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On the Posterior Distribution in Denoising: Application to Uncertainty Quantification
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Derives recursions linking posterior central moments of Gaussian denoising to derivatives of the posterior mean, and uses them for training-free uncertainty visualization (posterior principal components and marginal posteriors) of pre-trained denoisers.
Problem
Denoisers give only the MSE-optimal posterior mean, hiding uncertainty; prior higher-order score methods use non-central moments and are memory-prohibitive for large images, so uncertainty quantification of pre-trained denoisers at high resolution is hard.
Approach
Proves that posterior central moments (univariate, multivariate, and along any direction) satisfy simple recursions in terms of the derivatives of the posterior mean (Tweedie-based), enabling covariance eigenvectors via forward-pass finite differences and marginal densities via maximum-entropy fitting of moments.
Key results
- Theorem gives recursions mu_2 = sigma^2 mu_1', mu_{k+1} = sigma^2 mu_k' + k mu_{k-1} mu_2 for central moments, so marginal posteriors are recovered from forward-pass derivatives.
- Forward-pass finite-difference subspace iteration avoids storing the posterior covariance and reduces memory about 6x for an 80x92 patch with SwinIR (per paper).
- On a 2D Gaussian mixture, the moment-based maximum-entropy marginal closely matches the closed-form posterior along the principal component.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Approximate Jacobian-vector products with forward finite differences instead of computing them with backward passes | Posterior principal-component computation with SwinIR · 80×92 patch | memory footprint: 6× reduction | Helped |
| Use polynomial fitting rather than numerical derivatives for high-order derivatives | Marginal posterior estimation · Two-component GMM example | fit quality | Hurt |
| Use an estimated noise level rather than the true noise level | Uncertainty visualization method · Blind denoising | qualitative plausibility | No clear effect |
Surprising findings
- Extreme reconstructions along a posterior principal component can look realistic yet have near-zero posterior probability given the noisy input.
Limitations
- Theory applies only to additive white Gaussian noise with known sigma; real microscopy data uses estimated sigma from a blind denoiser.
- Relies on high-order numerical differentiation, which is unstable in low precision; double precision is needed in forward passes.
- Polynomial-fit alternative to numerical differentiation was sensitive to degree and fitting range.
Experimental setup
Datasets: DIV2K Flickr2k BSD500 WED FFHQ CelebAMask-HQ MNIST FMD CBSD68 Kodak
Benchmarks: CBSD68 Kodak FMD CelebAMask-HQ MNIST
Metrics: eigenvector accuracy eigenvalue accuracy marginal posterior density agreement with ground truth
Baselines: posterior sampler (naive PC estimation) closed-form posterior mean for toy GMM
Builds on: Tweedie's formula higher-order score estimation by denoising (Meng et al.) subspace iteration denoising score matching
Closest papers by meaning
- Sampling Binary Data by Denoising through Score Functions (ICML 2025 · similarity 0.87)
- Exact Posterior Score Estimation for Solving Linear Inverse Problems (NeurIPS 2026 · similarity 0.87)
- Diffusion Priors for Variational Likelihood Estimation and Image Denoising (NeurIPS 2024 · similarity 0.87)
- Noise Conditional Variational Score Distillation (ICML 2025 · similarity 0.87)
- MAP Estimation with Denoisers: Convergence Rates and Guarantees (NeurIPS 2025 · similarity 0.87)
- Provable Posterior Sampling with Denoising Oracles via Tilted Transport (NeurIPS 2024 · similarity 0.86)
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