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

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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.