UC Davis

The concept of fiducial inference was introduced by R. A. Fisher in the 1930s as a response to the limitations he perceived in Bayesian inference, notably the requirement for a subjective prior distribution on model parameters when no prior information exists. However, Fisher’s initial fiducial approach lost favor due to its complexity, especially in multi-parameter settings. A resurgence of interest in the early 2000s led to the development of generalized fiducial inference (GFI), which extends Fisher’s ideas and offers a promising framework for addressing a wide array of inferential challenges. Despite its potential, the adoption of GFI has been limited by its complex mathematical derivations and the need for sophisticated Markov Chain Monte Carlo (MCMC) algorithms.

This dissertation addresses these implementation challenges by proposing novel variants of GFI that simplify the sampling process and improve accessibility for researchers and practitioners. Specifically, Chapter 3 introduces AutoGFI, an intuitive algorithm that facilitates the application of GFI across diverse inference problems involving additive noise. Chapter 4 presents Fiducial Selector, a method specifically developed for high-dimensional linear regression within the GFI framework. Chapter 5 introduces AutoGFI-B and its regularized version AutoGFI-BR, which extend AutoGFI’s application to binary response models, thus broadening its use from continuous to binary data. Theoretical and empirical evaluations in this work validate the effectiveness of these innovative approaches, underscoring the significant potential of GFI in modern inference challenges. Overall, this research paves the way for a more accessible and powerful application of GFI across various practical domains, substantially expanding the toolkit available for robust statistical inference.