Hessian concept

Definition	The matrix of second derivatives of a score, whose off-diagonal entries are the interactions. Naive Bayes has a diagonal one. Chapter 6. Also cross-partial.
Example	For the score 3x1 + 2x2 + x1 x2 the Hessian has 1 off the diagonal; for naive Bayes it has 0.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `hessian`, kind concept.
Status	measures [predicted]. Corrections: none recorded.
Defining equation	none
Assumptions and scope	The matrix of second derivatives of a score, whose off-diagonal entries are the interactions. The second difference recovers a quadratic’s curvature exactly, and a score that is additive across features, naive Bayes, has a diagonal Hessian because one feature’s contribution is the same whatever the others are. The read operator is not the Hessian. Two features’ sensitivities can rise and fall together across rows even when neither changes the other’s effect, so a diagonal Hessian does not give a diagonal read operator.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:122`, `geometric-observation/claims/LEDGER.md`, `lean/DataMiningAsObservation/Curvature.lean`, `lean/DataMiningAsObservation/NaiveBayes.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

Second derivatives, with the interactions off the diagonal.

Equation

none

Conditions

The matrix of second derivatives of a score, whose off-diagonal entries are the interactions. The second difference recovers a quadratic’s curvature exactly, and a score that is additive across features, naive Bayes, has a diagonal Hessian because one feature’s contribution is the same whatever the others are.
The read operator is not the Hessian. Two features’ sensitivities can rise and fall together across rows even when neither changes the other’s effect, so a diagonal Hessian does not give a diagonal read operator.

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

Chapter 6 section 6.1 and chapter 7 section 7.2 of Data Mining as Observation, with the exact Hessian of a real logistic model 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/Curvature.lean, theorems second_quad, second_affine, linear_error, gradient_change, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

lean/DataMiningAsObservation/NaiveBayes.lean, theorems log_likelihood_sum, log_odds_sum, contribution_independent, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 6, 7, 14.

curvature; naive Bayes; boosting; read operator; Jacobian.

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.

← headhierarchical clustering →