The Fermi-Dirac distribution provides a calibrated probabilistic output for binary classifiers

Sung-Cheol Kim, Adith S Arun, Mehmet Eren Ahsen, Robert Vogel, Gustavo Stolovitzky

Research output: Contribution to journalArticlepeer-review


Binary classification is one of the central problems in machine-learning research and, as such, investigations of its general statistical properties are of interest. We studied the ranking statistics of items in binary classification problems and observed that there is a formal and surprising relationship between the probability of a sample belonging to one of the two classes and the Fermi-Dirac distribution determining the probability that a fermion occupies a given single-particle quantum state in a physical system of noninteracting fermions. Using this equivalence, it is possible to compute a calibrated probabilistic output for binary classifiers. We show that the area under the receiver operating characteristics curve (AUC) in a classification problem is related to the temperature of an equivalent physical system. In a similar manner, the optimal decision threshold between the two classes is associated with the chemical potential of an equivalent physical system. Using our framework, we also derive a closed-form expression to calculate the variance for the AUC of a classifier. Finally, we introduce FiDEL (Fermi-Dirac-based ensemble learning), an ensemble learning algorithm that uses the calibrated nature of the classifier's output probability to combine possibly very different classifiers.

Original languageEnglish (US)
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number34
StatePublished - Aug 24 2021


  • binary classification
  • distribution
  • Fermi–Dirac
  • calibrated probability
  • machine learning
  • ensemble learning


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