  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.
