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

Matias G. Delgadino, Sebastien Motsch, Advait Parulekar, William Porteous, Sanjay Shakkottai

arXiv:2605.06538v1cs.LG

TL;DR

Diffusion-based posterior samplers remain theoretically difficult because approximate guidance is biased and low-temperature discretizations can be unstable. The paper compares sampler and surrogate paths through a density-ratio PDE and Feynman-Kac representation, deriving bias corrections for DPS and STSL and analyzing early guidance-stopping. It identifies covariance–reward-curvature interactions behind DPS over- and under-sampling, interprets STSL as uncertainty-directed drift, and links instability to forward-Euler violations.

  • Problem

    Approximate posterior samplers can remain biased even with exact prior scores, while low-temperature DPS discretizations can become unstable and existing heuristics lack quantified effects.

    Source: §1 · pp. 1–2

  • Method

    The paper compares sampler paths with a tractable posterior-to-Gaussian surrogate using a density-ratio PDE and Feynman-Kac path expectations.

    Source: §1 · p. 2, §2 · p. 3, §3 · p. 4

  • Results

    DPS bias is characterized by covariance–reward-curvature alignment, STSL is an auxiliary drift toward low-uncertainty regions, and early guidance-stopping is analyzed under forward-Euler instability.

    Source: §1 · p. 2, §5 · p. 9

  • Takeaways & Limitations

    The analysis explains where DPS over- or under-samples, why STSL reduces spatial bias, and how guidance stopping relates to stable sampler design.

    Source: §1 · p. 2

    The exact posterior-score evolution remains intractable in polynomial time, so the analysis relies on a tractable surrogate path and cannot efficiently compute the corresponding exact correction.

    Source: §3 · p. 5

Abstract

from arXiv · show

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.

1 Introduction

The paper addresses two open questions about diffusion-based posterior sampling: how DPS bias selects posterior regions and why low-temperature implementations become unstable. It develops a Feynman-Kac analysis that quantifies bias, explains STSL, and characterizes early guidance-stopping.

Motivation

  • DPS is a widely used plug-and-play posterior-sampling heuristic for inverse problems, but its theoretical behavior remains insufficiently understood.It replaces the intractable conditional score with reward gradients evaluated at the Tweedie posterior mean.Source: §1 · pp. 1–2

Contributions

  • Feynman-Kac analysis closes open questions about which samples DPS over- or under-represents and how corrective methods affect this bias.The framework complements existing convergence analyses that rely on restrictive assumptions or treat the algorithm as a black box.Source: §1 · p. 2

Exact bias formula for DPS

  • DPS differs from the true posterior even with exact prior scores, and its bias is represented by a pointwise Radon-Nikodym weight.The weight admits an Ornstein-Uhlenbeck path-expectation form whose reaction term identifies over- and under-sampled regions through covariance–curvature alignment.Source: §1 · p. 2

STSL-type bias reduction

  • STSL reduces DPS bias by steering trajectories toward regions of lower conditional uncertainty, flattening the spatially varying reaction term.The trace-of-covariance potential recovers the empirically successful STSL correction.Source: §1 · p. 2

Quantifying low-temperature instability and early stopping

  • Standard DPS forward-Euler discretization violates the stability condition in low-temperature regimes, producing oscillations.The paper characterizes early-guidance-stopping as a weighted version of the prior.Source: §1 · p. 2

2 Background and Related Work

Score-based models use learned prior scores and reverse diffusion to generate samples, while posterior sampling modifies this process with a reward or likelihood. Feynman-Kac analysis instead tracks the explicit density mismatch introduced by approximate guidance.

Score Based Generative Models

  • Score-based generative models interpolate a noised prior from ρ∗ to a standard Gaussian and use reverse diffusion driven by a learned score.The reverse process can be sampled in polynomial time when the score network has been trained on target-distribution samples.Source: §2 · p. 3

Posterior Sampling

  • Posterior sampling tilts the prior by a log-likelihood or reward, but the posterior score is generally unavailable from the prior score alone.Tweedie’s formula supplies a tractable proxy using E[X0 | Xt = xt], yielding practical heuristic guidance whose samples are not formally drawn from the target posterior.Source: §2 · p. 3
  • DPS replaces the intractable conditional score with the reward gradient evaluated at the Tweedie posterior mean, enabling plug-and-play guidance with pretrained score networks.This simplicity made DPS a de-facto inverse-problem baseline and motivated numerous corrective methods.Source: §2 · p. 3

