Bayesian hierarchical models have been demonstrated to provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models comprise typically a conditionally Gaussian prior model for the unknown, augmented by a hyperprior model for the variances. A widely used choice for the hyperprior is a member of the family of generalized gamma distributions. Most of the work in the literature has concentrated on numerical approximation of the maximum a posteriori (MAP) estimates, and less attention has been paid on sampling methods or other means for uncertainty quantification. Sampling from the hierarchical models is challenging mainly for two reasons: The hierarchical models are typically high-dimensional, thus suffering from the curse of dimensionality, and the strong correlation between the unknown of interest and its variance can make sampling rather inefficient. This work addresses mainly the first one of these obstacles. By using a novel reparametrization, it is shown how the posterior distribution can be transformed into one dominated by a Gaussian white noise, allowing sampling by using the preconditioned Crank-Nicholson (pCN) scheme that has been shown to be efficient for sampling from distributions dominated by a Gaussian component. Furthermore, a novel idea for speeding up the pCN in a special case is developed, and the question of how strongly the hierarchical models are concentrated on sparse solutions is addressed in light of a computed example.

翻译：贝叶斯层次模型已被证明能够为非适定逆问题的稀疏解求解提供高效算法。此类模型通常包含一个条件高斯先验模型（用于未知量），并辅以方差超先验模型。广泛使用的超先验选择是广义伽马分布族中的成员。现有文献大多集中于最大后验（MAP）估计的数值近似，而较少关注采样方法或其他不确定性量化手段。从层次模型中采样主要面临两大挑战：其一，层次模型通常具有高维特性，因而受困于维度灾难；其二，待求未知量与其方差之间的强相关性可能导致采样效率低下。本文主要针对第一个障碍展开研究。通过引入一种新颖的重参数化方法，我们展示了如何将后验分布转化为以高斯白噪声为主导的分布形式，从而可借助预条件克兰克-尼科尔森（pCN）方案进行采样——该方案已被证明对高斯分量主导的分布采样具有高效性。此外，本文还提出一种在特殊情形下加速pCN方法的新思路，并通过计算实例探讨了层次模型对稀疏解的集中程度问题。