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.
