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.

翻译：马尔可夫链蒙特卡洛（MCMC）方法构成了贝叶斯反问题算法基础的重要一环。最近发展的似然信息子空间（LIS）方法通过利用底层反问题的内在低维结构，为设计高效探索高维后验分布的MCMC方法提供了可行路径。然而，现有LIS方法及其相关性能分析通常假定先验分布为高斯分布。这一假设在参数估计中旨在促进稀疏性的反问题中具有局限性，因为此类问题常需使用重尾先验（例如贝叶斯LASSO中常用的拉普拉斯分布或弹性网络）。为克服这一局限，我们考虑一种先验归一化技术，可将任何非高斯（例如重尾）先验变换为标准高斯分布，从而使得通过此类变换实现LIS方法以加速MCMC采样成为可能。我们还严谨地研究了这种变换与多种面向高维问题的MCMC方法的集成方案。最后，通过两个非线性反问题实例验证了理论主张的多个方面。