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

2026-05-07

AI Summary

Diffusion posterior samplers are widely used in inverse problems, yet their outputs suffer from bias and discretization instability at low temperatures, with the underlying mechanisms poorly understood. This work constructs a tractable surrogate path bridging the true posterior and a standard Gaussian distribution, leveraging the Feynman–Kac formula to express the density ratio as a path-space expectation. For the first time, it derives a partial differential equation that characterizes sampling bias. By integrating Radon–Nikodym derivatives, Ornstein–Uhlenbeck processes, and auxiliary drift reconstruction, the study reveals the origins and spatial distribution of bias in methods such as DPS and STSL: it precisely identifies regions of over- and under-sampling in DPS and explains how STSL enhances stability through a smoothed reaction term. This theoretical framework provides a foundation for designing stable and efficient posterior sampling algorithms.

Abstract

Diffusion-based posterior samplers use pretrained diffusion priors to sample from measurement- or reward-conditioned posteriors, and are widely used for inverse problems. Yet their theoretical behavior remains poorly understood: even with exact prior scores, their outputs are biased, and in low-temperature regimes their discretizations can become unstable. We characterize this bias by introducing a tractable surrogate path connecting the true posterior to a standard Gaussian and comparing it to the sampler's path. Their density ratio satisfies a parabolic PDE whose reaction term measures the accumulated bias. A Feynman-Kac representation then expresses the Radon-Nikodym correction as an explicit path expectation, identifying which posterior regions are over- or under-sampled. We apply this framework to DPS and STSL, a related sampler. For DPS, the correction is an Ornstein-Uhlenbeck path expectation coupling the data conditional covariance with the reward curvature, revealing where DPS over- or under-samples. Next, we reinterpret STSL as an auxiliary drift that steers trajectories toward low-uncertainty regions, flattening the spatially varying part of the DPS reaction term. Finally, we characterize early guidance-stopping, a common mitigation for low-temperature instabilities caused by forward-Euler integration of the vector field. Together, these results clarify sampler bias, explain existing correctives, and guide stable variant designs.

Problem

• posterior sampling
• diffusion models
• bias
• stability
• inverse problems

Innovation

• Feynman-Kac representation
• posterior sampling
• diffusion models
• bias characterization
• low-temperature stability