Bayesian hierarchical models can provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models typically comprise a conditionally Gaussian prior model for the unknown which is augmented by a generalized gamma hyper-prior model for variance hyper-parameters. This investigation generalizes these models and their efficient maximum a posterior (MAP) estimation using the iterative alternating sequential (IAS) algorithm in two ways: (1) General sparsifying transforms: Diverging from conventional methods, our approach permits the use of sparsifying transformations with nontrivial kernels; (2) Unknown noise variances: We treat the noise variance as a random variable that is estimated during the inference procedure. This is important in applications where the noise estimate cannot be accurately estimated a priori. Remarkably, these augmentations neither significantly burden the computational expense of the algorithm nor compromise its efficacy. We include convexity and convergence analysis for the method and demonstrate its efficacy in several numerical experiments.

翻译：贝叶斯分层模型可为病态反问题的稀疏解提供高效算法。该类模型通常包含一个条件高斯先验模型用于描述未知量，并通过广义伽马超先验模型对方差超参数进行增强。本研究从以下两方面推广了这些模型及其基于迭代交替顺序（IAS）算法的最大后验（MAP）高效估计方法：（1）广义稀疏化变换：与传统方法不同，本方法允许使用具有非平凡核的稀疏化变换；（2）未知噪声方差：我们将噪声方差视为随机变量，在推理过程中进行估计。在无法通过先验信息准确估计噪声的应用场景中，这一特性尤为重要。值得注意的是，这些扩展既未显著增加算法的计算负担，也未损害其有效性。我们对该方法进行了凸性与收敛性分析，并通过多项数值实验验证了其有效性。