Feynman-Kac formulas

  • Feynman-Kac tracks the Radon-Nikodym derivative between approximate and true posterior sampling rather than only bounding KL, total variation, or χ2 error.The reaction term creates path-dependent weighting that can be interpreted as killing or spawning before normalization.Source: §2 · p. 3
  • A reaction term changes transport-diffusion evolution into a weighted process whose terminal normalization yields the corresponding output density.Positive reaction corresponds to spawning, while negative reaction corresponds to killing or early termination.Source: §2 · pp. 3–4

3 Surrogate path and the Bias of DPS

The paper constructs tractable surrogate paths from the posterior to a Gaussian, then compares their reaction-free algorithmic paths with the intended posterior evolution using Feynman–Kac representations. For DPS, this yields an explicit density correction that explains its bias and identifies over- and under-sampled regions.

Surrogate-path framework

  • A surrogate path interpolates the exponentially tilted posterior µy with a standard Gaussian while retaining a tractable score derived from the noised prior.The framework represents the path as htρt and uses the prior score together with the chosen scalar field ht.Source: §3 · p. 4

Algorithmic path

  • Dropping the reaction term from the surrogate-path PDE produces the algorithmic path, whose terminal law generally differs from the true posterior by an unavoidable bias.The reaction term is the mechanism that separates the surrogate evolution from the implementable SDE path.Source: §3 · pp. 4–6

DPS construction

  • DPS arises from a surrogate path with the correct posterior and Gaussian endpoints, whose score is computable from Tweedie estimates of conditional means and covariances.The resulting reverse SDE uses the prior score and a reward-gradient guidance term evaluated through the denoising process.Source: §3 · pp. 5–6

DPS bias characterization

  • The DPS terminal law differs from the true posterior by a pointwise weight ω, expressible as a Feynman–Kac expectation over forward or backward diffusion paths.The path functionals use reaction terms obtainable from the score oracle and its Jacobian via Tweedie’s formula.Source: §3 · p. 6

Bias interpretation

  • Relative to the posterior, DPS under-samples where ω(x) > 1 and over-samples where ω(x) < 1, as illustrated by a Gaussian-mixture example.The example reports nearly absent extremal x1 modes, with x2 below the measurement value over-sampled and x2 above it under-sampled.Source: §3 · pp. 6–7, Figure 2 · p. 7

Geometric interpretation

  • The spatially varying DPS reaction is amplified where conditional uncertainty and reward curvature align along the same eigendirections.Large conditional-covariance eigenvalues indicate broad data-manifold directions, while reward curvature measures sensitivity along those directions.Source: §3 · pp. 6–7

4 Bias Reduction

The paper interprets bias reduction as modifying the sampler’s drift to reduce spatial variation in the DPS reaction term. STSL implements this idea by steering trajectories toward low-uncertainty regions, while excessive drift can obscure the original bias structure.

Bias-reduction principle

  • Reducing spatial variation in cDPS is a design goal because it corresponds to a smaller reaction term and therefore smaller implementation bias.The framework permits adding a potential vector field that drives trajectories toward regions where cDPS oscillates less.Source: §4 · p. 7

Drift–reaction trade-off

  • The added drift changes the effective reaction to ceff = cDPS + r cU, so excessively large r can dominate and erase the original DPS bias structure.The drift intensity r is a hyperparameter controlling this trade-off.Source: §4 · pp. 7–8

STSL as a special case

  • STSL chooses a potential that drives trajectories toward low-uncertainty initial-condition regions, flattening the spatially varying part of cDPS.Smaller uncertainty spread in dominant covariance eigendirections produces the flattening effect associated with improved output behavior.Source: §4 · p. 8

5 Numerical Instabilities of the DPS Algorithm

DPS becomes unstable near the data manifold because its guidance schedule violates forward-Euler stability, producing oscillations. Early guidance stopping mitigates this instability and admits an explicit weighted-prior characterization.

