Efficient Marginalization-Based MCMC Methods for Hierarchical Bayesian Inverse Problems
Efficient Marginalization-Based MCMC Methods for Hierarchical Bayesian Inverse Problems
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DOI：
10.1137/18m1220625
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发表时间：
2018-11
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影响因子：
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通讯作者：
A. Saibaba;Johnathan M. Bardsley;D. Brown;A. Alexanderian
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文献类型：
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作者：
A. Saibaba;Johnathan M. Bardsley;D. Brown;A. Alexanderian
贝叶斯反问题中的分层模型的特点是假设未知状态和测量误差精度的先验概率分布，以及先验参数的超先验。使用贝叶斯定律组合这些概率模型通常会产生无法直接采样的后验分布，即使是具有高斯测量误差和高斯先验的线性模型。吉布斯采样可以用来从后验中采样，但是当状态的维数很大时会出现问题。这是因为每次迭代所需的高斯样本计算起来可能非常昂贵，并且因为马尔可夫链的统计效率随着状态维度的增加而降低。后一个问题可以使用基于边缘化的技术来缓解，但这些技术也可能在计算上是禁止的。在本文中，我们将Brown，Saibaba和Vallelian（2018）的低秩技术与芸香和Held（2005）的边缘化方法相结合。我们认为这种方法的两个变种：延迟接受和伪边缘化。我们提供了一个详细的分析与我们提出的算法的接受率和计算成本，并比较其性能的两个数值测试案例-图像去模糊和逆热方程。
Hierarchical models in Bayesian inverse problems are characterized by an assumed prior probability distribution for the unknown state and measurement error precision, and hyper-priors for the prior parameters. Combining these probability models using Bayes' law often yields a posterior distribution that cannot be sampled from directly, even for a linear model with Gaussian measurement error and Gaussian prior. Gibbs sampling can be used to sample from the posterior, but problems arise when the dimension of the state is large. This is because the Gaussian sample required for each iteration can be prohibitively expensive to compute, and because the statistical efficiency of the Markov chain degrades as the dimension of the state increases. The latter problem can be mitigated using marginalization-based techniques, but these can be computationally prohibitive as well. In this paper, we combine the low-rank techniques of Brown, Saibaba, and Vallelian (2018) with the marginalization approach of Rue and Held (2005). We consider two variants of this approach: delayed acceptance and pseudo-marginalization. We provide a detailed analysis of the acceptance rates and computational costs associated with our proposed algorithms, and compare their performances on two numerical test cases---image deblurring and inverse heat equation.