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  • Open Access

Testing maximum entropy models with e-values

Francesca Giuffrida1,2,*, Diego Garlaschelli1,2, and Peter Grünwald3,4

  • *Contact author: francesca.giuffrida@imtlucca.it
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Phys. Rev. E 113, 054306 – Published 15 May, 2026

DOI: https://doi.org/10.1103/xhf5-1l7p

Abstract

E-values have recently emerged as a robust and flexible alternative to p-values for hypothesis testing, especially under optional continuation, i.e., when additional data from further experiments are collected. In this work we define optimal e-values for testing between maximum entropy models, in both the microcanonical (hard constraints) and canonical (soft constraints) settings. We show that, when testing between two hypotheses that are both microcanonical, the so-called growth-rate optimal e-variable admits an exact analytical expression, which also serves as a valid e-variable in the canonical case. For canonical tests, where exact solutions are typically unavailable, we introduce a microcanonical approximation and verify its excellent performance via both theoretical arguments and numerical simulations. We then consider constrained binary models, focusing on 2×k contingency tables—an essential framework in statistics and a natural representation for various models of complex systems. Our microcanonical optimal e-variable performs well in both settings, constituting a tool that remains effective even in the challenging case when the number k of groups grows with the sample size, as in models with growing features used for the analysis of real-world heterogeneous networks and time series.

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Phys. Rev. E 113, 054306– Published 15 May, 2026
  • Received 3 October 2025
  • Accepted 16 April 2026
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