Instability of DPS

  • DPS systematically violates the forward-Euler stability condition near the constraint manifold, causing oscillations in directions parallel to the data manifold.The instability occurs as trajectories approach the constraint manifold, where reward guidance is strongly enforced.Source: §5 · pp. 8–9

Early Guidance Stopping

  • Early guidance stopping removes reward guidance after t_stop, and the resulting output is characterized as an appropriately weighted tilt of the prior.The guidance is active until an intermediate time and then the diffusion follows only the untilted score.Source: §5 · pp. 8–9

Instability of DPS

  • A large effective guidance schedule triggers trajectory oscillations near the data manifold, including periodic oscillations in an MNIST posterior-sampling task.The reported MNIST behavior shows alternating deviations in pixel space around the trajectory mean.Source: Figure 3 · p. 9, §5 · p. 9

Inevitable Oscillations

  • For unsquared residual rewards, the gradient has an unbounded Lipschitz constant at the origin, so every positive step size produces a bounded but non-convergent limit cycle.The DPS bias integration is presented as the multidimensional analogue of forward Euler applied to |x|.Source: §5 · p. 9
  • For squared residual rewards, forward-Euler iterates enter a bounded limit cycle with amplitude proportional to σmax(∇A PT M)^2 and remain Lyapunov-semistable.The iterates settle within distance at most α∥∇A∥op of the constraint manifold.Source: §5 · p. 9
  • When the squared-residual stability criterion is violated, oscillations diverge rather than remaining bounded.Source: §5 · p. 9

A Feynman-Kac representations of the Radon-Nikodym derivative

The density ratio between surrogate and algorithm paths satisfies a parabolic PDE with a reaction term. Feynman–Kac converts this ratio into a path expectation, interpreting reaction as trajectory-dependent weighting.

A Feynman-Kac representations of the Radon-Nikodym derivative

  • Under stated regularity, positivity, growth, and initial-density assumptions, the density ratio is well defined and represented by a Feynman–Kac path expectation.The representation uses a diffusion driven by the PDE's drift and Brownian motion.Source: A Feynman-Kac representations of the… · p. 11
  • The Radon-Nikodym density ratio is formulated through a parabolic PDE whose transport, diffusion, and reaction terms determine its evolution.The PDE is written as ∂tgt = ∆gt + bt · ∇gt − ctgt.Source: A Feynman-Kac representations of the… · p. 11
  • Applying Itô's formula makes the relevant process a martingale, yielding the endpoint identity that establishes the Feynman–Kac representation.Source: A Feynman-Kac representations of the… · p. 12

B.1 The effective sample backward path

The effective backward path reverses an Ornstein–Uhlenbeck flow from a standard Gaussian toward the prior and supports score-based sampling. Tweedie identities express denoising quantities through the score, while the zero-noise limit gives manifold projection geometry.

B.1 The effective sample backward path

  • The forward Ornstein–Uhlenbeck path interpolates between the prior and a standard Gaussian, and reversing it yields the effective backward sampling path.The reverse evolution is expressed through the score of the forward noised prior.Source: §B.1 · p. 12
  • With negligible score-approximation error, denoising samples are practically indistinguishable from prior samples, while relative-entropy bounds quantify approximation error.The score error is measured by an integrated squared discrepancy between the true score and the learned score.Source: §B.1 · p. 13

B.2 Tweedie’s identities

  • Tweedie’s identities express the conditional mean and its spatial derivative using the score and Hessian of the noised prior.The derivative has a probabilistic interpretation as a rescaled conditional covariance.Source: §B.2 · p. 13

B.2.1 The zero-noise limit

  • Under the smooth-submanifold assumptions, the small-noise conditional law is asymptotically Gaussian in tangent directions with covariance 2t I_T.Normal fluctuations contribute only at higher order.Source: §B.2.1 · p. 14
  • Near the manifold, the Tweedie estimate approaches the orthogonal projection and its Jacobian approaches the tangent-space projection.This geometry drives the manifold-tangent oscillations analyzed for DPS.Source: §B.2.1 · pp. 14–15

C The DPS Algorithm [Chung et al., 2024]

DPS estimates posterior guidance from the reward gradient at the Tweedie posterior mean, using a problem-dependent bias schedule.

C The DPS Algorithm [Chung et al., 2024]

D Proof of Theorem 1

Theorem 1 compares the DPS path with a tilted-prior surrogate and expresses their discrepancy through a reaction term and Feynman–Kac density-ratio weights.

