Prevalence Threshold and bounds in the Accuracy of Binary Classification Systems
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The accuracy of binary classification systems is defined as the proportion of correct predictions - both positive and negative - made by a classification model or computational algorithm. A value between 0 (no accuracy) and 1 (perfect accuracy), the accuracy of a classification model is dependent on several factors, notably: the classification rule or algorithm used, the intrinsic characteristics of the tool used to do the classification, and the relative frequency of the elements being classified. Several accuracy metrics exist, each with its own advantages in different classification scenarios. In this manuscript, we show that relative to a perfect accuracy of 1, the positive prevalence threshold (ϕ_e), a critical point of maximum curvature in the precision-prevalence curve, bounds the F_β score between 1 and 1.8/1.5/1.2 for β values of 0.5/1.0/2.0, respectively; the F_1 score between 1 and 1.5, and the Fowlkes-Mallows Index (FM) between 1 and √(2)≈ 1.414. We likewise describe a novel negative prevalence threshold (ϕ_n), the level of sharpest curvature for the negative predictive value-prevalence curve, such that ϕ_n > ϕ_e. The area between both these thresholds bounds the Matthews Correlation Coefficient (MCC) between √(2)/2 and √(2). Conversely, the ratio of the maximum possible accuracy to that at any point below the prevalence threshold, ϕ_e, goes to infinity with decreasing prevalence. Though applications are numerous, the ideas herein discussed may be used in computational complexity theory, artificial intelligence, and medical screening, amongst others. Where computational time is a limiting resource, attaining the prevalence threshold in binary classification systems may be sufficient to yield levels of accuracy comparable to that under maximum prevalence.
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