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

1) The proposed method maps the probability of correct classification to the Fermi-Dirac distribution, enabling computation of optimal thresholds and statistical parameters;
2) The relationship between classification probabilities and the Fermi-Dirac distribution allows for calibrated probabilistic outputs in binary classifiers;
3) FiDEL, an ensemble learning algorithm, utilizes this calibration to combine different classifiers effectively.

Abstract

While it would be desirable that the output of binary classification algorithms be the probability that the classification is correct, most algorithms do not provide a method to calculate such a probability. We propose a probabilistic output for binary classifiers based on an unexpected mapping of the probability of correct classification to the probability of occupation of a fermion in a quantum system, known as the Fermi–Dirac distribution. This mapping allows us to compute the optimal threshold to separate predicted classes and to calculate statistical parameters necessary to estimate confidence intervals of performance metrics. Using this mapping we propose an ensemble learning algorithm. In short, the Fermi–Dirac distribution provides a calibrated probabilistic output for binary classification. 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.

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