A Gibbs posterior sampler for inverse problem based on prior diffusion model

Jean-Fran\c{c}ois Giovannelli

REVIEW 1 major objections 4 minor 26 references

A Gibbs posterior sampler for inverse problem based on prior diffusion model

T0 review · 1 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A Gibbs sampler that alternates backward and forward diffusion conditionals draws posterior samples for linear inverse problems with a diffusion prior.

desk verdict G-DPS's Gibbs idea is novel and clearly presented, but the convergence guarantee is vacuous for real diffusion priors and the experiments are minimal. read the letter →

arxiv 2602.11059 v2 pith:OFYUS4MD submitted 2026-02-11 stat.ML cs.LGstat.AP

Jean-François Giovannelli This is my paper · ORCID

classification stat.MLcs.LGstat.AP MSC 62F1565C0565C40

keywords inverseproblemsdiffusionmodelsBayesianinferenceGibbssamplingposteriordeconvolutionuncertaintyquantificationMarkovchainMonteCarlo

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tackles Bayesian inversion when the prior is a diffusion model: given a blurred, noisy image, sample from the posterior over the clean image and the latent variables the diffusion model introduces. Existing posterior samplers rely on guided ancestral sampling or particle methods; this paper instead proposes a block Gibbs sampler that treats each image in the hierarchy as a block. The key move is to sample the clean image from the backward conditional (the learned denoiser combined with the likelihood) and sample all latent images from the forward schedule's transition densities. Every conditional is Gaussian—diagonal in the Fourier domain for the image itself—so each iteration is cheap and requires a single network pass. The paper argues the sampler is guaranteed to converge when the learned backward joint density exactly equals the forward joint density, and demonstrates empirically that it recovers the true image and delivers coherent posterior intervals.

What carries the argument

The central object is the mixed set of conditional densities: the paper 'switches' between the backward joint prior for x0 and the forward joint prior for x1:T, treating them as the full conditionals of a single extended posterior. The practical workhorse is the Gaussian form of each conditional—for x0 it is a linear-Gaussian product combining the likelihood with the backward denoiser mean, with precision HtH/ve + I/v0−; for 1≤t<T it is p+t|t−1(xt|xt−1)p+t+1|t(xt+1|xt), also Gaussian; and for t=T just p+T|T−1. The identity p+0:T = p−0:T after learning is what would make these mixed conditionals the true conditionals of one joint distribution, licensing the Gibbs convergence guarantee.

What would settle it

Take a small linear inverse problem where the true posterior is computable (e.g., by exhaustive grid or a long particle-based run), and compare the empirical marginal distribution of x0 from G-DPS to the true marginal. If a discrepancy larger than Monte Carlo error persists after many iterations on a well-trained prior—or if the discrepancy grows when the forward/backward gap is artificially increased—the purported convergence guarantee is falsified.

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Extended reading notes

Core claim

The paper's central claim is that a Gibbs sampler alternating the backward conditional for x0 and the forward transition conditionals for x1:T is an effective, simple way to draw from the posterior of a linear inverse problem with a diffusion prior. For x0, the conditional is the product of a Gaussian likelihood and the backward denoiser at time 0, yielding a Gaussian with Wiener/Tikhonov form; for intermediate latents it is the product of two forward transition Gaussians; and for xT it is the final forward step. Writing all conditionals explicitly shows they are Gaussians with diagonal covariances in the appropriate domain, so sampling is straightforward. The authors show on a toy deconvolu

Load-bearing premise

The result rests on the assumption that the learned backward joint density is exactly equal to the forward joint density; if that equality fails, the mixed conditionals are not the full conditionals of any single posterior and the Gibbs convergence argument does not apply.

