  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.
