Confirmation is a useful concept for assessing the impact of the premise on the conclusion of a rule induced from data. Interpretation of probabilistic relationships between premise and conclusion of a rule led to four mathematical formulations of confirmation, called perspectives. The logical equivalence of these perspectives and the resulting general definition of confirmation underline the known qualitative aspect of the concept of confirmation. The quantitative aspect of confirmation is handled by definitions of particular confirmation measures. In this paper, we relate the qualitative and quantitative aspects by introducing a property of monotonicity of measures with respect to left- and right-hand side probabilities defining the perspectives. This new property permits consideration of confirmation measures in association with particular perspectives. We also identify several other properties that valuable confirmation measures should possess. A particular care is devoted to discussion of behavior of confirmation measures monotonic in different perspectives with respect to symmetry properties, taking also into account two new perspectives of Bayesian confirmation. We also prove that confirmation measures monotonic in the six perspectives are exhaustive in the sense that their set is closed under transformations related to symmetry properties. Finally, we verify which confirmation measures enjoy these properties.
- Rule interestingness measures
- Bayesian confirmation
- Evidential support
- Properties of measures