# Is your uncertainty map wrong, or is its target? Exact diagnostics for the Tweedie diagonal, and a gradient-free alternative

arXiv: https://arxiv.org/abs/2609.33786 · HTML: https://mlsanity.com/paper/2609.33786/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 33 of its 47 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 30 of 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [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)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Aleatoric uncertainty estimation with test-time augmentation for medical image segmentation with convolutional neural networks](https://mlsanity.com/paper/1807.07356/) (2018)
- [Brain Imaging Generation with Latent Diffusion Models](https://mlsanity.com/paper/2209.07162/) (2022)
- [Diffusion Models for Implicit Image Segmentation Ensembles](https://mlsanity.com/paper/2112.03145/) (2021)
- [Estimating the Optimal Covariance with Imperfect Mean in Diffusion Probabilistic Models](https://mlsanity.com/paper/2206.07309/) (2022)
- [Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion](https://mlsanity.com/paper/2312.00852/) (2023)
- [Generative AI for Medical Imaging: extending the MONAI Framework](https://mlsanity.com/paper/2307.15208/) (2023)
- [Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance](https://mlsanity.com/paper/2402.02149/) (2024)
- [BayesDiff: Estimating Pixel-wise Uncertainty in Diffusion via Bayesian Inference](https://mlsanity.com/paper/2310.11142/) (2023)
- [Enhancing Spatiotemporal Disease Progression Models via Latent Diffusion and Prior Knowledge](https://mlsanity.com/paper/2405.03328/) (2024)
- [Generative Uncertainty in Diffusion Models](https://mlsanity.com/paper/2502.20946/) (2025)
- [Shedding Light on Large Generative Networks: Estimating Epistemic Uncertainty in Diffusion Models](https://mlsanity.com/paper/2406.18580/) (2024)
- [Diffusion Model Guided Sampling with Pixel-Wise Aleatoric Uncertainty Estimation](https://mlsanity.com/paper/2412.00205/) (2024)
- [Treatment-aware Diffusion Probabilistic Model for Longitudinal MRI Generation and Diffuse Glioma Growth Prediction](https://mlsanity.com/paper/2309.05406/) (2023)
- [Unified Uncertainty-Aware Diffusion for Multi-Agent Trajectory Modeling](https://mlsanity.com/paper/2503.18589/) (2025)
- [Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs](https://mlsanity.com/paper/2410.11149/) (2024)
- [Improving Probabilistic Diffusion Models With Optimal Diagonal Covariance Matching](https://mlsanity.com/paper/2406.10808/) (2024)
- [Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling](https://mlsanity.com/paper/2412.20045/) (2024)
- [Bayesian computation with generative diffusion models by Multilevel Monte Carlo](https://mlsanity.com/paper/2409.15511/) (2024)
- [Temporally-Aware Diffusion Model for Brain Progression Modelling with Bidirectional Temporal Regularisation](https://mlsanity.com/paper/2509.03141/) (2025)

## Abstract (arXiv)

A diffusion model can predict a follow-up medical scan from a baseline, but a clinician needs a per-voxel map of where that prediction can be trusted. Many such maps approximate the diagonal of the Tweedie posterior covariance, and are evaluated against another approximation of it, so whether the estimator or the target limits them is unclear. We compute the exact diagonal on six checkpoints across fourteen model-corpus conditions. Hutchinson at M=200 tracks it at rank agreement of at least 0.92 everywhere, yet in four of the fourteen the exact diagonal is anti-correlated with the denoising error, reaching -0.13, so a faithful estimator reproduces that reversal. All four are real-image conditions; on the models' own samples the reversal does not appear, so evaluating on generated samples flatters this family. What limits these maps is the target, not the estimator. We then introduce Tweedie Probe-Tangent (T-PT), a gradient-free residual probe that corrupts one model-supported prediction repeatedly and measures the voxel-wise variance of the denoiser's response. T-PT reads a different functional of the same Jacobian, and its exact second-order form ranks with the diagonal wherever the diagonal reverses; at thirty probes it returns a map too unstable to reproduce that ranking, while Hutchinson at M=5 already reproduces it, so T-PT there is not evidence against the reversal. We offer it as an instrument, not a better approximation. On brain MRI at full resolution, where every Jacobian-based estimator we test runs out of memory, T-PT leads a twenty-chain Monte-Carlo ensemble on five of eight endpoints inside tissue and trails it on none, at 16x fewer network evaluations; over the whole volume the ensemble leads, and fifty chains close the tissue gap. On lung CT the ensemble is ahead throughout. Both lose most of their discrimination where the change is, which remains open.