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. Actually, 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. Numerical results illustrate the relevance of this correction 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。该方法依赖哈密顿动力学积分生成提议，并通过梅特罗波利斯过程决定接受或拒绝该提议。无偏采样的保证依赖于数值积分器保留哈密顿动力学的两个关键性质：体积守恒和动量反转条件下的可逆性。对于可分离哈密顿函数，某些标准显式数值格式（如斯托默-韦尔莱积分器）满足这些性质。然而，出于数值或物理原因，研究者可能考虑不可分离的哈密顿函数，此时能够保留体积守恒且满足动量反转可逆性的标准数值格式均为隐式格式。实际上，当此类隐式格式在实际应用时，尤其在时间步长过大时，可能产生多解或无解的情形。本文展示了在此背景下如何强制数值可逆性，从而确保HMC方案的无偏性。数值结果验证了该修正方法在简单问题上的有效性。