Precision

Hook problem: trust a spam alarm or not

Your model scans 12 emails with fixed fixture subjects. You need to know whether an alarm predicted = spam is trustworthy.

Accuracy says:

That gives 58.3%, but it mixes two question types:

• Did the model get the whole mail stream mostly correct?
• Did the model’s spam alarms stay correct?

First naive idea: keep using all examples

If we stay with the global fraction above, those five predicted alarms sit in denominator along with all non-alarms.

Why this hurts

e3 and e4 are both wrong, but they do not affect a trust metric in the same way:

• e3: actual not-spam, predicted spam (FP) → hurts trust directly.
• e4: actual spam, predicted not-spam (FN) → hurts recall, not spam-alarm trust.

Core invention: read only predicted positives

For precision, set the denominator to the count of predicted positives:

This gives the fraction of alarms that were right:

Interactive visual demo: predicted-positive trace

Each step below processes only e1, e3, e5, e9, e10 in fixture order:

1,2,3,4,5 → e1, e3, e5, e9, e10

The static ledger below is still useful when scripts are unavailable.

Correctness intuition

Each processed alarm is either TP or FP, so the denominator TP + FP exactly counts all alarm predictions.

The non-alarm examples are outside the denominator:

• TN + FN outside: e2, e4, e6, e7, e8, e11, e12
• FN missed positives outside: e4, e8, e12

At this fixture end:

• TP = 3
• FP = 2
• precision = 3 / 5 = 0.6

Implementation sketch

interface Counts {
  tp: number;
  fp: number;
}

function precisionFromCounts(counts: Counts): { numerator: number; denominator: number; value: number | null } {
  const denominator = counts.tp + counts.fp;
  if (denominator === 0) {
    return { numerator: counts.tp, denominator, value: null };
  }
  return { numerator: counts.tp, denominator, value: counts.tp / denominator };
}

Complexity

If confusion-matrix counts already exist:

• O(1) to compute precision from counts.
• O(n) + O(1) extra space if you scan one fixture and count tp/fp.

Common confusions

• Precision is not accuracy. Accuracy uses all 12 emails. Precision uses only 5 alarm predictions.
• Precision is not recall. Recall asks about FN coverage, which is a different denominator.
• High precision does not mean no missed spam. Precision can stay at 3/5 while e4,e8,e12 are still missed.

Zero-denominator convention

When there are no predicted = spam outputs:

• tp = 0
• fp = 0
• denominator tp + fp = 0
• precision = null (internally)

Rendered value should be:

• English: not available
• Chinese: 不可用

Graph connection

Edges and edges only

From this node:

• confusion-matrix -> precision (uses)

Exercises

1. Why does accuracy use all 12 emails but precision use only TP + FP?
2. Compute the first two running precision values by hand from e1 then e3.
3. In a dataset with no positive predictions, what should precision render, and why?
4. Which of e3, e4, and e8 changes the precision denominator?

Graph connections : Precision