F1 concept

Definition	Twice the product of precision and recall over their sum. The optimal thresholded F1 of a score is the best F1 over every threshold and depends only on the score's ranking. Equation 0.28.
Example	Precision 0.6 and recall 0.75 give F1 of 2 times 0.45 over 1.35, or 0.667.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `f1`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.28.
Assumptions and scope	Twice the product of precision and recall over their sum. The optimal thresholded F1 of a score is the best F1 over every threshold in both directions, and it depends only on the score’s ranking, so a strictly monotone transform of the score leaves it unchanged. Its ceiling from the Youden index holds for every score, and the AUROC form of that ceiling only for concave ROC curves, which is the theory-radar erratum.
Prior art	none recorded
Evidence	`theory-radar/paper/astar_paper.tex:94-170`, `theory-radar/paper/theory_radar_paper.tex:290-304`, `lean/DataMiningAsObservation/MonotoneInvariance.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The harmonic mean of precision and recall at a threshold.

Equation

Book equation 0.28.

\[P=\frac{TP}{TP+FP},\qquad R=\frac{TP}{TP+FN},\qquad F_1=\frac{2PR}{P+R}.\]

Book equation 6.2.

\[\begin{gathered} \max_{\tau,\ \mathrm{dir}}F_1\!\Big(\mathbf 1\big[\mathrm{dir}\big(g(f(X)),\tau\big)\big],\,y\Big)=\max_{\tau,\ \mathrm{dir}}F_1\!\Big(\mathbf 1\big[\mathrm{dir}\big(f(X),\tau\big)\big],\,y\Big) \\ \text{for every strictly monotone } g. \end{gathered}\]

Book equation 5.4.

\[\begin{gathered} F_1^{\max}\ \le\ \sup_{t\in[J,\,1]}\ \frac{2t\pi}{t\pi+\pi+(t-J)(1-\pi)},\qquad J=\max_{\tau}\big(\mathrm{TPR}-\mathrm{FPR}\big),\qquad \pi=\text{prevalence}, \\ J\le 2A-1\ \text{when the ROC curve is concave, and not in general.} \end{gathered}\]

Conditions

Twice the product of precision and recall over their sum. The optimal thresholded F1 of a score is the best F1 over every threshold in both directions, and it depends only on the score’s ranking, so a strictly monotone transform of the score leaves it unchanged.
Its ceiling from the Youden index holds for every score, and the AUROC form of that ceiling only for concave ROC curves, which is the theory-radar erratum.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Chapter 0 section 0.14 and chapter 6 section 6.3 of Data Mining as Observation, with the program’s optimal thresholded F1 in theory-radar, DOI 10.5281/zenodo.20660206.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 5 section 5.4	Monotone Invariance and AUROC invariance theorems, the AUROC to F1 bound via the Youden index, stated in the source for every ROC curve and corrected in the book to concave curves, with the counterexample at AUROC 0.75 and F1 0.857	`theory-radar/paper/astar_paper.tex:94-170`; `theory-radar/paper/theory_radar_paper.tex:290-304`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/MonotoneInvariance.lean, theorems aurocNum_comp, auroc_comp, predicted_comp, predictedBelow_comp, sweptF1_comp, sweptF1Below_comp, optF1_comp, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer S, chapters 0, 5, 6, 7.

precision, recall; Monotone Invariance Theorem; Youden F1 bound; formula search; threshold.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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