Spatial Prediction with Guided Diffusion

1 Introduction

• Sampling the predictive distribution is straightforward for Gaussian processes with linear-Gaussian data, but difficult for non-Gaussian processes or nonlinear and non-Gaussian data models.Source: §1 · p. 2
• Existing approaches have model-specific trade-offs: MCMC is widely applicable when the target density is evaluable up to a constant, while INLA targets latent Gaussian models and ABC scales poorly with summary-statistic dimension.Source: §1 · p. 2
• NCS trains directly for a specified data model, so changing or misspecifying that model requires retraining and can make the approach impractical.Source: §1 · p. 3
• Guided diffusion first learns an unconditional process model, then combines its score with a likelihood gradient at prediction time to sample from the predictive distribution.This separates the process model from the data model during training.Source: §1 · p. 3
• Guided diffusion is presented as a general, scalable approach across process and data models, with one trained process model reused for different data models.The process model need only be available through simulation.Source: §1 · p. 3
• The framework is demonstrated on Gaussian and Ginzburg–Landau processes and data models covering partial observations, counts, censoring, aggregates, and area-exceedance functionals.Source: §1 · p. 3

2 Score-Based Diffusion for Spatial Prediction

• The process is represented on a grid of n cells, and the forward diffusion evolves it with a deterministic noise schedule and diffusion time distinct from physical time.Source: §2 · p. 3
• At the terminal diffusion time, the process is approximately standard Gaussian regardless of its initial value, enabling unconditional sampling by integrating the reverse-time SDE.Source: §2 · p. 4
• For Gaussian processes, the marginal score is available in closed form, so the reverse-time system can be integrated without a neural network.Source: §2 · p. 4
• For general non-Gaussian processes, the marginal score is intractable and is estimated with a neural network trained using score matching.The network estimates the score of the noised process.Source: §2 · p. 4
• NCS trains a network to target the conditional score, but if prediction data do not follow its training data model, its samples do not target the correct predictive distribution.Retraining for each possible data model can be infeasible.Source: §2 · p. 5

3.1 Score decomposition and approximate guidance

• The predictive score decomposes into a process-model score and a data-model score, allowing a process-score network to be reused with different data models at prediction time.No additional neural network is needed to estimate the data-model score.Source: §3.1 · pp. 5–6
• DPS ignores the conditional variance of Y0 given Yt and can yield under-dispersed samples, while importance sampling weights degenerate at moderately high dimensions and sequential Monte Carlo is costly.ΠGDM instead uses a Gaussian surrogate for p(Y0 | Yt); its extra approximations can be more laborious, but its predictive intervals generally have empirical coverage closer to nominal.Source: §3.1 · pp. 6–7

3.2 Guided diffusion with linear-Gaussian data models

• For linear-Gaussian observations, the framework covers partial observations, averages over disjoint regions, and soft mass-conservation constraints.The noised likelihood for ΠGDM is available in closed form for this data model, so no further approximation is needed.Source: §3.2 · pp. 7–8

3.3 Guided diffusion with non-linear and non-Gaussian data models

• For non-linear or non-Gaussian observations, the section treats conditional Poisson counts, including log-Gaussian Cox process representations, without requiring a Gaussian latent field.The count-data likelihood is approximated for ΠGDM.Source: §3.3 · p. 8
• Right-censored observations contribute a smooth, differentiable likelihood based on whether noisy measurements exceed a known threshold.The censored log-likelihood is smooth in each latent field value, and its noised likelihood is approximated for ΠGDM.Source: §3.3 · p. 8
• Area-exceedance observations count threshold-exceeding cells across the whole field, forming a non-linear scalar functional distinct from linear spatial aggregates.A smoothed indicator and low-variance discrete distribution make the likelihood differentiable for guided diffusion.Source: §3.3 · pp. 8–9

4 Experiments

• Across the experiments, the Gaussian-process score and linear-Gaussian noised likelihood are available in closed form, while non-Gaussian cases use approximations.The grid has n = 1024 cells; the non-Gaussian process model uses a trained score network.Source: §4 · p. 9

4.1 Gaussian process model

• ΠGDM guidance is exact up to time discretization against direct Gaussian predictive sampling, while DPS loses some fidelity.Figure 1 compares predictive means, standard deviations, and covariance; Table 1 reports diagnostics averaged over 100 realisations.Source: §4.1 · p. 10, Figure 1 · p. 10, Table 1 · p. 11
• For Poisson counts, ΠGDM closely matches ESS under reported diagnostics, while DPS retains mild under-dispersion.Figure 2 compares predictive means, standard deviations, and covariance; Table 2 reports diagnostics averaged over 100 realisations.Source: §4.1 · p. 11, Table 2 · p. 11, Figure 2 · p. 12

