logistic regression instrument

Definition	A linear classifier whose score is the sigmoid of a weighted sum, strictly between zero and one, above one half exactly on one side of the hyperplane. Chapter 6. Also logistic, linear classifier, sigmoid.
Example	A weighted sum of 0 gives score 0.5, of 2 gives 0.881, and of −2 gives 0.119.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `logistic-regression`, kind instrument.
Status	measures [predicted]. Corrections: none recorded.
Defining equation	Book equation 6.1.
Assumptions and scope	A linear classifier whose score is the sigmoid of a weighted sum. The sigmoid lies strictly between zero and one, is one half at zero, is symmetric about one half, and is strictly increasing, so the decision at threshold one half is the sign of the weighted sum and the decision boundary is the hyperplane. Its score moves only along its weight vector, so its read subspace is one direction and its nuisance is everything orthogonal, and its read operator is the outer product of the weight vector with itself scaled by how steep the sigmoid is at each row.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:122`, `geometric-observation/claims/LEDGER.md`, `lean/DataMiningAsObservation/Logistic.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The sigmoid of a weighted sum, above one half on one side of the hyperplane.

Equation

Book equation 6.1.

\[s(x)=w\cdot x+b,\qquad P_C=\mathbb E\big[\sigma'(s)^{2}\big]\,w\,w^{\top},\qquad \operatorname{rank}P_C=1.\]

Conditions

A linear classifier whose score is the sigmoid of a weighted sum. The sigmoid lies strictly between zero and one, is one half at zero, is symmetric about one half, and is strictly increasing, so the decision at threshold one half is the sign of the weighted sum and the decision boundary is the hyperplane.
Its score moves only along its weight vector, so its read subspace is one direction and its nuisance is everything orthogonal, and its read operator is the outer product of the weight vector with itself scaled by how steep the sigmoid is at each row.

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

Ledger

measures. GO-B-optim-D4 (034 · D4) [predicted]. Optimization — gradient compression, curvature (Hessian) read operator, on a REAL model (logistic regression); optional stretch geometric-observation/claims/LEDGER.md:122.

First stated

Cox, the regression analysis of binary sequences, 1958, as chapter 6 section 6.1 of Data Mining as Observation reads it, with the real logistic model’s Hessian in geometric-observation/claims/LEDGER.md row GO-B-optim-D4.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 7 section 7.2	real logistic model, exact Hessian, anti 300 of 300, flip 82 of 300, coupling diagnosis, bound not refutation	`geometric-observation/claims/LEDGER.md` row GO-B-optim-D4

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Logistic.lean, theorems sigmoid_pos, sigmoid_lt_one, sigmoid_zero, sigmoid_neg, sigmoid_strictMono, decision_iff, decision_linear, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 0, 6, 7, 11, 12.

classifier; decision boundary; threshold; read subspace; Hessian.

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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