Score-Based Diffusion Priors

• Score-based diffusion priors are learned data-driven distributions that use diffusion models to approximate the gradient of a log-density for inverse problems.
• They integrate with Bayesian frameworks by combining a learned score field with forward models to produce measurement-consistent and plausible posterior samples.
• The approach enhances complex imaging tasks such as MRI, CT, and optical tomography by offering scalable, uncertainty-aware reconstruction compared to classical Gaussian priors.

Score-based diffusion priors are learned prior distributions in which a diffusion model supplies the score field of an image, signal, graph, or function-space law and that score is then combined with a forward model to solve Bayesian inverse problems. Across the literature, the prior may be used as an explicit probabilistic prior through a probability-flow ODE, as an implicit score field inside posterior samplers, as a denoiser-derived prior in plug-and-play algorithms, or as a task-dependent auxiliary score that reconstructs the conditional posterior score from unconditional training (Feng et al., 2023). In all cases, the central object is a learned approximation to a gradient of a log-density or smoothed log-density, and the practical goal is to regularize ill-posed inference toward samples or estimates that are both measurement-consistent and plausible under empirical data distributions (Schneider et al., 3 Feb 2026).

1. Bayesian formulation and the role of the learned prior

In Bayesian inverse problems, the unknown $x$ is inferred from measurements $y$ through a forward model and a likelihood, while the prior constrains solutions in directions not determined by the data. Several works in this area adopt the standard formulation

$p(x\mid y)\propto p(y\mid x)\,p(x),$

or, for linear Gaussian problems,

$$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$

with posterior defined relative to a prior $\mu$ on $x$ (Schneider et al., 3 Feb 2026). In classical model-based reconstruction, this prior is often Gaussian, such as an Ornstein–Uhlenbeck prior with covariance

$$s(r_1,r_2)=\sigma^2\exp\!\left(-\frac{\|r_1-r_2\|}{\ell}\right),$$

which yields a Gaussian posterior with closed-form mean and covariance (Schneider et al., 3 Feb 2026).

Score-based diffusion priors replace such handcrafted priors by a data-driven prior learned from examples. In this formulation, the diffusion model is trained on clean samples so that its score network approximates the gradient of the log-density of a noisy marginal or of a smoothed version of the prior. The learned score is then used either to define the prior density itself, to approximate $\nabla_x \log p(x)$, or to generate posterior samples through reverse diffusion dynamics (Feng et al., 2023). This makes the prior expressive in settings where smoothness or sparsity penalties are too restrictive, including diffuse optical tomography, MRI, CT, deblurring, interferometric imaging, multi-target detection, source separation, and Gaussian graphical model support estimation (Schneider et al., 3 Feb 2026).

A recurrent distinction in the literature is between using the diffusion model as a genuine prior and using it merely as a reconstruction heuristic. One line of work constructs an explicit prior density $p_\theta(\mathbf{x})$ from the probability-flow ODE and then performs variational inference in the posterior induced by that prior (Feng et al., 2023). Another line uses the learned score only through posterior-sampling dynamics or plug-and-play updates, which can yield effective reconstructions without necessarily corresponding to exact posterior inference (Feng et al., 2023). This suggests that “score-based diffusion prior” refers to a family of related constructions rather than a single algorithmic template.

2. Score representations, training objectives, and prior parameterizations

The basic finite-dimensional score-based diffusion prior learns

$s(x,t;\mu)=\nabla_x\log q_t(x),$

where $y$0 is the law obtained by perturbing the prior with a forward diffusion process (Schneider et al., 3 Feb 2026). For the variance-preserving Gaussian kernel used in diffuse optical tomography,

$y$1

the score governs the reverse-time SDE that samples from the posterior once the observation term is incorporated (Schneider et al., 3 Feb 2026).

A central training paradigm is denoising score matching. For whole-image priors, patch priors, and function-space priors, the network is trained to predict either the score directly or the corresponding denoised signal under Gaussian perturbations. For example, PaDIS trains a denoiser by

$y$2

with score estimate

$y$3

while patch scores are assembled into a full-image score using shifted partitions and positional encoding (Hu et al., 2024). In noisy-data training, SURE-Score couples Stein’s unbiased risk estimate with denoising score matching so that a score-based prior can be learned directly from noisy samples rather than clean targets (Aali et al., 2023).

