# A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

arXiv: https://arxiv.org/abs/2111.13329 · HTML: https://mlsanity.com/paper/2111.13329/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 45 references that are in the map.
- First posted on arXiv 26 Nov 2021.

## Builds on (most cited first, 8 of 8)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Inverse Problems and Data Assimilation](https://mlsanity.com/paper/1810.06191/) (2018)
- [Iterative Ensemble Kalman Methods: A Unified Perspective with Some New Variants](https://mlsanity.com/paper/2010.13299/) (2020)
- [Variational MCMC](https://mlsanity.com/paper/1301.2266/) (2013)
- [Brain activity mapping from MEG data via a hierarchical Bayesian algorithm with automatic depth weighting: sensitivity and specificity analysis](https://mlsanity.com/paper/1707.05639/) (2017)
- [A variational inference framework for inverse problems](https://mlsanity.com/paper/2103.05909/) (2021)
- [Computing the quality of the Laplace approximation](https://mlsanity.com/paper/1711.08911/) (2017)

## Built on by (most cited first, 2 of 2)

- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [Leveraging joint sparsity in hierarchical Bayesian learning](https://mlsanity.com/paper/2303.16954/) (2023)

## Abstract (arXiv)

Hierarchical models with gamma hyperpriors provide a flexible, sparse-promoting framework to bridge L1 and L2 regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to textitmaximum 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.