Why can a basket rule with 40% support and 67% confidence still be worthless?

## The blind spot in confidence `confidence(A -> B) = support(A and B) / support(A)`. Look at what is missing: `support(B)` appears nowhere. Confidence measures how reliably B follows A without ever asking how easy it is to find B in the first place. When the consequent is common, confidence is high for almost any antecedent you attach to it, and a rule list ranked by confidence turns into a list of the store's most popular items with random things bolted onto the left. ## The bread-and-milk case Weekly supermarket checkout data. Bread is in 60% of baskets, milk in 70%, and both together in 40%. ``` support({bread, milk}) = 0.40 confidence(bread -> milk)= 0.40 / 0.60 = 0.67 lift = 0.67 / 0.70 = 0.95 leverage = 0.40 - 0.60 * 0.70 = -0.02 ``` Support of 0.40 is enormous - the rule covers four baskets in ten. Confidence of 0.67 says two thirds of bread buyers take milk. Both look like a headline. But milk is in 70% of baskets full stop, so *knowing about the bread makes you slightly worse at predicting milk*. Lift 0.95 and negative leverage say the same thing twice. The pairing is frequent because both items are staples, not because the items have anything to do with each other. ## Leverage, and why it is not just lift again `leverage(A -> B) = support(A and B) - support(A) * support(B)` Lift is a **ratio**, leverage a **difference**, and the difference is measured in baskets. That distinction decides which rules survive contact with a business. - A staple pairing can have huge support and leverage near zero, as above: enormous volume, no excess. - A rare pairing can have spectacular lift and leverage near zero too, because the excess is a ratio over almost nothing. Leverage 0.0001 on ten million baskets is a thousand baskets; on five thousand baskets it is half of one. So lift ranks rules by strength of association and leverage ranks them by how much co-occurrence actually exists beyond chance. Reading both keeps you from shipping either a trivial staple rule or a lottery ticket. **Conviction** is a third correction sometimes seen: `(1 - support(B)) / (1 - confidence(A -> B))`. It also equals 1 under independence and grows without bound as the rule approaches being exception-free, which makes it sensitive to the rare counter-examples that lift ignores. ## Ubiquitous items Every real transaction log has items that are in most baskets and mean nothing: the plastic carrier bag, the loyalty-card line, the deposit on a returnable bottle, bananas in some chains. A bag at 60% frequency lands on the right of a large fraction of all high-confidence rules, because 60% is the floor for any antecedent. It also lands on the *left* of rules with near-total confidence for other staples. The practical handling: 1. Blacklist non-merchandise lines (bags, fees, deposits) before the mining run, not after. 2. Apply a maximum-support cut alongside the minimum, so items present in almost every basket are excluded from candidate generation. 3. Mine the remaining catalogue, then rank by lift with a leverage floor. That sequence typically removes more junk than any amount of post-hoc rule filtering, and it makes the run faster, because ubiquitous items are exactly the ones that generate the most frequent itemsets. ## Why frequent does not mean informative The underlying point generalises past shopping. Support and confidence are unconditional and conditional *frequencies*; neither is a comparison against a baseline. Any metric that never looks at the base rate of the thing being predicted will reward predicting whatever is common. That is the same shape as an accuracy score on an imbalanced problem: high, correct, and uninformative. Lift and leverage restore the baseline, one as a ratio, one as an absolute excess. ## Interpreting lift 0.95 Slightly below 1 is not 'almost strong' - it is on the wrong side of neutral. On a very large basket count it hints at mild substitution or at two items competing for the same trip, which can be worth a look as a cannibalisation question. On modest counts it sits inside the noise. Either way it is not a cross-sell, and presenting it as one because the support and confidence look impressive is the mistake this whole question is about.