Abstract

Various Markov chain Monte Carlo (MCMC) methods are studied to improve upon random walk Metropolis sampling, for simulation from complex distributions. Examples include Metropolis-adjusted Langevin algorithms, Hamiltonian Monte Carlo, and other algorithms related to underdamped Langevin dynamics. We propose a broad class of irreversible sampling algorithms, called Hamiltonian-assisted Metropolis sampling (HAMS), and develop two specific algorithms with appropriate tuning and preconditioning strategies. Our HAMS algorithms are designed to simultaneously achieve two distinctive properties, while using an augmented target density with a momentum as an auxiliary variable. One is generalized detailed balance, which induces an irreversible exploration of the target. The other is a rejection-free property for a Gaussian target with a prespecified variance matrix. This property allows our preconditioned algorithms to perform satisfactorily with relatively large step sizes. Furthermore, we formulate a framework of generalized Metropolis–Hastings sampling, which not only highlights our construction of HAMS at a more abstract level, but also facilitates possible further development of irreversible MCMC algorithms. We present several numerical experiments, where the proposed algorithms consistently yield superior results among existing algorithms using the same preconditioning schemes.

摘要

研究了各种马尔可夫链蒙特卡罗 (MCMC) 方法，以改进随机游走 Metropolis 采样，以进行复杂分布的模拟。示例包括 Metropolis 调整的 Langevin 算法、哈密顿蒙特卡罗以及与欠阻尼 Langevin 动力学相关的其他算法。我们提出了一类广泛的不可逆采样算法，称为哈密顿辅助大都市采样（HAMS），并开发了两种具有适当调整和预处理策略的特定算法。我们的 HAMS 算法旨在同时实现两个独特的属性，同时使用增强的目标密度和动量作为辅助变量。一是广义的细节平衡，引发对目标的不可逆转的探索。另一个是具有预先指定方差矩阵的高斯目标的无拒绝属性。此属性使我们的预处理算法能够以相对较大的步长令人满意地执行。此外，我们制定了广义Metropolis-Hastings采样的框架，这不仅突出了我们在更抽象的层面上构建HAMS，而且有利于不可逆MCMC算法的进一步发展。我们提出了几个数值实验，其中所提出的算法在使用相同预处理方案的现有算法中始终产生优异的结果。这不仅突出了我们在更抽象层面上构建的 HAMS，而且还促进了不可逆 MCMC 算法的进一步发展。我们提出了几个数值实验，其中所提出的算法在使用相同预处理方案的现有算法中始终产生优异的结果。这不仅突出了我们在更抽象层面上构建的 HAMS，而且还促进了不可逆 MCMC 算法的进一步发展。我们提出了几个数值实验，其中所提出的算法在使用相同预处理方案的现有算法中始终产生优异的结果。