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 Störmer–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.

哈密​​顿蒙特卡罗 (HMC) 是一种马尔可夫链蒙特卡罗方法，允许对高维概率度量进行采样。它依赖于哈密顿动力学的整合来提出一个举措，然后通过大都会程序来接受或拒绝该举措。无偏采样是通过数值积分器保留哈密顿动力学的两个关键属性来保证的：体积保留和动量反转的可逆性。对于可分离哈密顿函数，一些标准显式数值方案（例如 Störmer-Verlet 积分器）满足这些属性。然而，出于数值或物理原因，人们可以考虑不可分的哈密顿函数，在这种情况下，保留体积并满足动量反转的可逆性的标准数值方案是隐含的。在实践中实施时，这种隐式方案可能会接受许多解决方案，也可能没有解决方案，特别是当时间步长太大时。我们在这里展示如何通过引入可逆性检查来强制 HMC 方案的数值可逆性，从而实现无偏性。此外，对于哈密顿函数的某些特定形式，我们讨论了这些 HMC 方案与某些 Langevin 动力学的一致性，并特别表明我们的算法产生了具有位置相关扩散系数的都市化过阻尼 Langevin 动力学的有效离散化。数值结果说明了可逆性检查对简单问题的相关性。