures for model parameters such as the number of clusters. We offer theoretical investigations of the clustering effects and posterior concentration results. Finally, we illustrate the performance of the model with simulation studies and a real data application for anomaly detection on a road network. The results indicate substantial improvements over other competing methods such as the sparse fused lasso.
翻译:图表通常用于代表复杂的数据结构。 在涉及图形结构数据的模型中,多变量参数可能不仅表现出稀少的模式,而且具有结构化的宽度和光滑性,因为零参数和非零参数往往会聚集在一起。我们提出一个新的高维参数之前的图形关系,即以树为基础的低级马蹄(T-Lohoo)模型(T-Lohoo)模型,该模型将流行的单向贝亚赛马蹄木马缩缩影概括为通用。在多变量设置之前,可同时探测结构的宽度和顺畅度。T-LoHo 之前的T-LoHo 参数可以嵌入许多高维等级模型中。为了说明其实用性,我们应用它来规范巴伊西亚高维高度回归率的问题,因为图形将回归系数联系在一起,因此由此产生的组群具有灵活的形状,并满足与图形有关的集毗连性约束。我们设计了高效的Markov链 Monte Carlo 算法,以不确定性测量模型参数,例如集体数。我们提供了对组合效应效应的理论性研究,以及后方位浓度浓度浓度的模型的精确度分析,我们用模拟方法来说明如何模拟模拟,从而模拟地测测测测路结果。我们用其他数据。我们用其他的模型,用其他的方法演示了其他的模型,用来模拟。