# Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging

arXiv: https://arxiv.org/abs/2212.12499 · HTML: https://mlsanity.com/paper/2212.12499/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 64 references that are in the map.
- First posted on arXiv 23 Dec 2022.

## Builds on (most cited first, 30 of 30)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Least Ambiguous Set-Valued Classifiers with Bounded Error Levels](https://mlsanity.com/paper/1609.00451/) (2016)
- [Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control](https://mlsanity.com/paper/2110.01052/) (2021)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [A Conformal Prediction Approach to Explore Functional Data](https://mlsanity.com/paper/1302.6452/) (2013)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Bayesian Uncertainty Estimation of Learned Variational MRI Reconstruction](https://mlsanity.com/paper/2102.06665/) (2021)
- [Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimization](https://mlsanity.com/paper/1803.00889/) (2018)
- [Insights into analysis operator learning: From patch-based sparse models to higher-order MRFs](https://mlsanity.com/paper/1401.2804/) (2014)
- [Higher-order total variation approaches and generalisations](https://mlsanity.com/paper/1912.01587/) (2019)
- [A Generative Variational Model for Inverse Problems in Imaging](https://mlsanity.com/paper/2104.12630/) (2021)
- [Denoising Score-Matching for Uncertainty Quantification in Inverse Problems](https://mlsanity.com/paper/2011.08698/) (2020)
- [Computed Tomography Reconstruction using Generative Energy-Based Priors](https://mlsanity.com/paper/2203.12658/) (2022)
- [Distribution-Free Conditional Median Inference](https://mlsanity.com/paper/2102.07967/) (2021)

## Built on by (most cited first, 12 of 12)

- [FlowSDF: Flow Matching for Medical Image Segmentation Using Distance Transforms](https://mlsanity.com/paper/2405.18087/) (2024)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Score-Based Generative Models for Medical Image Segmentation using Signed Distance Functions](https://mlsanity.com/paper/2303.05966/) (2023)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [pcaGAN: Improving Posterior-Sampling cGANs via Principal Component Regularization](https://mlsanity.com/paper/2411.00605/) (2024)
- [Conformal Bounds on Full-Reference Image Quality for Imaging Inverse Problems](https://mlsanity.com/paper/2505.09528/) (2025)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Self-supervised conformal prediction for uncertainty quantification in Poisson imaging problems](https://mlsanity.com/paper/2502.19194/) (2025)
- [Minimax Multi-Target Conformal Prediction with Applications to Imaging Inverse Problems](https://mlsanity.com/paper/2511.13533/) (2025)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)
- [Bridging data-driven priors via the score function for posterior sampling -- Comparative review and experimental study](https://mlsanity.com/paper/2606.14800/) (2026)

## Abstract (arXiv)

In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, also a novel primal-dual Langevin algorithm for sampling from non-smooth distributions is introduced in this work.