SCIENCE · BENCHMARKS
Our scientific results, quantified, published, reproducible.
Maximum Entropy Relaxation is compared with generalized raking, the classic reference method, on standardized benchmarks: NPORS-derived, 4 to 40 attributes, ternary constraints. Beyond 28 attributes, its advantage is structural. Results published in April 2026 on ArXiv.
BENCHMARKS AS A PRINCIPLE
A method that is not benchmarked is not a scientific method.
A methodology cannot simply assert its superiority: it must demonstrate it on standardized benchmarks that allow adversarial comparison. Without a benchmark, a methodology remains a dressed-up hypothesis. This principle is rarely honored in decision intelligence: methodologies documented without confrontation with benchmarks, performance claimed without possible reproduction, qualitative superiority asserted without quantifiable measurement.
Imagine All The People publishes its results on standardized benchmarks, with reproducible measurements. The paper Maximum Entropy Relaxation of Multi-Way Cardinality Constraints for Synthetic Population Generation, published in April 2026 on ArXiv by François Pachet and Jean-Daniel Zucker, compares our approach with generalized raking, the classic reference method, on NPORS-derived benchmarks with 4 to 40 attributes and ternary constraints. These results are open to any challenge by the scientific community.
THE PUBLISHED RESULTS
Four structuring results from the NPORS 2026 benchmark.
MAXENT SUPERIORITY THRESHOLD
K ≥ 28
attributes with ternary constraints: MaxEnt structurally outperforms generalized raking
SCALE OF THE TESTED BENCHMARK
4 to 40
simultaneous attributes in the synthetic population
REFERENCE METHOD COMPARED
generalized raking
the standard of official statistical surveys
PUBLICATION AND REPRODUCIBILITY
ArXiv, April 2026
reproducible benchmarks, code and methodology published
THE BENCHMARK METHODOLOGY
How we rigorously compare Maximum Entropy Relaxation with the classic methods.
The choice of the NPORS-derived benchmark.
We built our benchmark from the National Public Opinion Reference Survey (NPORS), a public reference dataset used by the scientific community in population modeling. This choice is methodologically structuring: it lets any third-party researcher reproduce our tests from publicly available data. Our benchmarks are not proprietary or opaque: they are public, standardized and contestable.
The benchmark's parameters.
The benchmark tests the generation of synthetic populations with a number of attributes ranging from 4 to 40, systematically including ternary constraints (simultaneous satisfaction of unit, binary and ternary marginal distributions). This setup covers the full practical spectrum of real simulations: from simple cases with a few attributes to complex configurations with several dozen crossed attributes.
The comparison with generalized raking.
We compare Maximum Entropy Relaxation with generalized raking, the classic reference method in synthetic population generation, used for decades in statistical studies and official surveys. This adversarial comparison is structuring: we do not compare ourselves with dated or marginal methods, but with the standard method currently used by public and private players in quantitative sociology.
The performance measurements.
We measure both methods' performance on standardized quantitative criteria: average deviations from the target distributions at the unit, binary and ternary levels, solver convergence times, robustness to inconsistent constraints. These measurements are documented in the tables and figures of the ArXiv paper, open to any reproduction or challenge.
The central result.
Our results show that Maximum Entropy Relaxation becomes structurally superior to generalized raking from 28 simultaneous attributes with ternary constraints present. Below that threshold, the two methods remain comparable, and raking can even be more efficient on the simplest configurations. Above it, MaxEnt's advantage grows structurally: this is the practical regime of the real simulations we run for our clients, which typically involve 30 to 60 crossed attributes with numerous ternary constraints.
DETAILED TECHNICAL COMPARISON
Maximum Entropy Relaxation versus generalized raking across five dimensions.
| Generalized raking (classic method) | Maximum Entropy Relaxation (our approach) | |
|---|---|---|
| Optimal performance regime | Simple configurations, up to 20-25 attributes | Complex configurations, beyond 28 attributes with ternary constraints |
| Handling of ternary constraints | Approximate iterative handling, progressive degradation | Native handling through the exponential-family formulation |
| Theoretical guarantee | Convergence guaranteed under restrictive conditions | Convex optimization with convergence guarantees via L-BFGS |
| Robustness to inconsistent constraints | Possible divergence with mutually inconsistent constraints | Penalized convex extension for inconsistent targets |
| Reproducibility by independent third parties | Public, documented method, widely implemented | Public, documented method, reproducible NPORS benchmarks |
THE NEXT BENCHMARKS
Continuous scientific output, not a one-off artifact.
The NPORS 2026 benchmark is a milestone, not an endpoint. Our scientific teams work continuously on extending the benchmarks to more demanding regimes and targeted sector applications. Three work streams are open.
Extension to the K > 20 regime through Persistent Contrastive Divergence.
Beyond 20 simultaneous attributes, the exact computation of expectations becomes intractable. Our ongoing work extends the Pachet-Zucker framework with stochastic estimation via Gibbs sampling, preserving the convergence guarantees at significantly larger scales. This extension opens access to benchmarks with several hundred attributes: the relevant regime for the most complex sector simulations.
Sector benchmarks calibrated on real data.
Beyond standardized academic benchmarks, we are preparing sector benchmarks calibrated on real configurations: consumer populations for retail and consumer goods, patient populations for healthcare, voter populations for the political field, employee populations for HR questions. These sector benchmarks will document our approach's performance on the practical configurations our clients actually mobilize.
Temporal robustness benchmarks.
A new methodological axis: measuring the stability of synthetic populations over time, by submitting the same populations to successive scenarios with evolving calibration parameters. This temporal dimension of benchmarking is essential to validate the robustness of multi-month and multi-year projections, which make up a growing share of our client deliverables.
All of these upcoming publications will be referenced on the Scientific papers page as soon as they are available.
Want to go deeper into our benchmarks?
The April 2026 ArXiv paper documents the methodology, the quantified results, the comparative tables and the sensitivity analyses in detail. It is available from the Scientific papers page. Our scientific team is available to answer technical questions, enable the reproduction of our benchmarks, or discuss the configurations specific to your sector. The benchmarks are open to adversarial challenge: that is precisely the point of a scientific approach.
See the scientific papers →