High Dimensional Bayesian Model Discovery, Inference and Prediction

本研究一般涉及高维回归分析的新方法，其中参数数量与观测数量相比非常大。本研究的第一部分侧重于在这种情况下识别和推断低维结构的方法的开发和应用。为此，研究者进一步开发了一种新的非参数效应模型，称为BART（贝叶斯加性树模型），该模型使用树木的维度自适应动态随机基础。特别是，BART开辟了多种新的方式来进行变量选择，然后是模型选择。与传统的依赖于预先选择的参数模型类的变量选择方法不同，这些新方法将在选择模型之前选择变量，从而消除了以前方法的一个主要限制。本研究的第二部分侧重于发展新的贝叶斯程序，用于多元回归的预测密度估计。建立了在Kullback-Leibler损失下的密度估计的极小值，适用于各类标度超谐波先验。在这些估计的基础上，研究者进一步开发了最小最大倍数收缩预测密度估计，利用可变的选择不确定性来实现风险降低。在复杂环境中发现结构和预测未来不确定事件是大多数学术研究领域的基本统计挑战，包括商业、自然科学和社会科学、医学、人文科学、计算机科学和工程。通过建模和预测经济状况，经济学家可以更好地指导经济；通过模拟和预测气候变化，科学家可以更好地管理环境；通过建模和预测医疗保健需求，政策分析师可以更好地分配资源以满足需求。这项研究开发了全新的程序，将高维模型发现和预测估计建立为理论和方法研究的高度可见和有价值的领域，将吸引最聪明的学生和最聪明的经验丰富的学者。这将需要通过在世界各地的公开讲座，以及通过与同事和研究生的新合作，在顶级期刊上广泛传播这项工作。为了进一步促进这些新方法的使用，研究者将继续创建并公开提供开源软件实现以及完整的文档和示例。

This research is concerned generally with new methods for regression analysis in high dimensions, where the number of parameters is very large compared to the number of observations. The first part of this research focuses on the development and application of methods that identify and draw inference about low dimensional structure in this setting. For this purpose, the investigator further develops a new nonparametric effects model called BART (Bayesian Additive Tree Models) that uses a dimensionally adaptive dynamic random basis of trees. In particular, BART opens up a variety of new ways to perform variable selection followed by model selection. As opposed to traditional variable selection methods that rely on a preselected parametric model class, these new methods will select variables before the selection of a model, thus eliminating a major limitation of previous methods. The second part of this research focuses on the development of new Bayesian procedures for predictive density estimation in multiple regression. The minimaxity of such density estimates under Kullback-Leibler loss are established for various classes of scaled superharmonic priors. Based on these estimates, the investigator further develops minimax multiple shrinkage predictive density estimates that exploit variable selection uncertainty to achieve risk reduction. The discovery of structure in complex settings and the prediction of future uncertain events are fundamental statistical challenges in most areas of academic research including business, the hard and social sciences, medicine, humanities, computer science and engineering. By modeling and predicting economic conditions, the economist can better direct the economy; by modeling and predicting climate changes, the scientist can better manage the environment; by modeling and predicting health care needs, the policy analyst can better allocate resources to meet demand. This research develops brand new procedures that establishes high dimensional model discovery and predictive estimation as highly visible and valuable areas of theoretical and methodological research, one that will attract the brightest students and the wisest seasoned scholars. This will entail broad dissemination of the work in pinnacle journals, through public lectures around the world, and through new collaborations with colleagues and graduate students. To further facilitate the use of these new methods, the investigator will continue to create and make publicly available open-source software implementations along with full documentation and examples.