  There has been considerable interest in designing Markov chain Monte Carlo
algorithms by exploiting numerical methods for Langevin dynamics, which
includes Hamiltonian dynamics as a deterministic case. A prominent approach is
Hamiltonian Monte Carlo (HMC), where a leapfrog discretization of Hamiltonian
dynamics is employed. We investigate a recently proposed class of irreversible
sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS),
which uses an augmented target density similarly as in HMC, but involves a
flexible proposal scheme and a carefully formulated acceptance-rejection scheme
to achieve generalized reversibility. We show that as the step size tends to 0,
the HAMS proposal satisfies a class of stochastic differential equations
including Langevin dynamics as a special case. We provide theoretical results
for HAMS under the univariate Gaussian setting, including the stationary
variance, the expected acceptance rate, and the spectral radius. From these
results, we derive default choices of tuning parameters for HAMS, such that
only the step size needs to be tuned in applications. Various relatively recent
algorithms for Langevin dynamics are also shown to fall in the class of HAMS
proposals up to negligible differences. Our numerical experiments on sampling
high-dimensional latent variables confirm that the HAMS algorithms consistently
achieve superior performance, compared with several Metropolis-adjusted
algorithms based on popular integrators of Langevin dynamics.
