Low-Rank Independence Samplers in Hierarchical Bayesian Inverse Problems
Low-Rank Independence Samplers in Hierarchical Bayesian Inverse Problems
复制标题
DOI：
10.1137/17m1137218
复制
发表时间：
2018-07
期刊：
影响因子：
--
通讯作者：
A. Saibaba;Sarah Vall
中科院分区：
文献类型：
--
作者：
A. Saibaba;Sarah Vall
在贝叶斯逆问题中，后验分布用于量化重构解的不确定性。在先验参数被分配为超先验的完全贝叶斯方法中，马尔可夫链蒙特卡罗算法通常用于从后验分布中提取样本。然而，这样的算法的实现可能在计算上是昂贵的。我们提出了一个计算效率高的方案，用于采样高维高斯分布的不适定贝叶斯线性逆问题。我们的方法使用大都会-黑斯廷斯独立抽样的建议分布的基础上，一个低秩近似的先验预处理海森。我们显示的接受率的依赖保留的特征值的数量和讨论的条件下，接受率高。我们证明了我们提出的采样器，使用它与大都会-Hastings-within-Gibbs抽样在图像去模糊，计算机断层扫描，核磁共振弛豫数值实验。
In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In fully Bayesian approaches in which prior parameters are assigned hyperpriors, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution. However, implementations of such algorithms can be computationally expensive. We present a computationally efficient scheme for sampling high-dimensional Gaussian distributions in ill-posed Bayesian linear inverse problems. Our approach uses Metropolis--Hastings independence sampling with a proposal distribution based on a low-rank approximation of the prior-preconditioned Hessian. We show the dependence of the acceptance rate on the number of eigenvalues retained and discuss conditions under which the acceptance rate is high. We demonstrate our proposed sampler by using it with Metropolis--Hastings-within-Gibbs sampling in numerical experiments in image deblurring, computerized tomography, and NMR relaxometry.