A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors
This work addresses the need for uncertainty quantification in Bayesian inverse problems for researchers and practitioners in fields like signal processing and dynamical systems, though it is incremental as it builds on existing hierarchical models.
The paper tackled the problem of performing uncertainty quantification in hierarchical inverse problems with gamma hyperpriors, which was previously limited to maximum a posteriori estimation, and introduced a variational inference approach that yields accurate reconstruction and meaningful uncertainty quantification, as illustrated in computed examples like deconvolution and sparse identification of dynamical systems.
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.