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In this work, we aim to calibrate the score outputs of an estimator for thebinary classification problem by finding an 'optimal' mapping to classprobabilities, where the 'optimal' mapping is in the sense that minimizes theclassification error (or equivalently, maximizes the accuracy). We show thatfor the given target variables and the score outputs of an estimator, an'optimal' soft mapping, which monotonically maps the score values toprobabilities, is a hard mapping that maps the score values to $0$ and $1$. Weshow that for class weighted (where the accuracy for one class is moreimportant) and sample weighted (where the samples' accurate classifications arenot equally important) errors, or even general linear losses; this hard mappingcharacteristic is preserved. We propose a sequential recursive merger approach,which produces an 'optimal' hard mapping (for the observed samples so far)sequentially with each incoming new sample. Our approach has a logarithmic insample size time complexity, which is optimally efficient.

Optimally Efficient Sequential Calibration of Binary Classifiers to Minimize Classification Error