Localization for MCMC: sampling high-dimensional posterior distributions with local structure
Localization for MCMC: sampling high-dimensional posterior distributions with local structure
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DOI：
10.1016/j.jcp.2018.12.008
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发表时间：
2019-03-01
影响因子：
4.1
通讯作者：
Marzouk, Y. M.
中科院分区：
文献类型：
--
作者：
Morzfeld, M.;Tong, X. T.;Marzouk, Y. M.
我们研究了如何将数值天气预测中协方差局部化的想法用于贝叶斯逆问题中产生的高维后验分布的马尔可夫链蒙特卡罗（MCMC）采样。局部化反问题是通过（i）忽略先验精度和协方差矩阵的小的非对角元素;以及（ii）将观测的影响限制在其邻域内来强制执行预期的“局部”结构。对于线性问题，我们可以指定的条件下，本地化问题的后验矩接近的原始问题。我们解释物理解释我们的假设，局部结构和讨论的概念，高维的局部问题，这是不同于通常的概念，高维的函数空间MCMC。Gibbs采样器是MCMC算法求解局部化反问题的一个自然选择，我们证明了它的收敛速度与局部化线性问题的维数无关。非线性问题也可以有效地解决本地化，作为这些想法的一个简单的说明，我们提出了一个本地化的大都市内吉布斯采样器。几个线性和非线性的数值例子说明本地化的背景下，MCMC采样反问题。(C)2018爱思唯尔公司All rights reserved.
We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems. (C) 2018 Elsevier Inc. All rights reserved.