confidence concept

Definition	Of a rule, the support of the rule's itemset divided by the support of its antecedent. It is not symmetric and it misleads when the consequent is common. Chapter 5.
Example	If 30 baskets hold bread and 24 of them also hold butter, the confidence of bread implies butter is 0.8.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `confidence`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 5.1.
Assumptions and scope	The support of a rule’s itemset over the support of its antecedent, in the unit interval and not symmetric. Under independence the confidence equals the consequent’s support whatever the antecedent, so a consequent in nine of ten transactions gives every rule into it confidence 0.9 with no association at all. Lift is confidence over the consequent’s support and is one in that case, which is why chapter 5 sends rule evaluation to chapter 8’s nulls.
Prior art	none recorded
Evidence	`lean/DataMiningAsObservation/Confidence.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The fraction of transactions with the antecedent that also hold the consequent.

Equation

Book equation 5.1.

\[s(X)=\frac{\sigma(X)}{N},\qquad c(X\to Y)=\frac{\sigma(X\cup Y)}{\sigma(X)}=\frac{s(X\cup Y)}{s(X)}.\]

Book equation 5.2.

\[\mathrm{lift}(X\to Y)=\frac{c(X\to Y)}{s(Y)}=\frac{s(X\cup Y)}{s(X)\,s(Y)}.\]

Conditions

The support of a rule’s itemset over the support of its antecedent, in the unit interval and not symmetric. Under independence the confidence equals the consequent’s support whatever the antecedent, so a consequent in nine of ten transactions gives every rule into it confidence 0.9 with no association at all.
Lift is confidence over the consequent’s support and is one in that case, which is why chapter 5 sends rule evaluation to chapter 8’s nulls.

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

Ledger

none

First stated

TSK chapter 5, as chapter 5 section 5.1 of Data Mining as Observation states it.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 5 section 5.1 to 5.3, 5.5	support, confidence, lift, the Apriori principle, FP-growth, closed and maximal itemsets, objective measures, Simpson’s paradox, cross-support	TSK 2e chapter 5

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Confidence.lean, theorems confidence_mem_unit, confidence_asymm, confidence_indep, common_consequent, lift_of_indep, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 5, 6, 8, 9, 11, 13, 14.

lift; support; itemset; multiple comparisons.

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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