Confusion-matrix denominators

Figure 01 / 12

The evaluated population

The cohort contains 50,000 labeled payments, of which 300 have the defined synthetic outcome: Defined 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
Defined fraud outcome	300
Other labeled outcomes	49,700

Precision, recall, and false-positive rate describe different conditional probabilities. The same threshold can create very different review demand when prevalence changes.

The reference case starts with the stated population and a functioning evidence path. The owner is risk operations.

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 986 of 50,000 labeled payments. Of those flags, 240 meet the synthetic target, giving 24.34% precision. It misses 60 target events. Under the stated cost assumptions, residual loss and operating friction total $23,574. 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 240 synthetic positives and 746 negatives. It misses 60 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
Defined fraud outcome	240	60
Other outcome	746	48,954

Figure 03 / 12

Three rates with different denominators

Precision is 24.34%, recall is 80%, and the false-positive rate is 1.5%. 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	240	986	24.34%
Recall	240	300	80%
False-positive rate	746	49,700	1.5%

Figure 04 / 12

Precision changes with prevalence

This sensitivity plot holds recall at 80% and false-positive rate at 1.5%, 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%	5.07
0.5%	21.14
1%	35.01
2%	52.12
5%	73.73
10%	85.56

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	294	7,455
Band 2	282	3,976
Band 3	258	1,740
Band 4	216	596
Band 5	150	199
Band 6	75	50

Figure 06 / 12

A transparent loss-and-friction calculation

At $265 severity per missed synthetic positive, residual loss is $15,900. Review costs $3,944; lost contribution on false alarms is $3,730. 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	15,900
Review cost	3,944
False-alarm contribution	3,730

Figure 07 / 12

Observed outcomes mature over time

The final synthetic positive count is 300. 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	54
3	105
7	180
14	246
30	300

Figure 08 / 12

Review demand and available capacity

The flag count is 986. The comparison capacity is an illustrative 2,000 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	986
Available capacity	2,000
Excess demand	0

Figure 09 / 12

A feature is an observation with provenance

This evidence contract supports read the confusion matrix. 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	Confusion-matrix denominators	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	available	Evidence quality, not an outcome
label_definition	Defined 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	49,500	500
Activity history	49,000	1,000
Context signal	47,500	2,500

Figure 11 / 12

Evidence, score, and action remain separate

The policy can use read the confusion matrix 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	Confusion-matrix denominators: evidence as of the decision time
Evaluate	Rule flags 986 of 50,000 labeled payments
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	240 known synthetic positives flagged	Production labels may be delayed or wrong
Prevented loss	Assumed 63,600 USD	Requires an intervention-effect assumption
Customer impact	746 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

Report all four cells, outcome definitions, and the cost and capacity implications.

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.