Collaborative Research: Generalized Fiducial Inference in the Age of Data Science

数据及其使用在现代社会中变得非常重要。这就迫切需要研究统计和数据科学的数学基础。在这个项目中，PI探索广义置信推理与现代数据科学问题和技术的相互作用。拟议的研究有几个好处。首先，预计拟议的研究将增加我们对频率主义，基准和贝叶斯范式之间的推理和关系的理解，以及这些范式如何适应数据科学，旨在改善更好的数据科学实践。第二，它预计将导致新的和有效的程序量化的不确定性在一些应用程序。一个重要的例子是法医学中数据科学算法报告的似然比的校准，这对法庭上似然比的实际使用具有潜在的影响。此外，该项目还将为研究生提供研究机会，特别是帮助培养女性和少数民族研究生在对社会有巨大利益的领域。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Fisher的方法，如果适当推广，为解决不确定度量化的许多重要和困难问题打开了大门。经过多年的初步调查，该团队能够为该领域的系统研究计划制定一个连贯，深思熟虑的计划。PI将他们对Fisher思想的概括称为广义置信推理（GFI）。PI现在正致力于应用他们的GFI方法来处理由于我们快速收集大量数据的能力而出现的数据科学问题。具体而言，PI建议对以下主题进行研究：（1）深入研究GFI的基本问题，以便它们可以简单地用于流形，具有约束和惩罚。这对适用性至关重要。(2)无偏基准选择器的发展，使稀疏的基准分布诱导作为一个自然的结果的最小化问题和unbiasedness是使用一种新的去偏的方法来实现。(3)协方差估计的客观贝叶斯和置信解之间的相互作用。(4)基于网络内聚度的图子与回归的不确定性量化。(5)使用深度网络计算GFI。(6)GFI在各种实际问题中的应用;例如，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

Data and their use have become extremely important in modern society. This provides for an urgent need to study mathematical foundations of statistics and data science. In this project the PIs explore interaction of generalized fiducial inference with modern data science problems and techniques. There are several benefits of the proposed research. First, it is expected that the proposed research will increase our understanding of inference and relationships between the frequentist, fiducial and Bayesian paradigms and how do these paradigms fit into data science which the aim to improve better data science practice. Second, it is expected to lead to new and efficient procedures for quantifying uncertainty in a number of applications. An important example is the calibration of likelihood ratios reported by data science algorithms in forensic science that has potential implication for practical usage of likelihood ratios in courtroom. Additionally, the project will provide research opportunities to graduate students and, in particular, help train women and minority graduate students in the field that is of a great benefit to society.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, opens doors to solve many important and difficult problems of uncertainty quantification. After many years of preliminary investigations, the team was able to put together a coherent, well thought out plan for a systematic research program in this area. The PIs termed their generalization of Fisher's ideas as generalized fiducial inference (GFI). The PIs are now working towards applying their GFI methodology to handle data science problems that have emerged due to our ability to collect massive amounts of data rapidly. In particular the PIs propose to conduct research into the following topics: (1) In-depth investigation of fundamental issues of GFI so that they can be simply used on manifolds, with constraint, and penalties. This is essential for applicability. (2) Development of a bias free fiducial selector, so that a sparsity of the fiducial distribution is induced as a natural outcome of a minimization problem and unbiasedness is achieved using a novel de-biasing approach. (3) Interplay between objective Bayesian and fiducial solutions for covariance estimation. (4) Uncertainty quantification for graphon and regression with network cohesion. (5) Use of deep networks for computation of GFI. (6) Applications of GFI to a wide variety of practical problems; e.g., calibration of likelihood ratios used for quantifying evidence in forensic science.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.