论文标题
贝叶斯逆问题的认证知识分子空间检测的事先归一化
Prior normalization for certified likelihood-informed subspace detection of Bayesian inverse problems
论文作者
论文摘要
马尔可夫链蒙特卡洛（MCMC）方法构成了贝叶斯逆问题的算法基础之一。近期可能性信息的子空间（LIS）方法的开发为设计有效的MCMC方法提供了可行的途径，用于探索高维后验分布，通过利用基本反面问题的内在低维结构。但是，现有的LIS方法和相关的性能分析通常假定先前的分布是高斯。对于旨在促进参数估计的稀疏性的反问题，这种假设受到限制，例如，在这种情况下，通常需要进行重尾的先验，例如拉普拉斯分布或贝叶斯套索中常用的弹性网。为了克服这一限制，我们考虑了一种先前的归一化技术，该技术将任何非高斯（例如重尾）先验转换为标准高斯分布，这使得可以实施LIS方法以通过此类转换来加速MCMC采样。我们还严格研究了这种转化与多种MCMC方法的整合，以解决高维问题。最后，我们证明了关于两个非线性反问题的理论主张的各个方面。
Markov chain Monte Carlo (MCMC) methods form one of the algorithmic foundations of Bayesian inverse problems. The recent development of likelihood-informed subspace (LIS) methods offers a viable route to designing efficient MCMC methods for exploring high-dimensional posterior distributions via exploiting the intrinsic low-dimensional structure of the underlying inverse problem. However, existing LIS methods and the associated performance analysis often assume that the prior distribution is Gaussian. This assumption is limited for inverse problems aiming to promote sparsity in the parameter estimation, as heavy-tailed priors, e.g., Laplace distribution or the elastic net commonly used in Bayesian LASSO, are often needed in this case. To overcome this limitation, we consider a prior normalization technique that transforms any non-Gaussian (e.g. heavy-tailed) priors into standard Gaussian distributions, which makes it possible to implement LIS methods to accelerate MCMC sampling via such transformations. We also rigorously investigate the integration of such transformations with several MCMC methods for high-dimensional problems. Finally, we demonstrate various aspects of our theoretical claims on two nonlinear inverse problems.
