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Testing maximum entropy models with e-values
Francesca Giuffrida1,2, Diego Garlaschelli1,2, Peter Grünwald3,4
1IMT School for Advanced Studies, Lucca, Italy.
Physical Review. E
|June 19, 2026
Summary
E-values offer a robust alternative to p-values for hypothesis testing, especially with optional continuation. This study introduces optimal e-values for maximum entropy models, providing exact solutions for microcanonical tests and effective approximations for canonical tests.
Area of Science:
- Statistical theory
- Information theory
- Complex systems modeling
Background:
- P-values are standard for hypothesis testing but have limitations, particularly with sequential data collection.
- E-values provide a flexible and robust alternative, especially beneficial in scenarios with optional continuation.
- Maximum entropy models are crucial for statistical inference under constraints.
Purpose of the Study:
- To define and derive optimal e-values for hypothesis testing between maximum entropy models.
- To address both microcanonical (hard constraints) and canonical (soft constraints) settings.
- To develop a statistically sound and computationally efficient testing framework.
Main Methods:
- Derivation of optimal e-values for microcanonical maximum entropy models.
- Development of a microcanonical approximation for canonical maximum entropy model testing.
- Analysis of constrained binary models, specifically 2xk contingency tables.
- Theoretical analysis and numerical simulations to validate performance.
Main Results:
- An exact analytical expression for the growth-rate optimal e-variable in microcanonical tests.
- Demonstration that this expression is also valid for canonical tests.
- Validation of the microcanonical approximation for canonical tests, showing excellent performance.
- The proposed optimal e-variable is effective even when the number of groups (k) grows with sample size.
Conclusions:
- Optimal e-values provide a powerful tool for hypothesis testing between maximum entropy models.
- The microcanonical approximation offers a practical solution for canonical tests where exact solutions are difficult.
- The developed method is robust and applicable to complex systems, including those with growing features.
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