Title: An Introduction to Extended Fiducial Inference: Theory, Computation, and Applications

Abstract: Extended fiducial inference provides a general paradigm for statistical inference in modern data science. Its defining feature is to formulate statistical inference as the problem of solving data-generating equations under uncertainty. EFI first constructs a fiducial law for the unobserved random errors underlying the observations and then propagates this law to the parameter space, either through an inverse mapping or, more generally, through a compatibility measure defined on the inverse fiber. This formulation enables EFI to address several difficulties encountered in Fisher’s original fiducial argument, including issues related to conditioning, marginalization, and non-invertibility. We develop general theoretical foundations for EFI that accommodate both single-valued and set-valued inverse mappings and apply in both low- and high-dimensional regimes. We also develop an adaptive stochastic-gradient MCMC computational framework for simulating the latent random errors and learning neural inverse mappings. The theory and computational methodology are illustrated through a variety of examples, including complex hypothesis testing, semi-supervised learning and uncertainty quantification for physics-informed neural networks.