SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 1, Page 29-64, January 2021.
In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-dimensionality, nonlinear dependence, and nonconcave structures in the joint posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we develop scalable optimization-based Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems with nonlinear parameter-to-observable maps and a broader class of hyperparameters. Our algorithmic development is based on the recently developed scalable randomize-then-optimize (RTO) method [J. M. Bardsley et al., SIAM J. Sci. Comput., 42 (2016), pp. A1317--A1347] for exploring the high- or infinite-dimensional parameter space. We first extend the RTO machinery to the Poisson likelihood and discuss the implementation of RTO in the hierarchical setting. Then, by using RTO either as a proposal distribution in a Metropolis-within-Gibbs update or as a biasing distribution in the pseudomarginal MCMC [C. Andrieu and G. O. Roberts, Ann. Statist., 37 (2009), pp. 697--725], we present efficient sampling tools for hierarchical Bayesian inversion. In particular, the integration of RTO and the pseudomarginal MCMC has sampling performance robust to model parameter dimensions. Numerical examples in PDE-constrained inverse problems and positron emission tomography are used to demonstrate the performance of our methods.

中文翻译：

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非线性层次统计逆问题的基于优化的马尔可夫链蒙特卡罗方法

SIAM / ASA不确定性量化杂志，第9卷，第1期，第29-64页，2021年1月。
在许多层次逆问题中，我们不仅要估计参数到可观察映射中的高维或无量纲模型参数，而且还必须估计表示统计和数学建模过程中关键假设的超参数。在模型参数和超参数的联合后验分布中，由于高维，非线性相关性和非凹结构的联合效应，解决分层贝叶斯设置中的逆问题带来了巨大的计算挑战。在这项工作中，我们开发了基于可伸缩优化的马尔可夫链蒙特卡洛（MCMC）方法，用于解决带有非线性参数到可观察映射和更广泛的超参数类别的贝叶斯逆问题。我们的算法开发基于最近开发的可扩展的随机优化然后优化（RTO）方法[JM Bardsley等，SIAM J. Sci。计算（42）（2016），第A1317--A1347页]，用于探索高维或无限维参数空间。我们首先将RTO机制扩展到Poisson可能性，然后讨论在分层设置中RTO的实现。然后，通过将RTO用作“城域内城域”更新中的提议分布或用作伪边际MCMC中的偏差分布[C. 安德列（Andrieu）和GO罗伯茨（GO Roberts），安。Statist。，37（2009），pp。697--725]，我们提出了用于分层贝叶斯反演的有效采样工具。特别是，RTO和伪边际MCMC的集成具有对模型参数维模型鲁棒的采样性能。