# Spatial Prediction with Guided Diffusion

arXiv: https://arxiv.org/abs/2610.12967 · HTML: https://mlsanity.com/paper/2610.12967/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 14 of 14)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [The Principles of Diffusion Models](https://mlsanity.com/paper/2510.21890/) (2025)
- [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [Neural Conditional Simulation for Complex Spatial Processes](https://mlsanity.com/paper/2508.20067/) (2025)
- [Neural Parameter Estimation with Incomplete Data](https://mlsanity.com/paper/2501.04330/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)

## Abstract (arXiv)

Spatial prediction is a critically important task in data analyses that involve spatially or spatio-temporally-referenced data. It is commonly carried out using hierarchical models, which comprise a process model describing the latent process of interest, and a data model describing how the process is observed. The target quantity is the predictive distribution, which is the conditional distribution of the latent spatial process given the observed data. Unfortunately, the predictive distribution is rarely available in closed form and can be difficult to sample from. Here, we show how recent developments in guided diffusion modelling from machine learning can address a wide class of problems in spatial prediction when the likelihood function derived from the data model is known and differentiable, possibly after relaxation. To show the versatility of the approach, we apply it to both Gaussian and non-Gaussian process models and a variety of data models that describe direct partial observations, linear and areal aggregates, counts, right censoring, and area-exceedance functionals. Where possible, we validate the framework against exact and Markov-chain-Monte-Carlo (MCMC)-derived predictive distributions. We demonstrate our approach on several simulation studies and in a controlled experiment of methane-flux inversion over the UK and Ireland. We show that guided diffusion can sample from the correct predictive distribution in a variety of settings, that it can be applied to models where MCMC is intractable, and that it scales well with the dimensionality of the latent process.