Hamiltonian Monte Carlo (HMC) and its dynamic extensions, such as the No-U-Turn Sampler (NUTS), are powerful Markov chain Monte Carlo methods for sampling from complex, high-dimensional probability distributions. Riemannian manifold Hamiltonian Monte Carlo (RMHMC) extends HMC by allowing the mass matrix to depend on position, which can substantially improve mixing but also makes implementation considerably more challenging. In this paper, we study an adaptive hierarchical version of RMHMC that is well suited to many hierarchical sampling problems. A key feature of hierarchical RMHMC is that, unlike general RMHMC, it admits a closed-form explicit leapfrog integrator, enabling efficient implementation and direct use within dynamic HMC methods such as NUTS. We introduce an adaptive scheme that automatically tunes the parameters of the hierarchical mass matrix during simulation. Importantly, the target density need not exhibit any hierarchical or block structure; the hierarchy is instead imposed on the mass matrix as a modeling device to capture the local geometry of the target distribution. Numerical experiments demonstrate appealing empirical performance in high-dimensional Bayesian inference problems.

翻译：哈密顿蒙特卡洛及其动态扩展方法（如无回头采样器）是从复杂高维概率分布中采样的强大马尔可夫链蒙特卡洛方法。黎曼流形哈密顿蒙特卡洛通过允许质量矩阵随位置变化扩展了HMC，这能显著改善混合性能，但也使实现变得更具挑战性。本文研究了一种适用于许多层次化采样问题的自适应层次化RMHMC方法。层次化RMHMC的一个关键特征在于，与通用RMHMC不同，它允许使用封闭形式的显式跳跃积分器，从而能在NUTS等动态HMC方法中实现高效部署与直接应用。我们提出了一种自适应方案，可在模拟过程中自动调整层次化质量矩阵的参数。重要的是，目标密度无需具备任何层次或块状结构；相反，层次结构是作为建模手段施加于质量矩阵，用以捕捉目标分布的局部几何特性。数值实验在高维贝叶斯推断问题中展示了令人信服的实证性能。