Several papers reinterpret the learned object to improve tractability. Whitened Score diffusion replaces the raw score target

$y$4

by a covariance-preconditioned or “whitened” target, avoiding explicit covariance inversion for anisotropic Gaussian forward processes (Alido et al., 15 May 2025). In distribution matching, the prior density itself is not needed during encoder optimization; only the prior score $y$5 is required, and it can therefore be learned by denoising score matching without explicit likelihood evaluation (Gong et al., 17 Jun 2025).

A further parameterization is the explicit probabilistic prior derived from the probability-flow ODE. For score-based priors in inverse imaging, the prior log-density is written as

$y$6

where $y$7 is the probability-flow drift and $y$8 is the terminal Gaussian (Feng et al., 2023). Because exact ODE-based likelihood evaluation is expensive, an ELBO-based surrogate prior

$y$9

has also been proposed for scalable Bayesian imaging, replacing ODE solves and divergence estimation by cheaper noise-injection and score-evaluation terms (Feng et al., 2023).

3. Posterior inference: sampling, optimization, and offline–online decompositions

Once a score-based prior is available, posterior inference can proceed through several distinct mechanisms. The most direct is reverse-time diffusion. In the diffuse optical tomography formulation,

$p(x\mid y)\propto p(y\mid x)\,p(x),$0

and $p(x\mid y)\propto p(y\mid x)\,p(x),$1 has the target posterior law $p(x\mid y)\propto p(y\mid x)\,p(x),$2 (Schneider et al., 3 Feb 2026). Closely related posterior samplers include annealed Langevin dynamics, diffusion posterior sampling, score-ALD, SDE+Proj, DAPS, DCDP, SGS-EDM, FK-Corrector, and particle-based formulations derived from posterior-evolution PDEs (Chen et al., 4 Jun 2025).

A major branch of the literature addresses the cost of repeated forward evaluations during sampling. The unconditional representation of the conditional score function (UCoS) constructs a task-dependent unconditional score so that the posterior score can be recovered by an affine transformation rather than learned conditionally. For linear inverse problems, the conditional score is expressed as

$p(x\mid y)\propto p(y\mid x)\,p(x),$3

with $p(x\mid y)\propto p(y\mid x)\,p(x),$4 and $p(x\mid y)\propto p(y\mid x)\,p(x),$5 (Schneider et al., 3 Feb 2026). The corresponding infinite-dimensional theory proves that the conditional score can be derived exactly from a trained unconditional score using affine transformations, thereby shifting computational effort to offline training and eliminating forward evaluations inside the online sampler (Schneider et al., 2024).

A different strategy treats the score-based model as a denoiser prior inside optimization. Plug-and-play priors can be written in score form using Tweedie’s formula. If the denoiser is the MMSE denoiser under the corruption model

$p(x\mid y)\propto p(y\mid x)\,p(x),$6

then

$p(x\mid y)\propto p(y\mid x)\,p(x),$7

and pretrained score-based diffusion models can be inserted into PnP-ADMM, RED, and DPIR without retraining (Park et al., 2024). This perspective clarifies that classical PnP methods solve a regularized inverse problem, whereas diffusion posterior samplers such as DiffPIR operate as stochastic samplers rather than optimizers (Park et al., 2024).

Another family avoids heuristic conditional-score approximations by working at the PDE level. Approximation-Free Diffusion Posterior Sampler defines a time-dependent posterior

$p(x\mid y)\propto p(y\mid x)\,p(x),$8

and derives an exact modified PDE whose drift contains a likelihood-gradient correction and whose reweighting term is implemented by stochastic weighted particle methods and SMC-style resampling (Chen et al., 4 Jun 2025). This line of work is explicit that “approximation-free” refers to avoiding ad hoc approximations of the conditional score $p(x\mid y)\propto p(y\mid x)\,p(x),$9, not to avoiding numerical approximation altogether (Chen et al., 4 Jun 2025).

