Collaborative Research: Generalized Fiducial Inference for Massive Data and High Dimensional Problems

现代统计学之父费雪（R. A. Fisher）在20世纪30年代提出了“基准推理”（Fiducial Inference）的概念。虽然他的建议导致了一些量化不确定性的有趣方法，但当时其他著名的统计学家并不接受费雪的方法，因为它违背了当时的统计推断思想。从2000年左右开始，pi和合作者开始重新研究基准推理的思想，并发现费雪的方法，如果适当推广，将为解决许多重要和困难的不确定性量化问题打开大门。pi们把他们对费雪思想的概括称为广义基准推理。经过多年的初步调查，ppi为这一领域的系统研究计划制定了一个连贯的、深思熟虑的计划。这个项目的很大一部分为不同的建模问题开发了实际的解决方案，这些问题在不同的领域有自然的应用。金融（波动率估计）和测量科学（校准来自不同政府实验室的测量，例如，美国国家标准与技术研究院）是两个主要的例子，而其他包括基因表达数据、气候问题、推荐系统和计算机视觉。这个项目的动机是由pi引入的广义基准推理（GFI）的成功，作为费雪基准论证的推广。pi现在正在努力扩大他们的GFI方法，以处理大数据问题和其他由于我们快速收集大量数据的能力而出现的难题。特别是，pi计划对以下主题进行研究：(i)对GFI的基本问题进行彻底调查，包括与近似贝叶斯计算和高阶渐近的联系；（ii）在“大p小n”情况下使用GFI进行稀疏协方差估计；（iii）发展基准选择器的思想，使基准分布的稀疏性被诱导为最小化问题的自然结果；（iv）使用GFI对矩阵补全问题进行不确定性量化，(v)将GFI应用于各种实际问题，例如金融中的波动率估计和测量科学中的国际关键比较实验。

R. A. Fisher, the father of modern statistics, proposed the idea of Fiducial Inference in the 1930s. While his proposal led to some interesting methods for quantifying uncertainty, other prominent statisticians of the time did not accept Fisher's approach because it went against the ideas of statistical inference of the time. Beginning around the year 2000, the PIs and collaborators started to re-investigate the ideas of fiducial inference and discovered that Fisher's approach, when properly generalized, would open doors to solve many important and difficult problems of uncertainty quantification. The PIs termed their generalization of Fisher's ideas as generalized fiducial inference. After many years of preliminary investigations, the PIs developed a coherent, well thought out plan for a systematic research program in this area. A large part of this project develops practical solutions for different modeling problems that have natural applications in diverse fields. Finance (volatility estimation) and measurement science (calibration of measurements from different government labs, for example, US NIST) are two primary examples, while others include gene expression data, climate problems, recommender systems, and computer vision. This project is motivated by the success of generalized fiducial inference (GFI) as introduced by the PIs as a generalization of Fisher's fiducial argument. The PIs are now working towards scaling up their GFI methodology to handle big data problems and other difficult problems that have emerged due to our ability to collect massive amounts of data rapidly. In particular the PIs plan to conduct research into the following topics: (i) a thorough investigation of fundamental issues of GFI including connection with Approximate Bayesian Calculations and higher order asymptotics; (ii) sparse covariance estimation using GFI in the "large p small n" context; (iii) development of the idea of Fiducial Selector so that a sparsity of the fiducial distribution is induced as a natural outcome of a minimization problem; (iv) uncertainty quantification for the matrix completion problem using GFI, and (v) applications of GFI to a wide variety of practical problems, such as volatility estimation in finance and international key comparison experiments in measurement science.