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:贝叶斯分层模型已被证明可提供有效算法,用于找到针对不适定反问题的稀疏解。该模型通常由未知数的条件高斯先前模型构成,再加上方差的超先前模型。超先前的一个广泛使用的选择是广义 Gamma 分布。文献中大部分工作集中在数值近似最大后验(MAP)的估计上,而没有放太多注意力在采样方法或其他方式来进行不确定性量化方面。从分层模型的采样通常存在两个挑战:由于分层模型通常是高维度的,因此会受到维度灾难的影响,加之未知数与其方差之间存在强相关性,使得采样变得非常不高效。本文主要解决这两个障碍中的第一个。通过使用一种新的重新参数化方法,本文展示了后验分布如何转化为由高斯白噪声主导的分布,从而可以通过预处理的 Crank-Nicholson (pCN) 方案采样,该方案已被证明对于从由高斯分量主导的分布中采样非常有效。此外,本文还开发了一种在特殊情况下加速pCN的新方法,并以计算示例的形式讨论了分层模型集中于稀疏解的程度。