Editorial extensions

If this is right

Each iteration needs only one forward pass through the neural network, so the sampler scales to larger images and runs quickly on a standard computer.
Because all conditionals are Gaussian with diagonal covariances, implementation requires only FFTs and linear combinations—no tuning knobs beyond the iteration count.
The output directly gives posterior mean and posterior standard deviations, so uncertainty quantification is a by-product of the sampling run.
If the forward and backward joint priors coincide, the sampler is a standard Gibbs sampler on the extended posterior and inherits its convergence guarantees.
The alternating structure splits the difficult posterior sampling problem into simple Gaussian sub-problems, suggesting the approach could replace more approximate guided-diffusion schemes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: The sampler's practical robustness likely depends on the size of the forward/backward KL gap; a testable prediction is that the stationary distribution degrades gracefully as this gap grows, and that the degradation can be quantified by comparing the sampler's output to a reference posterior estimate.
Inference: The same switching trick may extend to nonlinear observation operators, but the x0 conditional would no longer be a linear-Gaussian Wiener step; one would need a Laplace approximation or an auxiliary variable, a direction the paper does not explore.
Inference: The scan order described (0..T then repeat) is a systematic scan; a random-scan version would be a more standard Gibbs sampler and might ease the theoretical proof, a concrete next step for the convergence guarantee the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The first thing to know about this paper is that the core construction is neat. The author proposes a Gibbs sampler that updates the latent diffusion states with the forward kernels and the image of interest with the backward kernel (the denoiser) plus the likelihood. Every conditional is Gaussian, so the algorithm is simple, and each sweep costs one network pass. That combination of forward and backward conditionals in a single Gibbs loop is, as far as I can tell, not in the cited literature. Section III correctly derives each conditional under the stated assumption that the forward and backward joint priors are identical.

That assumption is where the paper breaks. The stress-test note is on target: for any non-Gaussian data distribution, the exact reverse-time conditional of the forward chain is non-Gaussian, while the learned backward kernel is Gaussian with a fixed covariance. No neural network can make these equal. So the forward and backward joint priors are structurally different. The 'clearly identified situation' in which convergence is guaranteed never obtains for trained diffusion priors. The author says as much in Section III—'strictly speaking, the result does not apply here'—so the abstract overstates the support.

The numerical evidence doesn't fill the gap. We get three pixels from one MNIST deconvolution, no baseline, and no comparison to a known posterior. The chains look stable, but that only suggests the Markov chain has some invariant distribution, not that it is the intended posterior. Uncertainty quantification claims based on these samples are anecdotal.

Credit where due: the paper is honest about its own limitations. The writing is clear. The derivations are straightforward. The idea of a mixed Gibbs sweep is worth noting for people working on diffusion-based inverse problems. But as it stands, it is a heuristic, not a posterior sampler with guarantees.

Who should read it? Researchers experimenting with approximate diffusion-prior inference might find the algorithm easy to implement and worth testing. But I would not cite it as a method for exact Bayesian inference.

For peer review: I would send it to a serious referee only on the condition that the author either proves something about the stationary distribution of the mixed sampler, or demonstrates on a known posterior that the algorithm actually samples the right thing. Given the structural impossibility of the current justification, a clean proof seems unlikely without changing the algorithm. But the empirical question could be settled. So: send to review if you expect the author to reposition the paper as a pragmatic approximate sampler with strong experimental validation; otherwise desk reject.

In short: a promising idea, poorly supported as it stands.

Referee Report

1 major / 4 minor

Summary. The paper proposes G-DPS, a block-Gibbs sampler for Bayesian linear inverse problems with a diffusion-model prior. The sampler alternately draws x0 from a Gaussian conditional formed from the backward diffusion transition and the likelihood, and draws latent variables x1:T from Gaussian conditionals formed from the forward diffusion transitions. The method is simple, requires only Gaussian sampling, and all conditionals have diagonal or Fourier-diagonal covariances. The authors claim a convergence guarantee when the forward and backward joint priors coincide, and provide a numerical demonstration on a 32×32 MNIST-like deconvolution problem.

