Confident Bayes Regularization in Discrete Multi-way Layouts

在离散的多路布局中，影响被测响应的每个因素都具有有限数量的级别，这些级别可以是标称的（纯标签），也可以是有序的（其顺序和大小包含信息的实值）。通常，只有相对较少的噪声观测是在一些因素水平组合中进行的，这些因素水平组合构成了所有理论上可能的因素水平组合的子集。分析这类稀疏回归型数据的基本问题是如何有效地从观测因子水平组合的噪声中提取信号；可理解地评估提取信号的不确定性；并将提取的信号合理地外推到未观察到的因素水平组合。本项目将开发以下数据驱动的正则化方法：对不完全多路布局使用可处理的一般概率模型来激励多路布局均值的候选贝叶斯估计器类；B)在此一般模型下估计每个候选估计器的风险；C)定义正则化估计量为估计风险最小的候选估计量；D)从理论上证明正则化估计量的风险渐近收敛于最佳候选估计量的风险；E)建立以正则化估计量为中心的置信集，量化该估计量的不确定性；F)在案例研究数据和相关的人工数据上试验正则化估计器。本项目处理的多路布局数据的重要实例有空间数据、生物信息学的基因或蛋白质芯片数据、医学成像或工业质量控制的数字图像和视频。该项目将开发有效的半自动算法，将这些数据中的模式与不重要的细节或噪声分离开来。因此，在对大型多路布局进行有效的算法分析后，它将有助于将人工干预集中在仍然存在的微妙问题上。该项目将有助于解决2003年美国国家科学基金会报告“统计：21世纪的挑战和机遇”第4节中确定的几个核心研究挑战。这些技术挑战包括贝叶斯和有偏估计，数据简化和压缩，以及构建统计理论和计算实验之间的相互作用。项目结果将提交期刊发表，并将发布在PI的网站上。该方法的统一帐户计划为专著。该项目的部分内容，包括上述数据类型方法的案例研究应用，将指导PI学生的博士论文研究。

In a discrete multi-way layout, each of the factors affecting measured response takes on a finite number of levels, which may be either nominal (pure labels) or ordinal (real-values whose order and magnitude bear information). Often, only relatively few noisy observations are made at only some factor-level combinations that form a subset of all theoretically possible combinations of the factor levels. Basic problems in analyzing such sparse regression-type data are to extract efficiently signal from noise at the observed factor-level combinations; to assess intelligibly the uncertainty in the extracted signal; and to extrapolate plausibly the extracted signal to the unobserved factor-level combinations. This project will develop the following data-driven regularization methodology: use a tractable general probability model for the incomplete multi-way layout to motivate classes of candidate Bayes estimators for the means of the multi-way layout; b) estimate the risk of each candidate estimator under this general model; c) define the regularized estimator to be the candidate estimator with smallest estimated risk; d) prove theoretically that the risk of the regularized estimator converges asymptotically to that of the best candidate estimator; e) develop confidence sets centered at the regularized estimator that quantify the uncertainty of that estimator; f) experiment with the regularized estimators on case-study data and on pertinent artificial data.Important practical instances of the multi-way layout data treated in the project are spatial data, the gene or protein chip data of bioinformatics, and the digital images and videos of medical imaging or of industrial quality control. The project will develop effective, semiautomatic algorithms for separating pattern from unimportant details or noise in such data. Thereby, it will serve to focus human intervention on the subtle questions that remain after effective algorithmic analyses of large multi-way layouts. The project will contribute to solving several core research challenges identified in Section 4 of the 2003 NSF Report "Statistics: Challenges and Opportunities for the Twenty-First Century". Amongst these technical challenges are Bayes and biased estimation, data reduction and compression, and structuring the interaction between statistical theory and computational experiments. Project results will be submitted for journal publication and will be posted on the PI's website. A unified account of the methodology is planned for a monograph. Portions of the project, including case study applications of the methodology to the data types listed above, will guide Ph.D. thesis research by the PI's students.