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 for 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 behaviour of HMC as the dimensionality of the target distribution increases.

中文翻译：

---

几何积分器和哈密顿蒙特卡罗方法

本文详细调查了数值积分与哈密顿（或混合）蒙特卡洛方法（HMC）之间的关系。由于 HMC 的计算成本主要在于数值积分，因此这些应该尽可能高效地执行。然而，HMC 需要具有体积保持和可逆的几何特性的方法，这限制了可以使用的积分器的数量。另一方面，这些几何特性对积分误差具有重要的定量影响，进而影响提案的接受率。虽然目前速度 Verlet 算法是有充分理由的选择方法，但我们认为 Verlet 可以改进。我们还详细讨论了随着目标分布的维数增加 HMC 的行为。