Association Rules, and What Confidence Leaves Out

All twenty directed rules between pairs of items in the unit's ten transactions, ordered by confidence, which is how a mining tool would present them by default.

(The four rules pairing jam with eggs or butter have support $0$ and are omitted.)

The top four rules have perfect confidence and rest on two transactions and one transaction respectively. The most trustworthy rules in the table, bread and milk, on seven transactions, sit fifth and sixth. Sorting by confidence puts the flimsiest evidence at the top.

Five rules pass a $0.5$ confidence threshold with lift below one. Every rule involving eggs with bread or milk has lift $0.833333$, and three of them clear a conventional confidence bar. A pipeline filtering on confidence alone would surface all three as findings, each describing an association that runs the opposite way.

The symmetry is visible. Each lift value appears in both directions of its pair: $\text{bread}\to \text{eggs}$ and $\text{eggs}\to \text{bread}$ both show $0.833333$ while their confidences are $0.5000$ and $0.6667$. Milk and bread both show $1.093750$ and, because the two items have identical support, identical confidences of $0.8750$. A coincidence of this data, not a rule.

Only two distinct lifts appear, $1.250000$ and $0.833333$, plus $1.093750$ for the bread–milk pair. That is an artefact of a tiny dataset with few distinct support values, and it should be read as such rather than as structure.

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Confidence ranks rules by how common the consequent is, mostly. Lift separates them into those occurring more often than chance and those occurring less. Support says how much data stands behind either verdict. Three columns, three different questions, and a tool presenting only the second column, sorted, would mislead on all of them.

Two rules with comparable confidence and opposite verdicts.

Both confidences are respectable. The column that decides is the third, and neither confidence contains it. This is the single comparison worth taking from the unit: a confidence is only interpretable beside the consequent's base rate.

A rule against its converse.

$\text{bread}\to \text{eggs}$ has confidence $0.5000$; $\text{eggs}\to \text{bread}$ has $0.6667$. Same four transactions, same co-occurrence, different denominators, $0.8000$ against $0.6000$. The lift is $0.833333$ for both.

So the question "are bread and eggs associated" has one answer, and the question "given bread, expect eggs" has a different answer from "given eggs, expect bread". Choosing the measure means choosing which question is being asked.

Perfect confidence on two transactions against modest confidence on seven.

$\text{butter}\to \text{milk}$ scores confidence $1.0000$ and lift $1.250000$, resting on two transactions. $\text{bread}\to \text{milk}$ scores $0.8750$ and $1.093750$, resting on seven. The first ranks higher on both measures and is far weaker evidence: one butter purchase without milk would take its confidence to $0.6667$, while one bread purchase without milk barely moves the second.

Support is the column that separates them, which is why it is reported rather than used only as a filter.

A pruned branch against an examined one.

Butter has support $0.2000$, below the $0.3$ threshold, so every itemset containing butter is eliminated by that one measurement, provably, not probably. Bread has support $0.8000$, so every pair containing bread is a live candidate and must be counted.

One measurement settles a whole branch; another opens one. The asymmetry is what makes the search affordable, and it rests on support never rising as a set grows.

Exhaustive search against a selective one.

Apriori returns every rule above the thresholds, all of them, guaranteed complete. That completeness is a strength for the search and a burden afterwards: on real data it produces thousands of true statements, and nothing in the procedure ranks them by interest. A method returning fewer, better rules would be making judgements the algorithm has no basis for making.

Lift above one against a cause.

A lift of $1.093750$ says two items co-occur about nine percent more often than independence predicts, in this transaction set. It does not say that buying one leads to buying the other, that a shared cause drives both, or that the pattern will hold next month. Those are separate claims, and every one of them needs evidence the transaction counts do not contain.

That a rule with high confidence has found an association, so a threshold on confidence is enough to filter mined rules. Confidence is the proportion of transactions containing the antecedent that also contain the consequent, and it takes no account of how common the consequent already is. On the ten-transaction set of this unit, $\text{bread}\to \text{eggs}$ has confidence $1/2=0.5000$, which passes most conventional thresholds, yet eggs appear in $3/5=0.6000$ of all transactions. Among transactions containing bread, eggs appear less often than among transactions generally, and the lift makes that precise: $0.5000/0.6000=5/6=0.833333$, below one. The rule is real as a frequency and describes a negative association, so acting on it, placing eggs beside bread to encourage the pairing, would be acting against the evidence. A confidence threshold cannot detect this, because the quantity it thresholds does not contain the consequent's base rate. Comparing against that base rate is what lift does, and it is why lift rather than confidence decides whether a rule says anything.

That $A\to B$ and $B\to A$ are the same rule reported two ways, so establishing one establishes the other. Confidence is directional: it divides the support of the pair by the support of the antecedent alone, and the two antecedents differ. On the transaction set of this unit, $\text{bread}\to \text{eggs}$ has confidence $0.4000/0.8000=0.5000$ while $\text{eggs}\to \text{bread}$ has confidence $0.4000/0.6000=0.6667$. The same co-occurrence, two different numbers, because bread is the more common item. Lift, by contrast, divides the same pair support by the product of both individual supports, so it is symmetric: both directions give $0.833333$. This is why the two measures answer different questions. Confidence answers "given this antecedent, how often does the consequent follow", which is what a recommendation or a shelf placement needs and which depends on which item is the trigger. Lift answers "do these items occur together more or less often than independence predicts", which is a property of the pair and carries no direction at all. Reporting a directional confidence as though it described the pair, or a symmetric lift as though it justified one direction of intervention, mismatches the measure to the claim.