# Generalized Fiducial Inference on Differentiable Manifolds

arXiv: https://arxiv.org/abs/2209.15473 · HTML: https://mlsanity.com/paper/2209.15473/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Extended Fiducial Inference: Toward an Automated Process of Statistical Inference](https://mlsanity.com/paper/2407.21622/) (2024)
- [Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inference](https://mlsanity.com/paper/2505.19136/) (2025)
- [Extended Fiducial Inference for Individual Treatment Effects via Deep Neural Networks](https://mlsanity.com/paper/2505.01995/) (2025)

## Abstract (arXiv)

We introduce a novel approach to inference on parameters that take values in a Riemannian manifold embedded in a Euclidean space. Parameter spaces of this form are ubiquitous across many fields, including chemistry, physics, computer graphics, and geology. This new approach uses generalized fiducial inference to obtain a posterior-like distribution on the manifold, without needing to know a parameterization that maps the constrained space to an unconstrained Euclidean space. The proposed methodology, called the constrained generalized fiducial distribution (CGFD), is obtained by using mathematical tools from Riemannian geometry. A Bernstein-von Mises-type result for the CGFD, which provides intuition for how the desirable asymptotic qualities of the unconstrained generalized fiducial distribution are inherited by the CGFD, is provided. To demonstrate the practical use of the CGFD, we provide three proof-of-concept examples: inference for data from a multivariate normal density with the mean parameters on a sphere, a linear logspline density estimation problem, and a reimagined approach to the AR(1) model, all of which exhibit desirable coverages via simulation. We discuss two Markov chain Monte Carlo algorithms for the exploration of these constrained parameter spaces and adapt them for the CGFD.