Significance. The idea of mixing forward and backward diffusion conditionals inside a Gibbs loop is original and computationally attractive: it requires only a single neural-network pass per iteration and yields tractable Gaussian updates. If the underlying joint priors coincided, the sampler would be a legitimate Gibbs sampler with a standard convergence guarantee, and the linear-Gaussian structure of the x0 update is cleanly derived. However, the paper's central claim of a convergence guarantee is compromised by its own admission and by a fundamental structural mismatch: the backward conditional family (5) cannot represent the exact reverse-time conditionals of a non-Gaussian forward diffusion. The numerical evidence is too limited to establish that the approximate sampler targets the intended posterior. The significance is therefore conditional: the proposal is promising as a practical approximate sampler, but the current manuscript does not substantiate the guaranteed-convergence claim or provide sufficient empirical validation.

major comments (1)

[§IV-B, Table I and Fig. 2] The empirical validation is far too thin to support the claim that G-DPS 'correctly samples the posterior.' Only three pixels from a single MNIST-like image are examined; there is no comparison with an asymptotically exact sampler (e.g., TDS or DPMC), no known-posterior verification on a small-scale problem where exact computation is feasible, and no image-wide quantitative posterior diagnostic. The ±2PSD coverage check is a weak test that many approximate methods pass. The paper also states in §IV-D that no algorithm parameters require tuning, yet the stopping threshold (10^{-2}) and iteration count (N=1030) are user choices, and the scan order is fixed. To establish practical validity, the authors should provide baselines, repeated trials, and a more systematic convergence/accuracy assessment.

minor comments (4)

[§I, notation] Eq. (1) writes H ∈ R^P × R^M, but the usual linear operator from x ∈ R^P to y ∈ R^M should have dimensions M × P. This is likely a typo but should be corrected.
[§IV-A] Reference [18] is a MATLAB documentation page, not a citable description of the diffusion architecture or training procedure. Please provide a proper reference or algorithm description so that the experiment is reproducible.
[§IV-D] The phrase 'does not require the adjustment of any algorithm parameters or fine-tuning quantities (apart from the number of iterations)' is misleading: the stopping threshold is a parameter and the initialization/scan order are choices. This overstatement should be softened.
[General readability] Several sentences in the Introduction and Conclusion are awkward or contain typographical errors (e.g., 'G-DSP' appears instead of 'G-DPS' in several places). A careful copyedit is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mixed-conditionals sampler is not derived from its outputs; its convergence gap is explicitly acknowledged.

full rationale

The derivation chain is self-contained: the proposed G-DPS sampler takes a pre-trained diffusion prior as input and constructs Gaussian conditionals from the forward/backward transition kernels. The central simplifying premise — 'the two joint priors are similar thanks to the learning stage, and we consider here they are identical' — is an explicit approximation, not a hidden reduction: no fitted parameter is relabeled as a prediction, and no output quantity is defined in terms of the target posterior. The paper's own Convergence-property paragraph flags the limitation: 'strictly speaking, the result does not apply here since some conditionals are deduced from the forward (joint) pdf and others from the backward (joint) pdf.' This is an acknowledged correctness caveat about the Gibbs-convergence guarantee, not a circular step; it weakens the abstract's 'guarantee of convergence in a clearly identified situation,' but that overstatement is a validity concern rather than circularity. Self-citations ([16] for FFT-domain Gaussian sampling, [19]–[22] for future hyperparameter/PSF estimation, [23]–[24] for comparison) are not load-bearing: [16] is an implementation convenience and the others are forward-looking. The limited numerical study (MNIST toy, three pixels, no baseline) is an evidence-strength limitation, not circularity. Thus no step in the paper reduces, by construction or self-citation, to its own input; score 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The sampler itself has no fitted parameters; the only free choices are the stopping threshold and run length. The central assumptions are the equality of forward and backward diffusion joints and the applicability of the Gibbs convergence theorem under that equality. No invented entities are introduced.

free parameters (2)

stopping threshold = 1e-2
Ad hoc convergence criterion controlling when the empirical average stops updating (§IV-D).
number of iterations N = 1030
Set by the stopping rule in one run; not fitted to data but a user choice (§IV-D).

assumptions (4)

ad hoc to paper Forward and backward joint priors are identical after learning: p^+_{0:T} = p^-_{0:T}.
Stated in §III: 'we consider here they are identical', with work ongoing. This is load-bearing because the Gibbs conditionals are drawn from two different joints; exact stationary distribution requires equality.
standard math Standard Gibbs convergence theorem applies to a single joint density.
Invoked in §III (Convergence property); correctness of the sampler relies on the conditionals being full conditionals of one common joint.
domain assumption The diffusion prior's marginal p^+_0 equals the data distribution π_0.
Assumed from the learning stage; the posterior is defined with respect to this prior.
domain assumption The observation noise is zero-mean white Gaussian with known variance v_e.
Used to define the likelihood f(y|x0) in §I; standard but untested here.

