# Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability

arXiv: https://arxiv.org/abs/2605.06538 · HTML: https://mlsanity.com/paper/2605.06538/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 28 of its 38 references that are in the map.
- First posted on arXiv 7 May 2026.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

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

- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion](https://mlsanity.com/paper/2312.00852/) (2023)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [Variational Diffusion Posterior Sampling with Midpoint Guidance](https://mlsanity.com/paper/2410.09945/) (2024)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [Diffusion Posterior Sampling is Computationally Intractable](https://mlsanity.com/paper/2402.12727/) (2024)
- [Posterior Sampling with Denoising Oracles via Tilted Transport](https://mlsanity.com/paper/2407.00745/) (2024)
- [Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach](https://mlsanity.com/paper/2602.05533/) (2026)
- [How to Guide Your Flow: Few-Step Alignment via Flow Map Reward Guidance](https://mlsanity.com/paper/2604.27147/) (2026)
- [Steering diffusion models with quadratic rewards: a fine-grained analysis](https://mlsanity.com/paper/2602.16570/) (2026)
- [Fine-Tuning Diffusion Models via Intermediate Distribution Shaping](https://mlsanity.com/paper/2510.02692/) (2025)
- [Efficient Approximate Posterior Sampling with Annealed Langevin Monte Carlo](https://mlsanity.com/paper/2508.07631/) (2025)

## Built on by (most cited first, 2 of 2)

- [Posterior Information Dynamics of Diffusion Models for Linear Inverse Problems](https://mlsanity.com/paper/2608.21709/) (2026)
- [Generative Translation Priors: Bayesian Imaging with Cross-Modality Image Translation](https://mlsanity.com/paper/2608.28872/) (2026)

## Abstract (arXiv)

Diffusion-based posterior samplers draw from measurement- or reward-conditioned posteriors using pretrained diffusion priors. Though widely used for inverse problems, their output is biased even with exact prior scores. First, we quantify this bias for the DPS algorithm, the base method for a family of samplers, by introducing a surrogate curve of probability densities which interpolates between the true posterior and the standard Gaussian. The ratio of sampler to surrogate densities solves a parabolic PDE, and its Feynman--Kac formula yields an explicit representation of this ratio as the expectation of a path functional. This representation reveals which posterior regions are over- or under-sampled. This framework also provides a design principle for bias reduction that includes STSL, a related sampler. Second, in hard-constraint problems (e.g. the low measurement noise regime) these samplers are known to be numerically unstable. We find that this instability of discretized DPS arises from an effective guidance update whose strength scales inversely with step size, inevitably violating the forward-Euler stability criterion near the constraint. This finding motivates implicit (backward-Euler) guidance updates. Finally, we show that the standard practice of early guidance-stopping, used to avoid oscillations, introduces a secondary bias which we characterize through our framework.