4. Structured priors, conditioning, and infinite-dimensional extensions

Score-based diffusion priors have increasingly been adapted to structural constraints that are not well represented by isotropic whole-image models. Patch-based position-aware priors learn on image patches only, then synthesize a global score from shifted partitions. In PaDIS, the full-image distribution is written as a product over patch distributions of a zero-padded image, and the full score is approximated as a sum of patch-level scores over many shifted tilings (Hu et al., 2024). This yields improved memory efficiency and data efficiency while retaining the ability to solve CT reconstruction, deblurring, and superresolution with a plug-in diffusion prior (Hu et al., 2024).

Structured Gaussian forward processes motivate a different extension. Whitened Score diffusion allows arbitrary Gaussian noising with non-diagonal covariance and learns a covariance-preconditioned predictable process rather than the raw score. This supports anisotropic Gaussian paths, establishes equivalence with flow matching for arbitrary Gaussian noise, and yields priors better aligned with imaging inverse problems having correlated or structured noise (Alido et al., 15 May 2025).

Conditioning theory is particularly delicate in infinite-dimensional settings. Function-space diffusion models define the score not by a Lebesgue density gradient, which is unavailable, but by conditional expectation. For the Ornstein–Uhlenbeck diffusion

$$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$0

the infinite-dimensional score can be written as

$$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$1

(Baldassari et al., 2024). This enables function-space Langevin samplers for nonlinear inverse problems with convergence bounds that depend explicitly on step size, annealing, and score approximation error (Baldassari et al., 2024).

A complementary development derives posterior conditioning by an infinite-dimensional Doob $$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$2-transform. Under suitable assumptions, the conditional score decomposes as

$$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$3

so that the posterior reverse SDE is driven by an unconditional score plus a guidance term (Baker et al., 28 Jan 2026). Because the guidance is intractable, Supervised Guidance Training introduces a simulation-free denoising score matching objective whose unique optimum is the exact guidance term $$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$4 (Baker et al., 28 Jan 2026). This suggests that conditioning diffusion priors in function space need not rely solely on heuristic likelihood guidance.

Mixed or hybrid score priors address a different structural issue: robustness under limited data or model mismatch. For diffuse optical tomography, the learned posterior score $$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$5 is combined with the closed-form Gaussian score $$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$6 through

$$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$7

with theoretical justification via geometric mixtures of densities for small diffusion times (Schneider et al., 3 Feb 2026). This regularization is applied at sampling time rather than training time, so one trained diffusion prior can be reused for multiple $$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$8 values (Schneider et al., 3 Feb 2026).

5. Application domains and empirical behavior

A striking feature of score-based diffusion priors is their breadth of application. In diffuse optical tomography, a learned prior over absorption and reduced scattering maps produces posterior samples with low variance and centers them around the ground truth even in the presence of modeling errors (Schneider et al., 3 Feb 2026). The method was evaluated on full-view, limited-view, out-of-distribution, and experimental phantom data, with UCoS and regularized UCoS producing tighter uncertainty around expected target regions than DPS or Gaussian OU posteriors (Schneider et al., 3 Feb 2026).

In inverse imaging, score-based priors have been used for denoising, deblurring, superresolution, CT reconstruction, MRI reconstruction, interferometric imaging, inpainting, and image refinement. The explicit-prior variational framework for inverse imaging reported that score-based priors outperform TV, PCA-Gaussian, and RealNVP priors in denoising and improve MSE, PSNR, and SSIM in deblurring and interferometric imaging, while also producing posterior uncertainty maps (Feng et al., 2023). The ELBO-surrogate formulation scaled Bayesian inference to $$y=Ax+\epsilon, \qquad \epsilon\sim\mathcal N(0,\Gamma_{\mathrm{obs}}),$$9 MRI and reported optimization that was at least two orders of magnitude faster than exact ODE-prior inference, with reported PSNR gains ranging from about 2.7 to 8.5 dB across acceleration factors over non-variational diffusion baselines (Feng et al., 2023). In 3D brain MRI analysis, a diffusion prior trained on 7,383 high-quality 1 mm isotropic MR images supported super-resolution, bias correction, inpainting, and refinement using explicit forward models rather than paired task-specific training (Aguila et al., 16 Oct 2025).

