Recycling intermediate steps to improve Hamiltonian Monte Carlo

Hamiltonian Monte Carlo and related algorithms have become routinely used in Bayesian computation. The utility of such approaches is highlighted in the software package STAN, which provides a platform for automatic implementation of general Bayesian models. Hence, methods for improving the efficiency of general Hamiltonian Monte Carlo algorithms can have a substantial impact on practice. We propose such a method in this article by recycling the intermediate leap-frog steps used in approximating the Hamiltonian trajectories. Current algorithms use only the final step, and wastefully discard all the intermediate steps. We propose a simple and provably accurate approach for using these intermediate samples, boosting the effective sample size with little programming effort and essentially no extra computational cost. We show that our recycled Hamiltonian Monte Carlo algorithm can lead to substantial gains in computational efficiency in a variety of experiments.