Hamiltonian Monte Carlo (HMC) is a powerful tool for Bayesian statistical inference due to its potential to rapidly explore high dimensional state space, avoiding the random walk behavior typical of many Markov Chain Monte Carlo samplers. The proper choice of the integrator of the Hamiltonian dynamics is key to the efficiency of HMC. It is becoming increasingly clear that multi-stage splitting integrators are a good alternative to the Verlet method, traditionally used in HMC. Here we propose a principled way of finding optimal, problem-specific integration schemes (in terms of the best conservation of energy for harmonic forces/Gaussian targets) within the families of 2- and 3-stage splitting integrators. The method, which we call Adaptive Integration Approach for statistics, or s-AIA, uses a multivariate Gaussian model and simulation data obtained at the HMC burn-in stage to identify a system-specific dimensional stability interval and assigns the most appropriate 2-/3-stage integrator for any user-chosen simulation step size within that interval. s-AIA has been implemented in the in-house software package HaiCS without introducing computational overheads in the simulations. The efficiency of the s-AIA integrators and their impact on the HMC accuracy, sampling performance and convergence are discussed in comparison with known fixed-parameter multi-stage splitting integrators (including Verlet). Numerical experiments on well-known statistical models show that the adaptive schemes reach the best possible performance within the family of 2-, 3-stage splitting schemes.

翻译：哈密顿蒙特卡洛（HMC）是贝叶斯统计推断中的强大工具，因其能够快速探索高维状态空间，避免了马尔可夫链蒙特卡洛采样器中常见的随机游走行为。哈密顿动力学的积分器选择对HMC的效率至关重要。越来越多的证据表明，多阶段分裂积分器是传统Verlet方法（HMC中常用）的优秀替代方案。本文提出一种在2阶段和3阶段分裂积分器族中，针对特定问题寻找最优积分方案（以谐波力/高斯目标的最佳能量守恒为准则）的原则性方法。该方法被称为“面向统计的自适应积分方法”（s-AIA），利用HMC预热阶段获得的多变量高斯模型和模拟数据，识别系统特定的维度稳定区间，并在该区间内为用户选择的任意模拟步长分配最合适的2阶段或3阶段积分器。s-AIA已集成到内部软件包HaiCS中，且不增加模拟的计算开销。通过与已知固定参数的多阶段分裂积分器（包括Verlet）对比，讨论了s-AIA积分器的效率及其对HMC精度、采样性能和收敛性的影响。基于经典统计模型的数值实验表明，自适应方案在2阶段和3阶段分裂格式族中达到了最佳性能。