Data Mining: Association Rule Numerical Problem

 

Question: 

Consider the following two association rules: Rule1 : {bread, butter} → {milk, jam}, Rule2 : {bread, butter, milk} → {jam}. If {bread, butter} is a closed itemset, then which of the two association rules will have higher confidence and why?

Answer: The confidence of an association rule X→Y is defined as: Confidence = support(X union Y) / support (X)

Calculating Confidence:

For Rule1: $\{bread, butter\} \rightarrow \{milk, jam\}$

Confidence({bread, butter}→{milk, jam})  = Support({bread, butter, milk, jam}) / Support(bread, butter)

For Rule2:   $\{bread, butter, milk\} \rightarrow \{jam\}$

Confidence({bread, butter, milk}→{jam}) =Support({bread, butter, milk}) / Support({bread, butter, milk, jam})

Since $\{bread, butter\}$ is a closed itemset:  Support (bread, butter) > Support (bread, butter, milk)

The denominator for Rule1 is larger than the denominator for Rule2.

​So, Rule1:$\{bread,butter\}\to \{milk,jam\}$ will have higher confidence than Rule2.​

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