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Cite this review

Pith. "Pith review of A Gibbs posterior sampler for inverse problem based on prior diffusion model." pith.science (2026). https://pith.science/paper/OFYUS4MD

@misc{pith2026260211059,
  author       = {Pith},
  title        = {Pith review of: A Gibbs posterior sampler for inverse problem based on prior diffusion model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFYUS4MD}},
  note         = {Machine review of arXiv:2602.11059}
}

read the original abstract

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive error, (2) the problem is ill-posed and regularization relies on a Bayesian strategy, (3)~the prior is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one and the paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that it is particularly effective and remarkably simple. In addition, it provides clear elements regarding convergence guarantees in a specific case and arguments supporting such guarantees in practical cases. The results are clearly confirmed by numerical simulations based on a toy example.

Figures

Figures reproduced from arXiv: 2602.11059 by the authors.

**Figure 1.** Hierarchy: x0 is the image of interest, x1:T are the latent images and y is the measured image (blurred and noisy version of the true x0). This development is based on a model with a natural conditional independence property: the pdfs for (y|x0) and (y|x0:T ) are equal (see the hierarchical representation in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Samples provided by the Gibbs algorithm for three pixels of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Point clouds for two dimensional marginals pdfs for three pixels. The [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Left to right: true object x ⋆, measurements y and estimated object xb. The figure shows the images themselves (top) and cross-sections (bottom). provides not only optimal estimation for the unknowns (e.g., Posterior Mean as the MMSE), but also coherent tool for uncert…

**Figure 6.** Figure 6: Left part: empirical average of the samples provided by the Gibbs [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 5.** Figure 5: Cross-sections of x ⋆ (plain green) and the “uncertainty” intervals (dashed blue) with center the estimate and of width two standard deviations. C. Additional consideration regarding convergence This section explores the issue of speed of convergence in greater depth, …

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Giovannelli and J

J.-F. Giovannelli and J. Idier, Eds.,Regularization and Bayesian Methods for Inverse Problems in Signal and Image Processing. London: ISTE and John Wiley & Sons Inc., 2015

2015

[2] [2]

Kaipio and E

J. Kaipio and E. Somersalo,Statistical and computational inverse problems. Berlin, Germany: Springer, 2005

2005

[3] [3]

J. C. Santamarina and D. Fratta,Discrete Signals and Inverse Problems: An Introduction for Engineers and Scientists. Chichester, England: WileyBlackwell, 2005

2005

[4] [4]

P. C. Hansen,Discrete Inverse Problems: Insight and Algorithms. Philadelphia, USA: Society for Industrial and Applied Mathematics, 2010

2010

[5] [5]

S. H. Chan,Tutorial on Diffusion Models for Imaging and Vision, ser. Foundations and Trends in Machine Learning. Hanover,MA, USA: Now Publishers Inc, Jan. 2024

2024

[6] [6]

F. D. S. Ribeiro and B. Glocker,Demystifying Variational Diffusion Models, ser. Foundations and Trends in Machine Learning. Hanover, MA, USA: Now Publishers Inc, Jan. 2025

2025

[7] [7]

Nakkiran, A

P. Nakkiran, A. Bradley, and M. Zhou, Hattieand Advani,Step-by- Step Diffusion: An Elementary Tutorial, ser. Foundations and Trends in Machine Learning. Hanover,MA, USA: Now Publishers Inc, 2025

2025

[8] [8]

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H. Chung, J. Kim, M. T. Mccann, M. L. Klasky, and J. C. Ye, “Diffusion posterior sampling for general noisy inverse problems,” 2024

2024