Area-exceedance functional

• Under the single area-exceedance constraint, all samplers respect the realised counts, but DPS under-disperses relative to ESS on the hard count.The experiment uses Z = 410 exceedances at c = 1; Table 3 reports realised exact and soft counts.Source: Area-exceedance functional · p. 11, Table 3 · p. 12

4.2 Ginzburg–Landau process model

• The Ginzburg–Landau amplitude field has no analytic density, so its predictive results cannot be checked against an exact or MCMC-based distribution.The experiments instead use predictive diagnostics; the process-model score is estimated with a trained neural network.Source: §4.2 · pp. 12–13
• For areal aggregates on the Ginzburg–Landau field, DPS and ΠGDM differ little, and both recover tested functionals with small spread and slight global-mean bias.The functionals include global mean, minimum, maximum, the proportion below 0.5, and values at four grid cells.Source: §4.2 · pp. 13–14, Table 4 · p. 14
• For upper-tail censoring, ΠGDM achieves closer-to-nominal coverage than DPS, and its global mean has a small positive bias.The censoring threshold is c = 1, with measurement error standard deviation σ_ϵ = 0.03 for uncensored cells.Source: §4.2 · p. 14, Table 4 · p. 14

5.1 Trace-gas inversion using guided diffusion

• The inversion maps flux at 122 UK and Ireland grid cells to 3,409 two-hourly station observations using NAME-derived sensitivities and a linear-Gaussian observation model.The controlled experiment uses four stations over January–March 2014, with the inventory treated as the unknown true field.Source: §5.1 · pp. 15–16
• The flux process uses a Box–Cox spatial model with priors elicited from an MCMC fit to the inventory, while guided-diffusion training uses process simulations rather than mole-fraction data.The trained process model is conditioned on the observations through the likelihood gradient at inference.Source: §5.1 · pp. 17–18

5.2 Guided predictions and comparison with MCMC

• DPS and ΠGDM approximately match MCMC across predictive diagnostics, but guided approximations slightly overestimate the upper spatial percentile and all samplers’ domain totals are upward-biased.The MCMC domain total is closest to truth; guided and MCMC distributions also agree closely on regional totals, with shared territorial biases.Source: §5.2 · pp. 18–20, Figure 7 · p. 19, Figure 8 · p. 20

5.3 Computational efficiency

• The scaling assessment holds the observation count fixed across three resolutions and compares GPU reverse diffusion with ten parallel CPU MCMC chains.The resolutions use n = 122, 487, and 1948 flux cells.Source: §5.3 · p. 21, Figure 11 · p. 22
• At n=1948, ΠGDM takes 0.506 versus MCMC’s 5.45 seconds per effective sample; MCMC is faster at n=122 but roughly eleven times costlier at n=1948.The comparison is time per effective sample of domain total; Figure 11 plots this against the number of flux cells.Source: §5.3 · pp. 21–22, Figure 11 · p. 22

6 Discussion

• Guided diffusion applies when the process can be simulated and a differentiable likelihood can be constructed, and it can be reused across different data-model scenarios.The discussion also identifies scalability as a motivation, noting that guided diffusion often scales linearly with process dimension while MCMC often scales poorly.Source: §6 · p. 22
• Both guided samplers are slightly under-dispersed relative to MCMC and, unlike MCMC, lack asymptotic exactness.ΠGDM achieved coverage closer to nominal than DPS in the cases considered.Source: §6 · p. 22
• Vanilla guided diffusion fixes the process model, its prior distributions, and the grid, returns no posterior over process parameters, and may require many simulations for non-Gaussian score training.The simulation requirement may be prohibitive when process simulation is expensive.Source: §6 · p. 23

A Derivations of (3), (4) and (6)

• The appendix derives the conditional distribution in equation (3).The derivation follows Anderson (1982), Vincent (2011), and Song et al. (2021), rewritten in the paper’s notation and context.Source: A Derivations of (3), (4) and (6) · p. 27
• The appendix derives the reverse-time SDE in equation (4).The derivation follows Anderson (1982), Vincent (2011), and Song et al. (2021), rewritten in the paper’s notation and context.Source: A Derivations of (3), (4) and (6) · p. 27
• The appendix derives the denoising score-matching objective in equation (6).The derivation follows Anderson (1982), Vincent (2011), and Song et al. (2021), rewritten in the paper’s notation and context.Source: A Derivations of (3), (4) and (6) · p. 27

A.1 Conditional distribution (3) and the coefficients αt and σ2

• Conditional on the initial field, the linear-Gaussian forward diffusion is an affine functional of Brownian motion, so its mean and variance determine its conditional distribution.The appendix therefore reduces the derivation to computing those two moments.Source: §A.1 · p. 27
• Integrating the stated stochastic equation and multiplying by αt yields the explicit solution used to derive the conditional moments.The passage describes this integration as the step producing the solution.Source: §A.1 · p. 27
• The Itō integral’s zero mean gives the conditional mean, while its independent components and the Itō isometry yield the conditional variance.The resulting moments are identified as those in equation (3).Source: §A.1 · p. 27

