Foundations — Precision-recall tradeoff and curves

Before you can read the parent note Precision-Recall Tradeoff and Curves, you need to earn every symbol it throws at you. This page builds each one from nothing: plain words → a picture → why the topic cannot live without it. Read top to bottom; each brick rests on the last.

1. The raw ingredients: a prediction and a truth

Every idea here starts from a single act: the model looks at one item and guesses a label, while reality holds the true label. We need names for both.

Why the topic needs this: precision and recall are comparisons between $\hat{y}$ and $y$. With only one of them there is nothing to measure.

2. The score $s(x)$ — why a number, not a yes/no

Real classifiers rarely commit straight to $\hat{y}$. They first emit a number measuring how positive the item looks.

Look at the figure: every item is a dot placed on a horizontal number line from $0$ to $1$ by its score. Fraud dots (pink) tend to sit right, normal dots (blue) tend to sit left — but they overlap in the middle. That overlap is the reason no single cutoff can be perfect, and it is the seed of the entire tradeoff.

3. The threshold $\tau$ — turning a score into a decision

Now we turn the dial into a verdict. We pick a line on the number line; everything to its right is called positive.

See 3.2.04-Decision-thresholds-and-calibration for how $\tau$ is chosen in practice.

4. The four buckets: TP, FP, FN, TN

Once the bar is placed, every dot falls into exactly one of four boxes, depending on what it truly is ($y$) versus what we said ($\hat{y}$). These four counts are the atoms of every metric on the page.

The figure is the confusion matrix — a 2×2 grid, rows = truth, columns = prediction. Each dot from figure s02 lands in one cell. This grid is built fully in 2.6.01-Confusion-matrix-and-basic-metrics; here we only need its four numbers.

Why the topic needs these: precision and recall are just ratios of these four buckets. Nothing else. Learn the buckets and the formulas become obvious.

5. Precision and Recall — two ratios of the buckets

Now we can define the two stars of the topic. Both are fractions between $0$ and $1$; both use TP on top; they differ only in what they divide by.

We use the symbols $r$ for recall and $p$ for precision when drawing the curve — just shorthand for the two fractions above. The parent plots points $(r,p)$: recall across the bottom, precision up the side.

6. The curve and its summary number

Sweeping $\tau$ from high to low gives many $(r,p)$ pairs; joining them is the precision-recall curve. To compress a whole curve into one number we measure the area under it.

Related tools you'll meet next: 2.6.07-ROC-curves-and-AUC (a different curve using TN), 2.6.09-F1-score-and-Fbeta-score (blending $p$ and $r$ into one score), and 4.1.08-Cost-sensitive-learning (choosing $\tau$ by real-world costs).

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