# Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms

arXiv: https://arxiv.org/abs/2103.10182 · HTML: https://mlsanity.com/paper/2103.10182/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 63 references that are in the map.
- First posted on arXiv 18 Mar 2021.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Image2StyleGAN: How to Embed Images Into the StyleGAN Latent Space?](https://mlsanity.com/paper/1904.03189/) (2019)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Optimizing the Latent Space of Generative Networks](https://mlsanity.com/paper/1707.05776/) (2017)
- [Inverting The Generator Of A Generative Adversarial Network](https://mlsanity.com/paper/1611.05644/) (2016)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)
- [Bayesian Image Reconstruction using Deep Generative Models](https://mlsanity.com/paper/2012.04567/) (2020)
- [Auxiliary gradient-based sampling algorithms](https://mlsanity.com/paper/1610.09641/) (2016)
- [On the well-posedness of Bayesian inverse problems](https://mlsanity.com/paper/1902.10257/) (2019)
- [Perturbation theory for Markov chains via Wasserstein distance](https://mlsanity.com/paper/1503.04123/) (2015)
- [Uncertainty quantification for radio interferometric imaging: I. proximal MCMC methods](https://mlsanity.com/paper/1711.04818/) (2017)
- [Solving Inverse Problems by Joint Posterior Maximization with a VAE Prior](https://mlsanity.com/paper/1911.06379/) (2019)

## Built on by (most cited first, 24 of 24)

- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimization](https://mlsanity.com/paper/2503.12615/) (2025)
- [Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priors](https://mlsanity.com/paper/2203.10078/) (2022)
- [Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling](https://mlsanity.com/paper/2402.12292/) (2024)
- [Inverse problem regularization with hierarchical variational autoencoders](https://mlsanity.com/paper/2303.11217/) (2023)
- [Do Bayesian imaging methods report trustworthy probabilities?](https://mlsanity.com/paper/2405.08179/) (2024)
- [Learning few-step posterior samplers by unfolding and distillation of diffusion models](https://mlsanity.com/paper/2507.02686/) (2025)
- [Variational Bayes image restoration with compressive autoencoders](https://mlsanity.com/paper/2311.17744/) (2023)
- [Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation](https://mlsanity.com/paper/2404.05445/) (2024)
- [Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurements](https://mlsanity.com/paper/2510.11964/) (2025)
- [Enhancing Electrical Impedance Tomography reconstruction using Learned Half-Quadratic Splitting Networks with Anderson Acceleration](https://mlsanity.com/paper/2304.14491/) (2023)
- [How many measurements are enough? Bayesian recovery in inverse problems with general distributions](https://mlsanity.com/paper/2505.10630/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Advantage of Machine Learning over Maximum Likelihood in Limited-Angle Low-Photon X-Ray Tomography](https://mlsanity.com/paper/2111.08011/) (2021)
- [Generative models and Bayesian inversion using Laplace approximation](https://mlsanity.com/paper/2203.07755/) (2022)
- [Learned reconstruction methods for inverse problems: sample error estimates](https://mlsanity.com/paper/2312.14078/) (2023)
- [Deep priors for satellite image restoration with accurate uncertainties](https://mlsanity.com/paper/2412.04130/) (2024)
- [Sampling Strategies in Bayesian Inversion: A Study of RTO and Langevin Methods](https://mlsanity.com/paper/2406.16658/) (2024)
- [Hypothesis Testing in Imaging Inverse Problems](https://mlsanity.com/paper/2505.22481/) (2025)
- [Deep unfolding of MCMC kernels: scalable, modular & explainable GANs for high-dimensional posterior sampling](https://mlsanity.com/paper/2602.20758/) (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)

This paper proposes a new methodology for performing Bayesian inference in imaging inverse problems where the prior knowledge is available in the form of training data. Following the manifold hypothesis and adopting a generative modelling approach, we construct a data-driven prior that is supported on a sub-manifold of the ambient space, which we can learn from the training data by using a variational autoencoder or a generative adversarial network. We establish the existence and well-posedness of the associated posterior distribution and posterior moments under easily verifiable conditions, providing a rigorous underpinning for Bayesian estimators and uncertainty quantification analyses. Bayesian computation is performed by using a parallel tempered version of the preconditioned Crank-Nicolson algorithm on the manifold, which is shown to be ergodic and robust to the non-convex nature of these data-driven models. In addition to point estimators and uncertainty quantification analyses, we derive a model misspecification test to automatically detect situations where the data-driven prior is unreliable, and explain how to identify the dimension of the latent space directly from the training data. The proposed approach is illustrated with a range of experiments with the MNIST dataset, where it outperforms alternative image reconstruction approaches from the state of the art. A model accuracy analysis suggests that the Bayesian probabilities reported by the data-driven models are also remarkably accurate under a frequentist definition of probability.