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

TL;DR

This paper analyzes the bias and stability of diffusion-based posterior samplers using a Feynman-Kac framework, providing insights into their behavior and guiding improved design.

Contribution

It introduces a tractable surrogate path and PDE-based analysis to characterize bias and stability, offering a theoretical foundation for sampler correction and enhancement.

Findings

Bias is characterized by a PDE reaction term indicating over- or under-sampling.

The Feynman-Kac representation links bias correction to explicit path expectations.

Guidelines for stable sampler design and early stopping are derived from the analysis.

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.