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Equation 11 · Part 8 · Comparing the Main Approaches to AI and Cybersecurity

Denominator: pi × TPR + (1-pi) × FPR

P(attack∣flagged)=π⋅TPRπ⋅TPR+(1−π)⋅FPR.P(\text{attack} \mid \text{flagged}) = \frac{\pi \cdot \mathrm{TPR}}{\pi \cdot \mathrm{TPR} + (1-\pi)\cdot \mathrm{FPR}}.
π⋅TPR+(1−π)⋅FPR\pi \cdot \mathrm{TPR} + (1-\pi)\cdot \mathrm{FPR}

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

pi × TPR + (1-pi) × FPR occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

The arithmetic behind that claim generalizes directly to any detector, rule-based or AI-driven, and it is worth stating because it is the crux of why the two kinds of tooling are hard to compare on accuracy alone. If the base rate of genuine attacks among all monitored events is π\pi , and a detector has true-positive rate TPR\mathrm{TPR} and false-positive rate FPR\mathrm{FPR} , Bayes’ rule gives the probability that a flagged event is a real attack as P(attack∣flagged)=π⋅TPRπ⋅TPR+(1−π)⋅FPRP(\text{attack} \mid \text{flagged}) = \frac{\pi \cdot \mathrm{TPR}}{\pi \cdot \mathrm{TPR} + (1-\pi)\cdot \mathrm{FPR}}. When π\pi is very small, as it genuinely is in most enterprise traffic, even a detector with a low false-positive rate in isolation produces an alert stream that is mostly false alarms, because the enormous volume of benign…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.