Entailment and Basis Optimality in Confidence-Bounded Association Rules
We study the set of association rules which reach a certain confidence threshold in a given dataset. By splitting them into exact rules, that is, rules of confidence~1 characterized in a standard way by a closure operator, and partial rules, of confidence less than 1, we show how to obtain a basis of the partial rules (a nonredundant subset of these rules such that all the true partial rules can be derived from them) of absolutely minimum size with respect to a natural notion of semantic entailment. We develop this result from characterizations of the entailment property. Then, we propose and analyze an extension of this notion of entailment, identifying exactly the cases where a partial association rule is entailed jointly by two partial association rules, again in the presence of a closure operator capturing exact rules.