Science · Benchmarks
A publicly documented protocol, reproducible by third parties.
Maximum Entropy Relaxation is compared with generalized raking under a common protocol based on NPORS-derived data, with 4 to 40 attributes and different levels of constraints.
Publication: Frontiers in Artificial Intelligence — Peer-reviewed · Accepted for publication
What happens as complexity increases?
The benchmark does not seek to establish that one method is inherently better. It observes how two methods behave as the number of simultaneous attributes and the arity of constraints increase.
What is compared, and on which data.
| Method evaluated | Maximum Entropy Relaxation |
|---|---|
| Reference method | generalized raking |
| Data | NPORS-derived |
| Attributes | 4 to 40 |
| Constraints | unary, binary, ternary |
SourceFrançois Pachet, Jean-Daniel Zucker — Maximum Entropy Relaxation of Multi-Way Cardinality Constraints for Synthetic Population Generation, Frontiers in Artificial Intelligence (peer-reviewed · accepted for publication) — DOI 10.3389/frai.2026.1840945 View
The figure
−73%: what was measured.
| Data | NPORS-derived benchmark |
|---|---|
| Number of attributes | 40 |
| Size of the evaluated synthetic population | 5,626 individuals |
| Constraints | Up to third order |
| Method evaluated | MaxEnt |
| Comparator | Generalized raking |
| Metric | Mean relative error across constraints |
| MaxEnt value | 0.095 |
| Generalized raking value | 0.352 |
| Interpretation | Lower is better |
| Calculation | (0.352 − 0.095) / 0.352 × 100 ≈ 73% This is a relative reduction in error, not a 73-point drop. |
The metric averages the absolute relative differences between generated and target frequencies across the evaluated constraints.
This result concerns the reconstruction of a population’s statistical characteristics in this configuration. It does not measure behavioral forecasting accuracy. Relative performance varies across configurations.
Deviations from target distributions.
The metric averages the absolute relative differences between generated and target frequencies across the evaluated constraints.
K ≥ 28.
In this protocol, from 28 attributes onward in the presence of ternary constraints, Maximum Entropy Relaxation achieves better results according to the selected metric.
Summary reading of the published result. Detailed values are provided in the article. No measurement point is reconstructed on this page.
A threshold, not general superiority.
Below this threshold, the two methods remain comparable according to the same metric. The result describes a dependence on complexity, not an absolute ranking. It applies to the protocol described in the preprint and to this metric.
Limit of this result
What this benchmark does not validate.
- This benchmark validates only the behavior observed under this protocol.
- It validates neither every configuration, nor every dataset, nor the behavioral conclusions of simulations.
Maximum Entropy Relaxation of Multi-Way Cardinality Constraints for Synthetic Population Generation
François Pachet · Jean-Daniel Zucker
Frontiers in Artificial Intelligence — DOI 10.3389/frai.2026.1840945
Peer-reviewed · Accepted for publication
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