Suppose I have two sets, A and B where 10|A| = |B| and A is a subset of B. For the case where |A| = 1 the Jaccard similarity of the sets will be 0.1. For the case where |A| = 100 the Jaccard similarity of the sets would still be 0.1. Is there a way to factor in "confidence" of similarity into the measure? I'm thinking to reformulate Jaccard(A, B) = (log (|A ∩ B|) + 1) / (log (A U B) + 1). Does this make sense? Are there alternatives?


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