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  • Open Access

Constrained surrogates for arbitrary families of continuous probability distributions

Jack Murdoch Moore1,2 and Eduardo G. Altmann3,*

  • 1MOE Key Laboratory of Advanced Micro-Structured Materials, and School of Physical Science and Engineering, Tongji University, Shanghai 200092, People's Republic of China
  • 2National Key Laboratory of Autonomous Intelligent Unmanned Systems, MOE Frontiers Science Center for Intelligent Autonomous Systems, Tongji University, Shanghai 200092, People's Republic of China
  • 3School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia

  • *Contact author: eduardo.altmann@sydney.edu.au
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Phys. Rev. E 114, 044327 – Published 28 October, 2026

DOI: https://doi.org/10.1103/fs71-p5y8

Abstract

Empirical regularities described by parametrized families of probability distributions can shape how complex systems are interpreted and explained. A recurring problem is to determine the extent to which these descriptions are supported by observations, and how they can be used to characterize the underlying system. Here we introduce a general framework for constructing constrained surrogates capable of representing arbitrary families of continuous distributions. We demonstrate its utility by providing explicit algorithms for several canonical distributions in statistical physics, including continuous power-law, lognormal, exponential, and truncated Gaussian distributions. These constrained surrogates enable unbiased estimation and accurate tests of model validity based on arbitrary (physically meaningful) sample statistics, without requiring specification or estimation of model parameters. We show how these theoretical advantages translate into more informative and nuanced conclusions in applications ranging from bushfire magnitudes and word usage in texts to brain connectomes and city populations.

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Phys. Rev. E 114, 044327– Published 28 October, 2026
  • Received 5 February 2026
  • Accepted 29 September 2026
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