Statistical
Bayes' Theorem formula: Positive Predictive Value (PPV) & false-positive paradox
Use a hypothetical fraud detector to calculate predictive values from prevalence, sensitivity and specificity.
Why base rates matter for interpreting a positive flag
Suppose fraud affects 1.0% of transactions (Prevalence A2 = 0.01), and your ML detector boasts 99.0% Sensitivity (catches 99% of true fraud, B2 = 0.99) and 99.0% Specificity (clears 99% of legitimate orders, C2 = 0.99 -> 1% False Positive Rate). Intuitively, people guess a flagged order is 99% likely to be fraud—wrong!
In an illustrative population of 10,000 orders, 100 are fraudulent. At 99% sensitivity, 99 are flagged. At 99% specificity, 99 of the 9,900 legitimate orders are also flagged. Half the flags are therefore true positives. These are expected counts under the assumed rates, not measured performance.
Updating the 50% posterior with a second positive result gives 99% only when the tests are conditionally independent given fraud status and given non-fraud status, with the same sensitivity and specificity. Repeating a correlated detector does not justify that update. With a 50% prior, the negative predictive value is also 99%, not 99.99%.
| Bayesian Screening Metric (99% Sens / 99% Spec) | Prevalence = 1% | Prevalence = 10% | 2nd Independent Confirming Test |
|---|---|---|---|
| Positive Predictive Value P(D | +) | 50.00% (1-in-2 false alarm!) | 91.67% | 99.00% (Updated prior = 50%) |
| False Discovery Rate (1 - PPV) | 50.00% | 8.33% | 1.00% |
| Negative Predictive Value P(~D | -) | 99.99% | 99.89% | 99.00% (prior = 50%) |
Bayes' Theorem Positive Predictive Value (PPV) and Negative Predictive Value (NPV)
Computes exact posterior P(Condition | Positive Flag) from Prior Prevalence (A2), Sensitivity (B2), and Specificity (C2).
=IF(OR(COUNT(A2:C2)<>3,MIN(A2:C2)<0,MAX(A2:C2)>1),"Invalid probabilities",IF(A2*B2+(1-A2)*(1-C2)=0,"Undefined conditional probability",ROUND((A2 * B2) / (A2 * B2 + (1 - A2) * (1 - C2)), 4)))Negative Predictive Value P(Clear | Negative Result)
Measures confidence that a non-flagged item is genuinely clean: TrueNegatives / (TrueNegatives + FalseNegatives).
=IF(OR(COUNT(A2:C2)<>3,MIN(A2:C2)<0,MAX(A2:C2)>1),"Invalid probabilities",IF((1-A2)*C2+A2*(1-B2)=0,"Undefined conditional probability",ROUND(((1 - A2) * C2) / ((1 - A2) * C2 + A2 * (1 - B2)), 4)))Try the example
Use the original values or change them to test your understanding.