Bayesian Global-Local Shrinkage in High Dimensions

高维数据在许多现代应用中无处不在，如基因组学、金融和图像分析。开发在计算上可扩展到这些数据集的大小的方法，同时保留强有力的理论依据仍然是一个挑战。该项目的目标是开发新的方法，使现代高维数据集的易处理的分析。从这项研究中开发的软件将公开提供。贝叶斯方法的高维数据传统上依赖于点质量混合先验，具有吸引人的理论属性，但往往规模差，由于与搜索高维离散空间的计算困难。该项目的目标是探索使用全局-局部替代方案来解决高维问题。许多最近的调查使用全球本地的先验，而显示出的承诺的迹象，已被限制在研究简单的正态均值模型。PI将采用全局-局部收缩技术来解决统计学中的基本问题，例如回归，非线性函数估计和协方差估计。更具体地说，PI的目的是在回归问题中表明，使用全局-局部收缩而不是纯粹的全局收缩方法（如岭回归或主成分回归）可以改善预测。他们的目的是显示全局-局部收缩先验是高维参数的低维函数的非信息分析的良好候选者。PI还建议在协方差估计和联合均值-协方差估计问题中使用全局-局部收缩，并将所开发的方法应用于基因组学或金融领域的合适应用中。

High-dimensional data are ubiquitous in many modern applications such as genomics, finance, and image analysis. Developing approaches that are computationally scalable to the size of these data sets while retaining strong theoretical justifications remains a challenge. The goal of the project is to develop new methodology to enable tractable analysis of modern high-dimensional data sets. Software developed from this research will be made publicly available.Bayesian methodology for high-dimensional data traditionally relies on point mass mixture priors that have attractive theoretical properties but often scale poorly due to the computational difficulties associated with searching a high-dimensional discrete space. The goal of the project is to explore the use of global-local alternatives to high-dimensional problems. Many recent investigations using global-local priors, while showing signs of promise, have been restricted to studying the simple normal means model. The PIs will employ the techniques of global-local shrinkage to problems of fundamental interest in statistics, such as regression, nonlinear function estimation and covariance estimation. More specifically, the PIs aim to show in regression problems that using global-local shrinkage instead of purely global shrinkage methods such as ridge regression or principal components regression can result in improved prediction. They aim to show global-local shrinkage priors are good candidates for non-informative analysis of low-dimensional functions of high-dimensional parameters. The PIs also propose to use global-local shrinkage in covariance estimation and joint mean-covariance estimation problems and apply the developed methodology in suitable applications arising from genomics or finance.