In this work, we propose a scalable Bayesian procedure for learning the local dependence structure in a high-dimensional model where the variables possess a natural ordering. The ordering of variables can be indexed by time, the vicinities of spatial locations, and so on, with the natural assumption that variables far apart tend to have weak correlations. Applications of such models abound in a variety of fields such as finance, genome associations analysis and spatial modeling. We adopt a flexible framework under which each variable is dependent on its neighbors or predecessors, and the neighborhood size can vary for each variable. It is of great interest to reveal this local dependence structure by estimating the covariance or precision matrix while yielding a consistent estimate of the varying neighborhood size for each variable. The existing literature on banded covariance matrix estimation, which assumes a fixed bandwidth cannot be adapted for this general setup. We employ the modified Cholesky decomposition for the precision matrix and design a flexible prior for this model through appropriate priors on the neighborhood sizes and Cholesky factors. The posterior contraction rates of the Cholesky factor are derived which are nearly or exactly minimax optimal, and our procedure leads to consistent estimates of the neighborhood size for all the variables. Another appealing feature of our procedure is its scalability to models with large numbers of variables due to efficient posterior inference without resorting to MCMC algorithms. Numerical comparisons are carried out with competitive methods, and applications are considered for some real datasets.

翻译：在这项工作中,我们提出了一个可伸缩的贝叶西亚程序,用于在变量具有自然顺序的高维模型中学习本地依赖性结构,变量具有自然顺序。变量的顺序可以按时间、空间位置的频度等等进行索引。变量的顺序可以按时间、空间位置的宽度等进行。自然假设变量相距甚远的自然假设具有薄弱的关联性。在金融、基因组协会分析和空间建模等多个领域应用这些模型。我们采用了一个灵活的框架,每个变量都取决于其邻居或前辈,而每个变量的相邻规模可能不同。通过估计共变或精确的矩阵,得出本地依赖性结构,同时得出对每个变量不同区域大小的一致估计值。关于带宽度的共变异矩阵估计的现有文献,假定固定带宽度的带宽度无法适应这一总体设置。我们采用了修改后的Choloysky脱色矩阵,并通过适当的前期时间,设计一个灵活的模型,即街坊大小和Cholosky因素的比较。Cholosky因素的后端缩率率是近乎或确切的近或确切的缩缩缩度模型。我们整个区域变式模型的缩缩缩缩缩图的流程,是整个的图的缩缩缩缩缩图。我们最优度的图的图的图的缩缩缩缩。