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

• URL: http://arxiv.org/abs/2605.06538v1
• Date: Thu, 07 May 2026 16:37:29 GMT
• Title: Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability
• Abstract summary: Diffusion-based posterior samplers use pretrained diffusion priors to sample from measurement- or reward-conditioned posteriors.<n>This paper introduces a tractable surrogate path connecting the true posterior to a standard Gaussian and comparing it to the sampler's path.<n>A Feynman-Kac representation then expresses the Radon-Nikodym correction as an explicit path expectation, identifying which posterior regions are over- or under-sampled.
• Score: 13.640937605089865
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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