Geometric integrators and the Hamiltonian Monte Carlo method
For practitioners of HMC, this paper provides insights into improving integration efficiency, though it is a survey without new empirical results.
This survey examines the relationship between numerical integration and Hamiltonian Monte Carlo (HMC), arguing that while the velocity Verlet algorithm is currently standard, it can be improved. It also analyzes HMC's performance as dimensionality increases.
This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications on the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behavior of HMC as the dimensionality of the target distribution increases.