  Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC)
algorithm for estimating expectations with respect to continuous un-normalized
probability distributions. MCMC estimators typically have higher variance than
classical Monte Carlo with i.i.d. samples due to autocorrelations; most MCMC
research tries to reduce these autocorrelations. In this work, we explore a
complementary approach to variance reduction based on two classical Monte Carlo
"swindles": first, running an auxiliary coupled chain targeting a tractable
approximation to the target distribution, and using the auxiliary samples as
control variates; and second, generating anti-correlated ("antithetic") samples
by running two chains with flipped randomness. Both ideas have been explored
previously in the context of Gibbs samplers and random-walk Metropolis
algorithms, but we argue that they are ripe for adaptation to HMC in light of
recent coupling results from the HMC theory literature. For many posterior
distributions, we find that these swindles generate effective sample sizes
orders of magnitude larger than plain HMC, as well as being more efficient than
analogous swindles for Metropolis-adjusted Langevin algorithm and random-walk
Metropolis.
