  The Hamiltonian Monte Carlo method generates samples by introducing a
mechanical system that explores the target density. For distributions on
manifolds it is not always simple to perform the mechanics as a result of the
lack of global coordinates, the constraints of the manifold, and the
requirement to compute the geodesic flow. In this paper we explain how to
construct the Hamiltonian system on naturally reductive homogeneous spaces
using symplectic reduction, which lifts the HMC scheme to a matrix Lie group
with global coordinates and constant metric. This provides a general framework
that is applicable to many manifolds that arise in applications, such as
hyperspheres, hyperbolic spaces, symmetric positive-definite matrices,
Grassmannian, and Stiefel manifolds.
