Class Imbalance, Calibration, and PU Learning

Three intertwined topics that dominate fraud/abuse interviews.

1. Class imbalance

Fraud base rates are 10⁻³–10⁻⁶. Naive training on 1:10⁵ data → the model predicts “never fraud,” achieves 99.999% accuracy, and is useless. (Also why accuracy is never the metric — use PR-AUC, recall at fixed precision, or precision at a fixed review budget.)

Standard playbook:

Downsample negatives to something like 1:10 or 1:100 at training time. Keeps every rare positive, slashes compute. But this distorts probabilities (see calibration below).
Upweight positives in the loss (pos_weight in BCE) — gradient-equivalent to oversampling without duplicating data.
Focal loss when easy negatives swamp the gradient — see Loss Functions Field Guide.
Don’t bother with SMOTE-style synthetic oversampling for deep models on industrial data; mention it only to dismiss it.

Fixing probabilities after downsampling. If you kept all positives and a fraction $\beta$ of negatives, the model’s predicted odds are inflated by $1/\beta$. Closed-form correction of predicted probability $p’$ back to true probability $p$:

\[p = \frac{\beta p'}{\beta p' - p' + 1}\]

Worked example: train at 1:100 sampling ($\beta = 0.01$) on a true base rate of 0.1%. The model sees ~9% positives in training, so an “average” example gets $p’ \approx 0.09$; the formula gives $p = \frac{0.01 \cdot 0.09}{0.01\cdot0.09 - 0.09 + 1} \approx 0.00099$ — the true 0.1%. Knowing this formula cold is a strong signal in ads/fraud interviews.

2. Calibration

A model is calibrated if among all events predicted at 0.8, ~80% actually happen. Ranking metrics (AUC) are invariant to any monotone distortion of scores — a model can rank perfectly and be wildly miscalibrated. Calibration matters whenever the score is consumed as a probability:

Ads: bid = pCTR × value — 2× miscalibration = 2× misbid, real money.
Abuse enforcement: “auto-ban above P(same owner) = 0.99” is meaningless if 0.99 really means 0.7. See AM-06 The Pairwise Matcher.
Thresholded review queues, risk scores, medical triage.

Measure: reliability diagram (bin predictions, plot mean prediction vs empirical frequency) + ECE (expected calibration error). Always check calibration per slice (per country, per device type) — globally calibrated, per-slice biased is common and harmful.

Fix (post-hoc, on a held-out calibration set):

Platt scaling: fit logistic regression $\sigma(as + b)$ on the model score $s$. Two parameters; good for small data.
Isotonic regression: fit a monotone step function; more flexible, needs more data; the industry default.
Temperature scaling: divide logits by learned $T$ (one parameter); the go-to for neural nets/softmax.

Causes of miscalibration: downsampling (above), focal loss, modern over-parameterized nets (overconfident), distribution shift after deployment (recalibrate on recent data, monitor ECE in prod).

3. PU learning (Positive–Unlabeled)

In abuse problems your “negatives” are really unlabeled: an account not flagged for ban evasion may be an undetected evader; a pair of accounts not labeled same-owner may simply never have been investigated. Treating unlabeled as negative:

biases probabilities downward,
punishes the model for correctly scoring undetected positives,
and poisons hard-negative mining (the “hardest negatives” are disproportionately the undiscovered positives!).

Framing: let $c$ = label frequency = P(labeled

positive). Under the Selected Completely At Random (SCAR) assumption, a classifier trained on labeled-vs-unlabeled predicts $P(\text{labeled}

x) = c \cdot P(\text{positive}

x)$ — so dividing by an estimate of $c$ recovers the true positive probability (Elkan & Noto). Estimating $c$: average score over a held-out set of known positives.

Practical techniques:

Unbiased/non-negative PU risk estimators (uPU/nnPU) — reweight the loss using an estimate of the class prior.
Spy / two-step: plant known positives among the unlabeled; anything the first-pass model scores below the spies is treated as a reliable negative; retrain on positives vs reliable negatives.
In system design terms: route a small random sample of “negatives” to human review to estimate the contamination rate and create gold negatives; treat reversed appeals as gold labels; report metrics with the contamination caveat. Saying “our negatives are unlabeled, so I’d estimate label frequency and use a PU-aware estimator, plus buy gold labels via sampled human review” is a senior-level answer.

Related: label delay (chargebacks arrive 30–90 days late; evasion confirmations take weeks) → train on data old enough for labels to mature, or model label maturation explicitly; survivorship/feedback bias (you only get labels where the previous system looked) → keep an exploration/holdout slice, see Monitoring, Drift, and Feedback Loops.