sensitivity concept

Definition	The vector of partial derivatives of a consumer with respect to each coordinate of its input, at one point. Equation 0.8.
Example	For C(x1, x2) equal to 3x1 + 4x2 the sensitivity is (3, 4) at every row.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `sensitivity`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.8.
Assumptions and scope	The sensitivity of a consumer at a row is the vector of partial derivatives of its output with respect to each coordinate, measurable without the formula by a central finite difference at two calls per coordinate. The read operator is the workload average of the sensitivity’s outer product, so its diagonal is the expected squared sensitivity to each feature and its off-diagonal entries are co-sensitivities, not interactions. A gradient-based attribution estimates the sensitivity. Permutation importance, partial dependence, and Shapley values measure other things.
Prior art	The gradient of the consumer at a row, the quantity active-subspace methods average.
Evidence	`readscope/readscope/regimes.py:1-60`, `lean/DataMiningAsObservation/ReadOperator.lean`
Reviewed	semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page.

The gradient of the consumer at a row.

Equation

Book equation 0.8.

\[g_j\;\approx\;\frac{C(x+h\,e_j)-C(x-h\,e_j)}{2h},\qquad j=1,\dots,d.\]

Book equation 0.11.

\[P_C(x)=J(x)^{\top}G\big(C(x)\big)\,J(x),\qquad J(x)=\frac{\partial C}{\partial x}(x),\qquad \bar P_{C,\mu}=\mathbb E_{\mu}\!\left[P_C(x)\right].\]

Book equation 14.4.

\[\text{attribution}_j(x)\approx g_j(x)\,\delta_j,\qquad \overline{\text{importance}}_j\approx\big(P_C\big)_{jj}=\mathbb E\big[g_j^{2}\big].\]

Conditions

The sensitivity of a consumer at a row is the vector of partial derivatives of its output with respect to each coordinate, measurable without the formula by a central finite difference at two calls per coordinate.
The read operator is the workload average of the sensitivity’s outer product, so its diagonal is the expected squared sensitivity to each feature and its off-diagonal entries are co-sensitivities, not interactions.
A gradient-based attribution estimates the sensitivity. Permutation importance, partial dependence, and Shapley values measure other things.

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

Ledger

none

First stated

Volume 14, chapter 5, geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48, DOI 10.5281/zenodo.21776291, as the gradient of the consumer at a row.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 6 section 6.1	selection consumers have zero sensitivity almost everywhere	`readscope/readscope/regimes.py:1-60`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/ReadOperator.lean, theorems rank_one_reads_one_direction, readOp_mulVec, quad_readOp, quad_readOp_nonneg, readOp_mulVec_eq_zero_iff, readOp_diag, readOp_offdiag, readOp_symm, readOp_neg, affine_const_along_nuisance, readOp_affine, readOp_sqLength_basis, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primers L and S, chapters 0, 1, 2, 4, 6, 7, 8, 11, 14.

read operator; consumer; blind probe; read subspace.

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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