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Riemannian Manifold Hamiltonian Monte Carlo

机译：黎曼流形哈密尔顿蒙特卡罗

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The paper proposes a Riemannian Manifold Hamiltonian Monte Carlo sampler toresolve the shortcomings of existing Monte Carlo algorithms when sampling fromtarget densities that may be high dimensional and exhibit strong correlations.The method provides a fully automated adaptation mechanism that circumvents thecostly pilot runs required to tune proposal densities for Metropolis-Hastingsor indeed Hybrid Monte Carlo and Metropolis Adjusted Langevin Algorithms. Thisallows for highly efficient sampling even in very high dimensions wheredifferent scalings may be required for the transient and stationary phases ofthe Markov chain. The proposed method exploits the Riemannian structure of theparameter space of statistical models and thus automatically adapts to thelocal manifold structure at each step based on the metric tensor. Asemi-explicit second order symplectic integrator for non-separable Hamiltoniansis derived for simulating paths across this manifold which provides highlyefficient convergence and exploration of the target density. The performance ofthe Riemannian Manifold Hamiltonian Monte Carlo method is assessed byperforming posterior inference on logistic regression models, log-Gaussian Coxpoint processes, stochastic volatility models, and Bayesian estimation ofparameter posteriors of dynamical systems described by nonlinear differentialequations. Substantial improvements in the time normalised Effective SampleSize are reported when compared to alternative sampling approaches. Matlab codeat \url{http://www.dcs.gla.ac.uk/inference/rmhmc} allows replication of allresults.
机译：提出了一种黎曼流形汉密尔顿蒙特卡洛采样器，以解决现有的蒙特卡洛算法从目标密度采样中可能存在的高维和强相关性的缺点。适用于Metropolis-Hastingsor，实际上是混合Monte Carlo和Metropolis调整的Langevin算法。即使在非常高的尺寸（对于马尔可夫链的瞬态和平稳相可能需要不同的缩放比例）的情况下，也可以进行高效采样。该方法利用统计模型参数空间的黎曼结构，从而基于度量张量自动适应每一步的局部流形结构。用于不可分哈密顿量的半显式二阶辛辛积分器，用于模拟跨此流形的路径，从而提供了高效的收敛性和对目标密度的探索。通过对逻辑回归模型，对数高斯Coxpoint过程，随机波动率模型以及用非线性微分方程描述的动力系统参数后验的贝叶斯估计进行后验推断，从而评估了黎曼流形哈密顿蒙特卡罗方法的性能。与其他抽样方法相比，报告了时间标准化有效样本量的显着改善。 Matlab代码\ url {http://www.dcs.gla.ac.uk/inference/rmhmc}允许复制所有结果。

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