A.2 Reverse-time SDE (4)

• The reverse-time SDE uses the forward process’s diffusion coefficient and selects its drift so the reverse marginals match the forward marginals.Comparing the forward and reverse Fokker–Planck equations identifies the required reverse drift.Source: §A.2 · p. 28

A.3 Denoising score-matching objective (6)

• The denoising score-matching loss differs from the explicit score-matching loss only by a constant, so both share the same minimizer.The denoising objective uses a conditional score available in closed form, unlike the unknown marginal score in the explicit loss.Source: §A.3 · pp. 28–29
• To obtain an optimal ϕ across all times, it is sufficient to minimize JDSM(ϕ) uniformly over t; integrating (A.13) with respect to time gives LS(ϕ) in (6).Source: §A.3 · p. 29

B ΠGDM approximations for non-linear data models

• The appendix details noised-likelihood approximations for count, right-censoring, and area-exceedance data models used with ΠGDM.These data models are introduced in Section 3.3.Source: B ΠGDM approximations for non-linear… · p. 29

B.1 Poisson data

• For Poisson counts, a second-order Taylor expansion around the Tweedie estimate approximates each likelihood factor as a Gaussian pseudo-response.Stacked pseudo-responses yield a linear-Gaussian likelihood, whose guidance gradient is differentiated through the Tweedie estimate.Source: §B.1 · pp. 29–30

B.2 Right censoring

• For right-censored data, assuming Vt = vtIn makes the noised-likelihood integral a product of one-dimensional integrals for uncensored and censored cells.Conditioning on Yt rather than Y0 effectively inflates the variance.Source: §B.2 · pp. 30–31

B.3 Area-exceedance functional

• For area-exceedance observations, the approximation fixes the small variance at the Tweedie estimate and linearises the nonlinear mean there.The discrete Gaussian is assumed to concentrate almost all its mass within one unit of the soft count when ξ2 is small.Source: §B.3 · pp. 31–32

S.1 Computational details

• Reverse sampling uses Euler–Maruyama updates on a log signal-to-noise-ratio grid, with guidance norm clipping when needed.The reported DPS guidance norm grew by several orders of magnitude near t = 0 in other experiments, whereas ΠGDM rarely required clipping; methane fields are embedded in 20 × 20 arrays and predictions are restricted back to land cells.Source: §S.1 · p. 33

S.2 Comparison with neural conditional simulation

• NCS is trained on MAR Gaussian data with observed proportions of 0.02–0.10, then applied without retraining to MAR data and upper-tail censoring.The MAR test setting uses a 5% observed proportion; the censoring setting observes a cell only when its noisy measurement is below c = 1.Source: §S.2 · p. 34
• Under matched MAR data NCS is similar to guided diffusion, but under upper-tail censoring its predictive accuracy deteriorates; guided samplers use the correct likelihood without retraining.The comparison reports RMSE, 90% coverage, ES, and CRPS over 50 realisations; the table defines these diagnostics and their censoring-cell scope.Source: Table S.1 · p. 34, §S.2 · p. 34
• Figure S.1 lays out the true field, observations, and predictive means and standard deviations for NCS and ΠGDM in one realisation per model setting.Its top and bottom rows show MAR and upper-tail censoring, respectively.Source: Figure S.1 · p. 35

• Figures S.2 and S.3 display functional residuals for 100 realisations of the areal-aggregation and censored-data experiments, with DPS in blue, ΠGDM in red, and zero marked by a dashed line.Source: Figure S.2 · p. 36, Figure S.3 · p. 36
• Figure S.4 shows methane-experiment model-scale parameter priors induced from independent informative priors on natural-scale parameters elicited from the inventory alone.Source: Figure S.4 · p. 37
• Figure S.5 presents five unconditional reverse-diffusion draws from p(Y0), annotated with domain flux totals, alongside the inventory total of 2.73 Tg yr−1 for context.Source: Figure S.5 · p. 37
• Table S.2 lays out region-level total methane flux in g s−1, including the inventory value and predictive mean ± standard deviation for DPS, ΠGDM, and MCMC.Source: Table S.2 · p. 38
• Figures S.6 and S.8 show predictive distributions for eight methane-flux functionals for DPS, ΠGDM, and MCMC at n = 487 and n = 1948, respectively.Source: Figure S.6 · p. 38, Figure S.8 · p. 39
• Figures S.7 and S.9 show per-cell predictive means and standard deviations for DPS, ΠGDM, and MCMC at n = 487 and n = 1948, respectively.Source: Figure S.7 · p. 39, Figure S.9 · p. 40