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）因能快速探索高维状态空间、避免许多马尔可夫链蒙特卡洛采样器典型的随机游走行为，而成为贝叶斯统计推断的有力工具。正确选择哈密顿动力学的积分器是HMC效率的关键。日益明晰的是，多阶段分裂积分器是HMC传统使用的Verlet方法的良好替代方案。本文在二阶段和三阶段分裂积分器族中，提出了一种寻找最优问题特定积分方案（基于谐波力/高斯目标的最佳能量守恒）的原则性方法。该方法称为自适应积分统计方法（s-AIA），利用多元高斯模型和HMC预热阶段获得的模拟数据，识别系统特定的维度稳定性区间，并在该区间内为用户选择的任意模拟步长分配最合适的二/三阶段积分器。s-AIA已在内部软件包HaiCS中实现，且未引入额外计算开销。通过与已知固定参数多阶段分裂积分器（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。对经典统计模型的数值实验表明，自适应方案能够在二、三阶段分裂方案族中达到最佳性能。