threshold concept
| Definition | The value at which a score becomes a positive decision. Chapter 6. |
|---|---|
| Example | Scores 0.3, 0.6, and 0.9 at threshold 0.5 give one negative and two positives; at 0.7, two negatives and one positive. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id threshold, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 6.2. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Threshold.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
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 0.28.
\[P=\frac{TP}{TP+FP},\qquad R=\frac{TP}{TP+FN},\qquad F_1=\frac{2PR}{P+R}.\]
Conditions
- The value at which a score becomes a positive decision. Raising it can only shrink the predicted set, so true positives and false positives can only fall, which is why the ROC curve is traced by a single sweep, and below every score everything is positive while above every score nothing is.
- A strictly increasing recalibration of the score with the matching recalibration of the threshold leaves every decision unchanged, which is the Monotone Invariance Theorem at the level of one decision.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 6 section 6.1 of Data Mining as Observation, with the program’s threshold sweeps in theory-radar.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Threshold.lean,
theorems predicted_anti, tp_anti,
fp_anti, decision_comp,
predicted_extremes, 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, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.
Related
Monotone Invariance Theorem; Youden F1 bound; reliability weight; calibration.
See also
none
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.