What do support, confidence and lift measure for a basket association rule?

## The object being scored Association rule mining runs over a transaction log: each row is a **basket**, an unordered set of items bought in one trip. A rule is written `{A} -> {B}` and read as *baskets containing A tend to also contain B*. The left-hand side is the **antecedent**, the right-hand side the **consequent**; either side may hold several items. Nothing about a rule is causal and nothing is ordered in time - it is a statement about items landing in the same set. Three numbers score every rule, and all three come from counting baskets in that one log. ## Support `support(X) = baskets containing every item of X / total baskets` For the rule `{A} -> {B}`, the rule's support is `support(A and B)`: the share of baskets containing the entire itemset. Support answers *how often does this situation even arise*. A rule at 0.00001 support can be perfectly reliable and still not worth a line of code, because it fires a handful of times a year. Support is **symmetric**: `{A} -> {B}` and `{B} -> {A}` have identical support, because set membership has no direction. It is also the quantity every mining algorithm prunes on, which is why the minimum-support threshold is the single knob that decides how long a run takes. ## Confidence `confidence(A -> B) = support(A and B) / support(A)` Of the baskets that contain A, what fraction also contain B. This is the rule's reliability *given that the antecedent fired*, and it is what a trigger-based cross-sell cares about: the shopper has A in the cart, how often is B there too. Dividing by `support(A)` rather than `support(B)` is what makes confidence **directional**. A rule from a rare item to a common one scores high confidence; reverse it and the number collapses. ## Lift `lift(A -> B) = confidence(A -> B) / support(B)` B appears in `support(B)` of all baskets. Among the A baskets it appears in `confidence` of them. Lift is the ratio of those two rates: how much more often B turns up when A is present than it turns up in the population at large. - **lift = 1** - the A baskets look exactly like every other basket with respect to B; the rule adds nothing. - **lift > 1** - positive association; the items co-occur more than their individual frequencies alone would produce. - **lift < 1** - negative association; seeing A goes with *not* seeing B, which is what substitutes look like. Lift, like support, is symmetric: swapping the sides leaves it unchanged. ## A worked example 10,000 baskets. Coffee in 1,200. Filters in 400. Both in 300. ``` support = 300 / 10000 = 0.03 confidence = 300 / 1200 = 0.25 support(filters)= 400 / 10000 = 0.04 lift = 0.25 / 0.04 = 6.25 ``` Read it out loud: the rule fires on 3% of trips; when coffee is in the basket, filters are there a quarter of the time; and a quarter is more than six times the 4% rate at which filters appear generally. Now reverse it: `confidence({filters} -> {coffee}) = 300/400 = 0.75`, three times the other direction, while support and lift do not move at all. That asymmetry is the most useful single fact about confidence. ## Why you need all three Each metric answers a different question and each is useless alone. - Support alone finds the obvious: the most frequent itemsets in a supermarket are the things everyone buys, and a rule between two staples is common without being informative. - Confidence alone is inflated by a popular consequent. If an item is in most baskets, almost any antecedent reaches high confidence for it. - Lift alone rewards rarity: two obscure items that happened to land in the same three baskets can score a spectacular ratio off almost no data. So mining is usually run as *filter by minimum support, then rank by lift, then sanity-check the absolute basket count behind each rule*. ## What counts as a transaction Before any of these numbers mean anything, someone decided what a row is: one checkout, one online session, one customer-month. Widen the window and almost every pair of items co-occurs somewhere, so support and confidence inflate while the rule stops describing a single shopping decision. The definition of a transaction moves every metric on this page and it is a modelling choice, not a data-engineering detail. ## Common mistakes Reporting support as a raw count rather than a fraction makes thresholds meaningless across datasets of different sizes. Treating confidence as symmetric produces recommendations pointed the wrong way. And treating lift near 1 as *nearly strong* inverts the scale: 1 is the neutral point, not the floor.