  The Hamiltonian Monte Carlo (HMC) method has been recognized as a powerful
sampling tool in computational statistics. We show that performance of HMC can
be significantly improved by incorporating importance sampling and an
irreversible part of the dynamics into a chain. This is achieved by replacing
Hamiltonians in the Metropolis test with modified Hamiltonians, and a complete
momentum update with a partial momentum refreshment. We call the resulting
generalized HMC importance sampler---Mix & Match Hamiltonian Monte Carlo
(MMHMC). The method is irreversible by construction and further benefits from
(i) the efficient algorithms for computation of modified Hamiltonians; (ii) the
implicit momentum update procedure and (iii) the multi-stage splitting
integrators specially derived for the methods sampling with modified
Hamiltonians. MMHMC has been implemented, tested on the popular statistical
models and compared in sampling efficiency with HMC, Riemann Manifold
Hamiltonian Monte Carlo, Generalized Hybrid Monte Carlo, Generalized Shadow
Hybrid Monte Carlo, Metropolis Adjusted Langevin Algorithm and Random Walk
Metropolis-Hastings. To make a fair comparison, we propose a metric that
accounts for correlations among samples and weights, and can be readily used
for all methods which generate such samples. The experiments reveal the
superiority of MMHMC over popular sampling techniques, especially in solving
high dimensional problems.
