Label-Error Correction (Rogan–Gladen)

By the end of this article, you should be able to: describe a labeller by its two error rates, explain why the false-positive rate dominates when the harm is rare, correct a measured rate with the Rogan–Gladen formula, and decide when the correction is worth its cost.

Two error rates

Every prevalence number rests on labels. A label might come from a trained reviewer, a classifier or an LLM, and none of them is perfect. A labeller that says "violating" or "safe" can be wrong in two ways, and each has a rate:

• Sensitivity, $r$: of the items that truly violate, the share the labeller flags. A sensitivity of 0.9 means it catches 9 in 10.
• False-positive rate, $f$: of the items that are truly safe, the share it wrongly flags. Its complement, $1 - f$, is called specificity.

The names come from medical screening, where they describe how well a test finds disease. The problem is the same one: a test that sorts people (or posts) into two groups, and makes mistakes in both directions.

"95% accurate" hides both numbers, and for counting something rare, they matter very differently.

Why rare harms are fragile

Write $\theta$ for the true prevalence: the real share of items that violate. The share the labeller flags, $\theta^{(L)}$, mixes two groups:

$\theta^{(L)} = r\,\theta + f\,(1 - \theta)$

The first term is real violations that were caught. The second is safe items wrongly flagged. And when the harm is rare, the safe items are almost everything.

Take a true prevalence of 1%, sensitivity 0.9, and a false-positive rate of 2%:

$\theta^{(L)} = 0.9 \times 0.01 + 0.02 \times 0.99 = 0.009 + 0.0198 = 2.88\%$

The measured rate is almost three times the truth. More than two thirds of the flags are false. A 2% error sounds small, but it's 2% of the 99% of content that's safe, and that's more than the real violations.

Pinterest's measurement team spells out the consequence for very rare policies.2 At a true prevalence of 0.1%, a false-positive rate of just 0.1% makes "roughly half of all measured positives false detections", an error of +90% at a sensitivity of 0.9. Missed violations only ever take away a bounded share. False positives scale with the safe majority. So for rare harms, the false-positive rate, not sensitivity, decides the error, and Pinterest tunes its labelling prompts to minimise false positives first.