Mix & Match Hamiltonian Monte Carlo

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 dramatically improved by incorporating importance sampling and an irreversible part of the dynamics into the 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 has been further complemented by (i) the efficient algorithms for computation of modified Hamiltonians; (ii) the implicit momentum update procedure as well as (iii) the two-stage splitting integration schemes specially derived for the methods sampling with modified Hamiltonians. MMHMC has been implemented in the in-house software package HaiCS (Hamiltonians in Computational Statistics), tested on the popular statistical models and compared in sampling efficiency with HMC, Generalized Hybrid Monte Carlo (GHMC), Riemann Manifold Hamiltonian Monte Carlo (RMHMC), Metropolis Adjusted Langevin Algorithm (MALA) and Random Walk Metropolis-Hastings (RWMH). To make a fair comparison, we propose a metric that accounts for both 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.