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

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.