Data efficiency and prior structure have also been central themes. PaDIS reported that the patch-based model trained in about 12 hours versus 24–36 hours for the whole-image model and outperformed the whole-image diffusion baseline in PSNR and SSIM across CT, deblurring, and superresolution, especially when training data were limited (Hu et al., 2024). SURE-Score showed that useful score-based generative priors can be learned from noisy training data; in multi-coil MRI denoising, SURE-Score reported $\mu$0 NRMSE at 0 dB training SNR versus $\mu$1 for supervised training, and in 5× accelerated MRI reconstruction it reported $\mu$2 NRMSE at 0 dB versus $\mu$3 for supervised score learning (Aali et al., 2023).

Beyond conventional imaging, the same prior concept has been transferred to more specialized inverse problems. In multi-target detection, a score-based diffusion prior integrated into approximate EM substantially reduced estimation error in high noise regimes, and a later moment-based formulation showed that diffusion priors can stabilize ill-conditioned polynomial inversion and make super-resolution multi-target detection feasible (Zabatani et al., 2023). In RF source separation, diffusion priors over each source yielded a reported BER reduction of 95% over classical and existing learning-based methods by optimizing a smoothed MAP-like objective across multiple Gaussian noise levels (Jayashankar et al., 2023). In Gaussian graphical model support estimation, a graph prior was learned through score estimation with graph neural networks and then sampled by annealed Langevin dynamics, shifting the prior from the precision matrix to the graph topology itself (Sevilla et al., 2024).

6. Evidence estimation, misconceptions, limitations, and open questions

A recurrent misconception is that any diffusion-based inverse solver automatically samples from the correct posterior. Multiple papers argue otherwise. Classical PnP optimization with a score-derived denoiser is not a sampler and therefore does not provide posterior samples or uncertainty quantification (Park et al., 2024). Several guided-diffusion inverse solvers modify reverse diffusion with measurement projection or guidance and may require inference-time measurement-weight tuning, which weakens a strict Bayesian interpretation (Feng et al., 2023). By contrast, approximation-free PDE methods, explicit ODE-based priors, and exact conditioning identities seek to re-establish a clearer posterior meaning, albeit at higher computational or modeling cost (Chen et al., 4 Jun 2025).

Another misconception is that prior choice is secondary once a strong diffusion model is available. Evidence-based model comparison directly challenges that view. DiME estimates the model evidence of a diffusion prior by integrating along posterior time-marginals of reverse diffusion and requires only a handful of posterior sample paths, for example 20 in experiments (Wang et al., 24 Feb 2026). It was shown to match analytic evidence where available, to select the correct digit diffusion prior in Gaussian and Fourier phase retrieval, and to diagnose prior misfit in a real-world black hole imaging problem (Wang et al., 24 Feb 2026). This suggests that score-based diffusion priors can themselves be objects of Bayesian model selection rather than fixed ingredients.

The main limitations reported across the literature are consistent. Exact ODE-based prior evaluation is prohibitively slow and memory-intensive for large images (Feng et al., 2023). Patch-based priors require careful patch-size selection and can produce boundary artifacts under aggressive accelerated samplers (Hu et al., 2024). VP-based plug-and-play adaptation requires approximate noise-level or time-step matching (Park et al., 2024). In infinite-dimensional nonlinear inverse problems, posterior sampling quality depends explicitly on score approximation quality, and poorly learned scores can lead the chain toward the wrong target (Baldassari et al., 2024). In real diffuse optical tomography, some inclusions are still missed or hallucinated, and scattering estimation remains harder than absorption estimation (Schneider et al., 3 Feb 2026).

Open questions follow naturally from these limitations. A plausible implication is that future work will continue to focus on principled conditioning, discretization-invariant formulations, sample-efficient evidence estimation, and priors adapted to structured corruption rather than isotropic image generation alone. The literature already points in that direction through function-space conditioning (Baker et al., 28 Jan 2026), anisotropic Gaussian forward processes (Alido et al., 15 May 2025), offline–online score factorizations (Schneider et al., 2024), and explicit evidence estimation for prior selection (Wang et al., 24 Feb 2026). Collectively, these developments position score-based diffusion priors not merely as generative heuristics, but as a broad probabilistic framework for learned regularization, posterior sampling, uncertainty quantification, and model criticism in inverse problems.