Cost-sensitive threshold

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Problem

A calibrated classifier outputs $p=P(y=1\mid x)$ . A false positive costs $1$ , a false negative costs $4$ , and correct decisions cost $0$ . Derive the threshold for predicting $1$ . With ties assigned to class $1$ , classify $p=(0.1,0.2,0.7)$ . If the labels are $(0,1,0)$ , compute mean cross-entropy.

Reveal answer or reference solution

Predicting $1$ costs $1-p$ , while predicting $0$ costs $4p$ . Thus, predict $1$ when

1-p\le 4p,

or $p\ge\frac{1}{5}$ . The predictions are $(0,1,1)$ . The mean cross-entropy is

-\frac{\ln(0.9)+\ln(0.2)+\ln(0.3)}{3}\approx 0.9729237441.

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Cost-sensitive threshold

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