Hierarchical models with gamma hyperpriors provide a flexible, sparse-promoting framework to bridge $L^1$ and $L^2$ regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to \textit{maximum a posteriori} estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. We illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.

翻译：带有伽马超优质的等级模型提供了一个灵活、稀疏的促进框架,使巴伊西亚配方的正规化达到1美元和2美元,从而解决反面问题。尽管巴伊西亚的动机是这些模型,但现有方法仅限于\ textit{ 最大后遗症估计。进行不确定性量化的潜力尚未实现。本文件为伽马超优质的等级反向问题引入了一种变换迭代办法。提议的变换推理方法可以进行准确的重建,提供有意义的不确定性量化,并且易于实施。此外,它自然地为选择超比分计进行模型选择。我们用几个计算的例子来说明我们的方法的绩效,包括变异问题和从时间序列数据中很少发现动态系统。