Hybrid Monte Carlo on Hilbert spaces
Hybrid Monte Carlo on Hilbert spaces
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DOI：
10.1016/j.spa.2011.06.003
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发表时间：
2011-10-01
影响因子：
1.4
通讯作者：
Stuart, A. M.
中科院分区：
文献类型：
--
作者：
Beskos, A.;Pinski, F. J.;Stuart, A. M.
混合蒙特卡罗（HMC）算法为从复杂的高维目标分布中采样提供了一个框架。与标准的马尔可夫链蒙特卡罗（MCMC）算法相比，它在状态空间中产生非局部、非对称的移动，减轻了模拟轨迹的随机行走行为。然而，与基于随机漫步或Langevin建议的算法类似，探索目标分布所需的步数通常随着状态空间的维数而增长。我们定义了一种广义HMC算法，该算法克服了目标测度在无限维希尔伯特空间上以相对于高斯测度具有密度的测度π的有限维近似产生的问题。关键思想是构造一个在希尔伯特空间上定义良好的MCMC方法。我们在Hilbert空间的无限维设置中依次解决了以下问题：(i)在以目标Pi为边缘的扩大相空间中构造概率测度Pi，以及保持Pi的哈密顿流；（ii）发展适合哈密顿流的几何数值积分器；（iii）推导一个接受/拒绝规则，以确保在使用上述数值积分器而不是实际的哈密顿流时保持Pi。实验报告将新算法与标准HMC和定义在Hilbert空间上的Langevin MCMC方法的一个版本进行了比较。(C) 2011 Elsevier B.V.版权所有
The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, high-dimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures pi which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself.We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure Pi in an enlarged phase space having the target pi as a marginal, together with a Hamiltonian flow that preserves Pi; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of Pi when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space. (C) 2011 Elsevier B.V. All rights reserved.