## Abstract

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 language | English (US) |
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Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 118 |

Issue number | 34 |

DOIs | |

State | Published - Aug 24 2021 |

## Keywords

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