  Riemannian manifold Hamiltonian Monte Carlo is traditionally carried out
using the generalized leapfrog integrator. However, this integrator is not the
only choice and other integrators yielding valid Markov chain transition
operators may be considered. In this work, we examine the implicit midpoint
integrator as an alternative to the generalized leapfrog integrator. We discuss
advantages and disadvantages of the implicit midpoint integrator for
Hamiltonian Monte Carlo, its theoretical properties, and an empirical
assessment of the critical attributes of such an integrator for Hamiltonian
Monte Carlo: energy conservation, volume preservation, and reversibility.
Empirically, we find that while leapfrog iterations are faster, the implicit
midpoint integrator has better energy conservation, leading to higher
acceptance rates, as well as better conservation of volume and better
reversibility, arguably yielding a more accurate sampling procedure.
