Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that allows to sample high dimensional probability measures. It relies on the integration of the Hamiltonian dynamics to propose a move which is then accepted or rejected thanks to a Metropolis procedure. Unbiased sampling is guaranteed by the preservation by the numerical integrators of two key properties of the Hamiltonian dynamics: volume-preservation and reversibility up to momentum reversal. For separable Hamiltonian functions, some standard explicit numerical schemes, such as the St\"ormer-Verlet integrator, satisfy these properties. However, for numerical or physical reasons, one may consider a Hamiltonian function which is nonseparable, in which case the standard numerical schemes which preserve the volume and satisfy reversibility up to momentum reversal are implicit. When implemented in practice, such implicit schemes may admit many solutions or none, especially when the timestep is too large. We show here how to enforce the numerical reversibility, and thus unbiasedness, of HMC schemes in this context by introducing a reversibility check. In addition, for some specific forms of the Hamiltonian function, we discuss the consistency of these HMC schemes with some Langevin dynamics, and show in particular that our algorithm yields an efficient discretization of the metropolized overdamped Langevin dynamics with position-dependent diffusion coefficients. Numerical results illustrate the relevance of the reversibility check on simple problems.

翻译：哈密顿蒙特卡洛（HMC）是一种马尔可夫链蒙特卡洛方法，可用于对高维概率测度进行采样。该方法依赖于哈密顿动力学的积分来生成提议状态，并通过梅特罗波利斯过程接受或拒绝该提议。无偏采样的保证在于数值积分器需保留哈密顿动力学的两个关键性质：体积守恒和（除动量反转外）可逆性。对于可分离的哈密顿函数，某些标准显式数值格式（如斯托默-韦莱积分器）满足这些性质。然而，出于数值或物理原因，可能需考虑不可分离的哈密顿函数，此时保留体积并满足动量反转可逆性的标准数值格式均为隐式格式。实际应用中，此类隐式格式可能存在多个解或无解，尤其在时间步长过大时。本文展示了在此背景下如何通过引入可逆性检查来强制实现HMC格式的数值可逆性，从而保证无偏性。此外，针对特定形式的哈密顿函数，我们讨论了这些HMC格式与某些朗之万动力学的一致性，并特别指出我们的算法可实现对具有位置相关扩散系数的梅特罗波利斯化过阻尼朗之万动力学的高效离散化。数值结果在简单问题中验证了可逆性检查的有效性。