D Proof of Theorem 1

D.4 Step 3: Feynman-Kac two ways

  • Feynman–Kac converts the density-ratio PDE into equivalent backward and forward path expectations, including an Ornstein–Uhlenbeck representation of ω.The path functionals use quantities obtainable from the score oracle and its Jacobian via Tweedie’s formula, enabling importance weighting of DPS samples.Source: D Proof of Theorem 1 · pp. 15–16, §D.4 · p. 18

D.1 Two Tweedie identities and the Kolmogorov backward equation

  • Tweedie identities and the Kolmogorov backward equation characterize the conditional mean x̂t and covariance Σt used in the analysis.The Jacobian of x̂t is symmetric positive semidefinite, providing the structural identities used later.Source: §D.1 · p. 16

D.2 Step 1: Evolution of tilted prior paths

  • A general tilted-prior path evolves by an OU Fokker–Planck equation with a scalar reaction coefficient c[h].Source: §D.2 · pp. 16–17

D.3 Step 2: Specialization to the DPS tilt

  • For the DPS tilt, cancellation from the backward equation leaves a reaction term determined by reward gradients, reward curvature, and the conditional covariance.The resulting cDPS spatial variation captures alignment between conditional covariance and reward curvature.Source: §D.3 · p. 17

E Early Guidance Stopping, Proof of Theorem 2

Theorem 2 characterizes early guidance stopping as a modified posterior path, while the DDPM analysis maps discrete noise increments to continuous OU time.

E Early Guidance Stopping, Proof of Theorem 2

F Time discretization of DDPM

  • The DDPM discretization matches the continuous OU transition when its mean and variance coefficients satisfy the derived relationship between βi and Δti.Source: F Time discretization of DDPM · p. 21
  • For the linear DDPM noise schedule, the continuous-time increment is approximately related to βi through a near-linear mapping with a nonzero origin floor.Source: F Time discretization of DDPM · p. 21

G Forward Euler instability

Forward-Euler integration of DPS guidance becomes unstable near the measurement constraint because the unsquared residual produces a nonvanishing, rapidly varying drift.

G Forward Euler instability

  • The DPS bias term is integrated explicitly by forward Euler, while the denoising diffusion term is integrated implicitly through DDPM.The annealing schedule compensates for the missing continuous-time step in the explicit bias update.Source: G Forward Euler instability · p. 22
  • The analysis interprets the implementation as using a modified effective reward, linking discretization choices to the resulting guided path.Source: G Forward Euler instability · p. 22
  • Near the constraint set, the forward-Euler stability condition is inevitably violated, causing overshoot and oscillations parallel to the data manifold.The unsquared residual gradient renormalizes instead of vanishing as the constraint is approached.Source: G Forward Euler instability · p. 22

H Empirical Evidence of Instability

Experiments reveal sustained oscillations in DPS trajectories under constant reward guidance, while early guidance stopping removes the oscillations but changes reward behavior. Step-alignment diagnostics connect the instability to near-opposite consecutive updates at low noise levels.

Conditional Guidance for MNIST digits

Step Alignment

  • Figure 5 reports α_t for standard DPS and early guidance stopping, showing the comparison across intermediate and low-noise steps.Source: Figure 5 · p. 23

Conditional Guidance for MNIST digits

  • Constant reward guidance sustains oscillations throughout the trajectory, whereas stopping guidance at i_stop = 100 eliminates them during the stopped period.The stopped schedule no longer pulls the reward toward 0, indicating that the instability is associated with reward guidance.Source: H Empirical Evidence of Instability · p. 23
  • Despite the oscillations, applying softmax to classifier logits still produces very high confidence in the correct class.Source: H Empirical Evidence of Instability · p. 23

Step Alignment

  • The trajectory diagnostic measures alignment between consecutive update vectors in a 50-step subspace, using α_t to quantify their projected alignment.The recent-step matrix is formed from ℓ = 50 updates, and the top-k projection is used for the alignment measure.Source: H Empirical Evidence of Instability · p. 23
  • For purely oscillating trajectories, consecutive updates are approximately opposite and α_t approaches −1, whereas unrelated updates yield α_t near 0.Source: H Empirical Evidence of Instability · p. 23
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