One cutoff, two kinds of mistake
Predict the positive class when the received value X is at least the cutoff c. Moving c left catches more positives, but also creates more false alarms.
The fraction of all cases that are truly positive, before seeing X.
Shaded tails are mistakes within each class
Blue shading to the right is a false positive (FP). Green shading to the left is a false negative (FN). Height is density; probability is area. Each full curve has area 1.
Calculate each shaded tail
Subtract the class mean μ and divide by its standard deviation σ to measure distance in standard deviations. Φ(z) is the area to the left of z under a standard normal curve.
z₀ = (c − μ₀) / σFPR = 1 − Φ(z₀)
z₁ = (c − μ₁) / σFNR = Φ(z₁)
μ₀ is the negative mean; μ₁ is the positive mean. Each class has its own reference point, even though the cutoff c is shared.
Read one actual class at a time
| Actual class (the denominator) | Predict negative | Predict positive |
|---|---|---|
| Negative: Y = 0 | True negativeTNR | False positiveFPR |
| Positive: Y = 1 | False negativeFNR | True positiveTPR |
Each row sums to 100%. For example, FPR = FP / (FP + TN); TPR = TP / (TP + FN); FNR = FN / (TP + FN). These are model probabilities, not counts from a sample.
What fraction of all cases are wrong?
Weight each mistake rate by how often its actual class occurs.
error = π₀ × FPR + π₁ × FNR
π₁ is the positive prior; π₀ = 1 − π₁ is the negative prior.
The ROC curve: every possible cutoff
ROC means receiver operating characteristic. Each point pairs a false-alarm rate (FPR) with a catch rate (TPR). The orange point uses the current cutoff. Moving toward the upper left means fewer false alarms and more catches.
Changing the prior alone does not move this ROC point: both rates condition on the actual class. Changing the cutoff does move it.
The minimum-error cutoff
With equal mistake costs and the same standard deviation σ in both classes:
c* = (μ₀ + μ₁) / 2+ [σ² / (μ₁ − μ₀)] ln(π₀ / π₁)
μ₀ and μ₁ are the negative and positive class means. σ is their common standard deviation. ln is the natural logarithm.