# On the Posterior Distribution in Denoising: Application to Uncertainty Quantification

arXiv: https://arxiv.org/abs/2309.13598 · HTML: https://mlsanity.com/paper/2309.13598/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 28 of its 66 references that are in the map.
- First posted on arXiv 24 Sep 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [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)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [MaskGAN: Towards Diverse and Interactive Facial Image Manipulation](https://mlsanity.com/paper/1907.11922/) (2019)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Label-Efficient Semantic Segmentation with Diffusion Models](https://mlsanity.com/paper/2112.03126/) (2021)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [GENIE: Higher-Order Denoising Diffusion Solvers](https://mlsanity.com/paper/2210.05475/) (2022)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [A Poisson-Gaussian Denoising Dataset with Real Fluorescence Microscopy Images](https://mlsanity.com/paper/1812.10366/) (2018)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Probabilistic Noise2Void: Unsupervised Content-Aware Denoising](https://mlsanity.com/paper/1906.00651/) (2019)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Conffusion: Confidence Intervals for Diffusion Models](https://mlsanity.com/paper/2211.09795/) (2022)
- [Semantic uncertainty intervals for disentangled latent spaces](https://mlsanity.com/paper/2207.10074/) (2022)
- [Interval Neural Networks: Uncertainty Scores](https://mlsanity.com/paper/2003.11566/) (2020)

## Built on by (most cited first, 30 of 33)

- [Zero-Shot Unsupervised and Text-Based Audio Editing Using DDPM Inversion](https://mlsanity.com/paper/2402.10009/) (2024)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure](https://mlsanity.com/paper/2410.24060/) (2024)
- [Uncertainty Quantification via Neural Posterior Principal Components](https://mlsanity.com/paper/2309.15533/) (2023)
- [Exploring Low-Dimensional Subspaces in Diffusion Models for Controllable Image Editing](https://mlsanity.com/paper/2409.02374/) (2024)
- [Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs](https://mlsanity.com/paper/2410.11149/) (2024)
- [Uncertainty Quantification for Machine Learning in Healthcare: A Survey](https://mlsanity.com/paper/2505.02874/) (2025)
- [Object-AVEdit: An Object-level Audio-Visual Editing Model](https://mlsanity.com/paper/2510.00050/) (2025)
- [InvFusion: Bridging Supervised and Zero-shot Diffusion for Inverse Problems](https://mlsanity.com/paper/2504.01689/) (2025)
- [What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Models](https://mlsanity.com/paper/2408.08307/) (2024)
- [When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets](https://mlsanity.com/paper/2506.19031/) (2025)
- [Self-Supervised Learning from Noisy and Incomplete Data](https://mlsanity.com/paper/2601.03244/) (2026)
- [Hierarchical Uncertainty Exploration via Feedforward Posterior Trees](https://mlsanity.com/paper/2405.15719/) (2024)
- [EigenScore: OOD Detection using Covariance in Diffusion Models](https://mlsanity.com/paper/2510.07206/) (2025)
- [Uncertainty Visualization via Low-Dimensional Posterior Projections](https://mlsanity.com/paper/2312.07804/) (2023)
- [PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inference](https://mlsanity.com/paper/2510.13763/) (2025)
- [Uncertainty in a Single Pass: A Closed-Form Identity for One-Step Flow Matching](https://mlsanity.com/paper/2605.00941/) (2026)
- [Amortized Moment Matching for Visual Generation](https://mlsanity.com/paper/2607.26860/) (2026)
- [Shallow Diffuse: Robust and Invisible Watermarking through Low-Dimensional Subspaces in Diffusion Models](https://mlsanity.com/paper/2410.21088/) (2024)
- [A Simple Combination of Diffusion Models for Better Quality Trade-Offs in Image Denoising](https://mlsanity.com/paper/2503.14654/) (2025)
- [Heteroscedasticity of Denoising Score Matching with Generalised Smooth Noise](https://mlsanity.com/paper/2508.01597/) (2025)
- [Spatially Adaptive Noise Injection](https://mlsanity.com/paper/2609.18466/) (2026)
- [Conditional Distribution Quantization in Machine Learning](https://mlsanity.com/paper/2502.07151/) (2025)
- [QUTCC: Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problems](https://mlsanity.com/paper/2507.14760/) (2025)
- [The Value of Covariance Matching in Gaussian DDPMs and the Lanczos Sampler](https://mlsanity.com/paper/2605.22723/) (2026)
- [PFM-HR: Pose Flow Matching for Humanoid Robots](https://mlsanity.com/paper/2608.03227/) (2026)
- [Improved denoising diffusion probabilistic models with efficient non-diagonal covariance modeling](https://mlsanity.com/paper/2608.21972/) (2026)
- [The Loss Floor of Denoising Score Matching: Fisher Geometry from Schr"odinger Bridges](https://mlsanity.com/paper/2608.23916/) (2026)
- [A Lagrangian View of Flow Matching](https://mlsanity.com/paper/2609.00198/) (2026)
- [Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity](https://mlsanity.com/paper/2609.15048/) (2026)

## Abstract (arXiv)

Denoisers play a central role in many applications, from noise suppression in low-grade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (ie the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .