Approval-rate denominator

Figure 01 / 12

The evaluated population

The cohort contains 32,000 card attempts, of which 384 have the defined synthetic outcome: Synthetic fraud outcome. The outcome is known by construction here. In production, label uncertainty and selection must be recorded separately.

Figure data and text version

Outcome	Count
Synthetic fraud outcome	384
Other labeled outcomes	31,616

A card dashboard mixes attempts, approved payments, and value-weighted loss. The resulting percentage appears precise while its population changes between reports.

Publish numerator, denominator, cohort, time basis, and label definition together.

All amounts, rates, capacity limits, and outcomes in this case are synthetic. The three conditions are separate assumptions for comparison. A better result in the response condition is not measured proof that the proposed control causes that improvement. The figures expose the calculation and its limits; a real deployment needs its own evidence.

Read the result

The rule flags 709 of 32,000 card attempts. Of those flags, 330 meet the synthetic target, giving 46.54% precision. It misses 54 target events. Under the stated cost assumptions, residual loss and operating friction total $9,212. The important result is the connection between the population, action, capacity, and outcome—not one isolated score.

Figure 02 / 12

Four outcomes of the rule

The rule flags 330 synthetic positives and 379 negatives. It misses 54 positives. A flagged item is a decision to intervene; it is not proof of fraud, a legal prohibition, or any other real-world conclusion.

Figure data and text version

Known outcome	Flagged	Not flagged
Synthetic fraud outcome	330	54
Other outcome	379	31,237

Figure 03 / 12

Three rates with different denominators

Precision is 46.54%, recall is 85.94%, and the false-positive rate is 1.2%. Changing the denominator changes the meaning. This record keeps each numerator attached to the population from which it came.

Figure data and text version

Metric	Numerator	Denominator	Result
Precision	330	709	46.54%
Recall	330	384	85.94%
False-positive rate	379	31,616	1.2%

Figure 04 / 12

Precision changes with prevalence

This sensitivity plot holds recall at 86% and false-positive rate at 1.2%, then changes prevalence. It is an algebraic comparison, not a forecast. Even unchanged detection quality can produce a very different review queue when the base rate changes. Horizontal positions are the labeled observations or scenarios; equal spacing does not imply equal numerical increments.

Figure data and text version

Assumed prevalence	Precision %
0.1%	6.69
0.5%	26.48
1%	41.99
2%	59.39
5%	79.04
10%	88.84

Figure 05 / 12

The threshold trade-off

Six illustrative score bands use a stated pair of detection rates. Lower sensitivity can reduce false alarms but miss more target events. These points do not come from a trained model and do not establish the best operating threshold. Horizontal positions are the labeled observations or scenarios; equal spacing does not imply equal numerical increments.

Figure data and text version

Score band	True positives	False positives
Band 1	376	4,742
Band 2	361	2,529
Band 3	330	1,107
Band 4	276	379
Band 5	192	126
Band 6	96	32

Figure 06 / 12

A transparent loss-and-friction calculation

At $90 severity per missed synthetic positive, residual loss is $4,860. Review costs $2,836; lost contribution on false alarms is $1,516. The calculation assumes intervention prevents all flagged-positive loss and each false alarm loses the stated contribution. Relax those assumptions before applying it to a real policy.

Figure data and text version

Cost component	USD
Missed-positive loss	4,860
Review cost	2,836
False-alarm contribution	1,516

Figure 07 / 12

Observed outcomes mature over time

The final synthetic positive count is 384. Earlier observations reveal only a stated fraction. Comparing a day-1 cohort with a day-30 cohort would confuse label age with control quality. This curve models observation delay only; it does not change the final outcome. Horizontal positions are the labeled observations or scenarios; equal spacing does not imply equal numerical increments.

Figure data and text version

Days after event	Observed positives
1	69
3	134
7	230
14	315
30	384

Figure 08 / 12

Review demand and available capacity

The flag count is 709. The comparison capacity is an illustrative 1,280 reviews per cohort window. A mathematical rule can be coherent while its resulting workload exceeds the operating team’s capacity. Capacity is not permission to ignore an applicable mandatory control.

Figure data and text version

Queue measure	Items
Flagged for review	709
Available capacity	1,280
Excess demand	0

Figure 09 / 12

A feature is an observation with provenance

This evidence contract supports keep denominators honest. A value needs its event time, arrival time, scope, and source. Keeping unavailable evidence distinct from a measured zero prevents an outage from becoming a falsely reassuring feature.

Figure data and text version

Field	Example	Meaning
entity_ref	Approval-rate denominator	Subject of this case
event_time	2026-09-18T09:00:00Z	When the event occurred
received_time	2026-09-18T09:00:02Z	When the system learned it
signal_status	repaired	Evidence quality, not an outcome
label_definition	Synthetic fraud outcome	The target used in these calculations

Figure 10 / 12

Missing evidence changes the observed population

The cells show an explicitly constructed completeness profile for three signal groups. The stress condition removes more history and device evidence. Missingness does not prove the target outcome; it changes what the decision process knows.

Figure data and text version

Signal group	Available	Missing
Identity evidence	31,840	160
Activity history	31,520	480
Context signal	31,040	960

Figure 11 / 12

Evidence, score, and action remain separate

The policy can use keep denominators honest only within its approved scope. The action record must retain which evidence was available, which model or rule ran, and which action was actually applied. The final action can differ from the score recommendation when a separate constraint applies.

Figure data and text version

Stage	Record
Observe	Approval-rate denominator: evidence as of the decision time
Evaluate	Rule flags 709 of 32,000 card attempts
Apply	Record action, reason, owner, and expiry
Reconcile	Join the action to later outcomes without overwriting history

Figure 12 / 12

What the result cannot establish

Observed classifications do not reveal every counterfactual. The synthetic labels make arithmetic possible, but production decline data is selected by prior policy. Keep measured outcomes, assumed prevention, and unknown alternatives separate when reporting impact.

Figure data and text version

Claim	Evidence in this case	Limit
Detected target	330 known synthetic positives flagged	Production labels may be delayed or wrong
Prevented loss	Assumed 29,700 USD	Requires an intervention-effect assumption
Customer impact	379 synthetic negatives flagged	Not every flag causes abandonment
Unobserved alternative	Outcome without the action	Needs a valid evaluation design

Connect the result to the system

Publish numerator, denominator, cohort, time basis, and label definition together.

Check the population, evidence, permitted action, and actual effect together. A balanced calculation can still use the wrong population; a successful response can still leave an unknown financial outcome. The case’s numerical result applies only to its stated assumptions.