Diffusion models have become the leading method for sampling from complex high-dimensional distributions, and a natural next question is posterior sampling: generating samples consistent with a partial or noisy observation — inpainting and super-resolution being prototypical examples — while reusing a pre-trained score network to enforce the prior. Diffusion Posterior Sampling (DPS) achieves this by "hacking" the conditional expectation through the likelihood via Tweedie's formula, which is biased even for Gaussian targets. Our main result is an exact characterization of this bias: the DPS surrogate path solves a Fokker–Planck equation with an explicit reaction term, which yields Feynman–Kac representations, along either the forward or backward path, of an importance weight that corrects DPS samples to the true posterior. Numerical experiments reveal that meaningless trajectories carry astronomically large weights. More surprisingly, the analysis of the implementation further uncovers a hidden, singular annealing schedule that violates the stability criterion of forward Euler resulting in numerical oscillations. These insights provide first-principles justification for heuristics used in deployed systems, such as switching off guidance in the final steps and implicit integration schemes. Joint work with Sebastian Motsch, Advait Parulekar, Will Porteous, and Sanjay Shakkottai.