Relative Entailment Among Probabilistic Implications

TL;DR

This paper explores relative entailment in probabilistic implications, providing a comprehensive characterization of when a set of partial implications entails another, using linear programming duality to improve decision algorithms and identify intrinsic confidence thresholds.

Contribution

It extends the understanding of entailment among probabilistic implications to arbitrary premises, offering new decision algorithms and insights into confidence thresholds.

Findings

Characterizes entailment from arbitrary premises using linear programming duality.

Provides decision algorithms with improved complexity.

Identifies intrinsic confidence thresholds for entailment cases.

Abstract

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion.