Puffin: Probabilistic Learning of Spatial Detail From Coarse Observations

The gist The PUFFIN framework introduces a probabilistic method for spatial disaggregation that uses high-resolution satellite embeddings to estimate fine-scale socioeconomic variables from coarse regional totals, which is crucial for applications like urban planning and public health where fine-scale data is unavailable

How it works

PUFFIN treats spatial disaggregation as probabilistic constraint satisfaction by learning a prior distribution over every subregion from regional totals alone, which is conditioned on the observed total to yield a posterior that satisfies the constraint exactly The model predicts a mean and variance for each subregion based on its context embeddings, where the prior is interpreted as what the model expects for each subregion before considering the aggregate total

Key methodological steps

The process involves two main stages:

1. Prior Learning: A neural network predicts a Gaussian distribution over the target variable for each subregion based on its context covariates, denoted as y˜i,j ∼ N (µi,j, σ2i,j) where µi,j = fθ(ci,j) and σ2i,j = ai,j softplus(vi,j) + ε

2. Posterior Conditioning: The model conditions this prior on the observed regional aggregate Ybi = PM j=1 yi,j to obtain a posterior distribution that conserves the total exactly

Mass Conservation and Uncertainty

The resulting posterior is closed-form and consistent with every aggregate, meaning it reproduces each published total exactly, and it provides calibrated uncertainty for each subregion The estimate yields a non-negative value, which is achieved by clipping negative estimates to zero and rescaling the region so that the estimates still sum to Ybi Furthermore, the method provides posterior variances that are then rescaled into intervals using a two-stage calibration process

Evaluation and Performance

PUFFIN is evaluated on German and US census, employment, and election data across population and jobs variables The metric used for comparison is the within-region R2, which measures how well each total is split among its subregions Experimental results show that PUFFIN reaches within-region R2 between 0.12 and 0.84 on German census data, higher than bicubic interpolation everywhere When applied to the US population and jobs, PUFFIN outperforms baselines in certain configurations, demonstrating its ability to recover fine structure from embeddings alone

Limitations and Calibration

The method's success is tied to where the fine variation is real and tied to what the embedding can see, as it falls back to or below the regional average where the fine signal is absent The calibrated intervals are not label-free, requiring a small labeled sample to set their width, indicating that deploying them without such samples would require transferring factors from other datasets Structural limits include treating subregions as independent and describing each by a single averaged embedding, which can hide small settlements inside large rural units The Gaussian prior also places mass on negative counts, which is handled by clipping and rescaling rather than using a count likelihood Future work suggests extending the posterior to a grid of cells rather than one averaged embedding per unit for greater variance and joint conditioning of correlated variables

The broad lesson is that a single coarse total, read together with a satellite embedding, already pins down a large part of the fine-scale map, but only where the fine variation is real and tied to what the embedding can see PUFFIN casts spatial disaggregation as probabilistic constraint satisfaction. Trained only on coarse totals, it learns a prior over every subregion and conditions it on the observed totals, which gives an estimate that is non-negative, reproduces each total exactly, and carries a calibrated uncertainty The point estimates use no fine labels at any stage The intervals are the one part of the method that is not label-free Skill grows with the size of the subregion, which fits a hard limit at the finest scale, where one sees roofs from above but not how many people are inside Where no fine-resolution reference grid exists, its point estimates apply where disaggregation is needed most The intervals additionally need a small labeled sample to set their width The case for the variance weighting therefore rests on the prior column and the area ablation The model also has structural limits. The Gaussian prior places mass on negative counts, which we remove by clipping and rescaling the mean rather than by a count likelihood It treats subregions as independent, and it describes each subregion by a single averaged embedding, which can hide a small settlement inside a large rural unit Its variance scales with area, which suits sparse land but understates the uncertainty of small, dense subregions It disaggregates one variable at a time The training loss improves the fine-scale split, showing that training on totals alone helps reduce the fine-scale error The interval coverage is 0.897 to 0.902 across the Germany configurations and 0.871 to 0.926 across the Berlin ones This work uses only publicly released data (census counts, workplace job counts, publicly reported incident records, and precinct turnout) together with public satellite embeddings PUFFIN redistributes totals that are already published, uses no data about identified individuals, and keeps every official total unchanged Fine-scale estimates of sensitive variables such as election turnout or violent incidents can nevertheless be mistaken for observations or misused to single out small areas, and Experiment 3 shows that such estimates can be unreliable This is why every estimate comes with a calibrated uncertainty, and why the method is intended for planning and research at an aggregate level, not for inferring anything about individuals Every data source is public: the Zensus 2022 grid (Statistisches Bundesamt, 2024a), the 2020 US census (U.S. Census Bureau, 2021), LODES (U.S. Census Bureau, 2024), the Gun Violence Archive (Gun Violence Archive, 2013), the 2024 precinct returns (New York Times, 2025), and the AlphaEarth embeddings (Brown et al., 2025) The code that produced every number in this paper is linked in the abstract Every method still adds error, and the less it moves away from the Preprint. Under review The region average PUFFIN estimate Ground truth 0 10 20 30 40 50 60 Population (count per cell) Population disaggregation Regionaverage baseline (left), PUFFIN estimate (middle), and ground truth (right), for population Figure 2 The region average baseline scores zero by construction, and it does better the larger the subregion, from 0.12 at 200 m regions with 100 m subregions to 0.68 at 10 km regions with 5 km subregions PUFFIN (prior) is the prior before conditioning, which the posterior improves on everywhere The region average PUFFIN estimate Ground truth 0 10 People per block (log) New York City: Population disaggregated to census blocks Region average (left), the PUFFIN estimate (middle), and ground truth (right) Figure 3 The region average gives every block of a tract the same count, the tract total divided by its number of blocks, while PUFFIN recovers the block-level structure The US population in New York City, with census tracts as regions and census blocks as subregions (83,774 tracts, 8.0M blocks), where every method uses the area term Subregion-R2 on Berlin for representative configurations, inhabited cells and all 100 m cells Figure 5 PUFFIN std ≤ 0.01 Table 3 Within-region R2 on Berlin, all 100 m cells, per configuration and method Best method per configuration in bold, and PUFFIN (prior) is the prior alone, before conditioning on the totals Table 4 Within-region R2 on Germany, all 35.7M cells, per configuration and method PUFFIN std ≤ 0.017 Table 9 US, census tracts as regions and census blocks as subregions, inhabited blocks only, read as Table 2 PUFFIN std ≤ 0.02 The region average the better it does (Table 5), whether it redistributes by neighboring totals or by the embedding Best per group in bold, and PUFFIN std ≤ 0.

1. The PUFFIN framework can perform spatial disaggregation on coarse totals using only a single satellite embedding and observed regional totals, which is applicable where such labels do not exist. This allows for high-resolution estimates for areas like many countries with no fineresolution census grid.

2. The system provides uncertainty quantification by yielding calibrated uncertainty, which means the posterior variances are rescaled using a factor derived from validation subregions, offering confidence intervals that are between 0.87 and 0.93 at the nominal 90% level in every configuration, across resolutions, countries, and targets.

3. PUFFIN can handle irregular spatial units by incorporating an area term into the prediction via Equation (4), allowing it to account for Uneven Subregion Sizes, which is crucial when dealing with units like US census blocks whose areas vary by four orders of magnitude between the 1st and 99th percentile.

4. The system learns from aggregates under an exact constraint, as described by the contribution Disaggregation as Probabilistic Constraint Satisfaction, meaning it learns a per-subregion predictive distribution from regional totals alone, with the aggregation constraint as the only supervision.

5. PUFFIN can be trained without fine-resolution labels (Training needs no fine-resolution labels), making its point estimates usable where they are needed most, such as in applications like urban planning, public health, disaster response.

High-resolution socioeconomic variables are important for applications such as urban planning, public health, disaster response, and resource allocation. In practice, however, these variables are often observed only at a coarse spatial resolution. We introduce Puffin, a probabilistic framework for statistical disaggregation that raises the resolution of coarse totals using high-resolution satellite embeddings as covariates. Instead of predicting a single value for each fine-resolution subregion, Puffin learns a probability distribution and is trained through an aggregation-aware likelihood. At inference, Puffin conditions these predictions on the observed regional total and splits it among the subregions. The resulting fine-scale estimates are consistent with the observed aggregate and come with calibrated uncertainty, without requiring fine-resolution labels for training. We evaluate Puffin on German and US census, employment, and election data across population, jobs, and other count variables, and study when statistical disaggregation succeeds or fails across regions, countries, and targets.