A Differential Geometric Perspective on Generalized Fiducial Inference

Generalized fiducial inference (GFI) produces Bayesian-like, post-data probabilistic statements without specifying a prior distribution for model parameters. In the current article, we propose a new characterization of the generalized fiducial distribution (GFD) with the aid of differential geometry. Under suitable smoothness conditions, we establish that the GFD has an absolutely continuous density with respect to the intrinsic measure of a smooth manifold that is uniquely determined by the data generating equation. The geometric analysis also sheds light on the connection and distinction between GFI and Bayesian inference. Compared to the explicit expression of the fiducial density given by Theorem 1 of Hannig et al. (2016), our new results can be applied to a broader class of statistical models, including those with additional random effects. Furthermore, Monte Carlo approximations to the GFD can be conveniently constructed via manifold Markov chain Monte Carlo samplers. We provide an illustration using repeated-measures analysis of variance.