  Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that
allows to sample high dimensional probability measures. It relies on the
integration of the Hamiltonian dynamics to propose a move which is then
accepted or rejected thanks to a Metropolis procedure. Unbiased sampling is
guaranteed by the preservation by the numerical integrators of two key
properties of the Hamiltonian dynamics: volume-preservation and reversibility
up to momentum reversal. For separable Hamiltonian functions, some standard
explicit numerical schemes, such as the St\"ormer-Verlet integrator, satisfy
these properties. However, for numerical or physical reasons, one may consider
a Hamiltonian function which is nonseparable, in which case the standard
numerical schemes which preserve the volume and satisfy reversibility up to
momentum reversal are implicit. When implemented in practice, such implicit
schemes may admit many solutions or none, especially when the timestep is too
large. We show here how to enforce the numerical reversibility, and thus
unbiasedness, of HMC schemes in this context by introducing a reversibility
check. In addition, for some specific forms of the Hamiltonian function, we
discuss the consistency of these HMC schemes with some Langevin dynamics, and
show in particular that our algorithm yields an efficient discretization of the
metropolized overdamped Langevin dynamics with position-dependent diffusion
coefficients. Numerical results illustrate the relevance of the reversibility
check on simple problems.
