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）方法因其能够快速探索高维状态空间、避免传统马尔可夫链蒙特卡洛采样器常见的随机游走行为，成为贝叶斯统计推断的重要工具。哈密顿动力学积分器的合理选择是影响HMC效率的关键。越来越多的研究表明，多级分裂积分器可替代传统HMC中使用的Verlet方法。本文在二级和三级分裂积分器框架下，提出了一种基于问题特性的最优积分方案（以谐波力/高斯目标能量守恒最优性为准则）。该方法称为自适应统计积分方法（s-AIA），利用多变量高斯模型和HMC预热阶段获得的模拟数据，识别系统特有的维度稳定区间，并在此区间内为用户选择的任意模拟步长分配最合适的二级/三级积分器。s-AIA已集成至自研软件包HaiCS中，且不增加模拟计算开销。通过与已知固定参数多级分裂积分器（含Verlet）的对比，本文讨论了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。在经典统计模型上的数值实验表明，自适应方案在二级和三级分裂方案中达到了最优性能。