Generalizing Probabilistic Material Implication and Bayesian Conditionals


Journal: International Journal of Approximate Reasoning
Pages: forthcoming

Michael Jahn and Matthias Scheutz

Conditional statements in natural language of the form "if A then B" have multiple interpretations that require different logical treatment. In this paper, we focus on probabilistic "if A then B" rules that are given either a Bayesian interpretation via conditional probabilities P(B|A), or couched as probabilistic material implication. While some have argued that Bayesian conditionals are the correct way to think about such rules, there are challenges with standard in- ferences such as modus ponens and modus tollens that might make probabilistic material implication a better candidate at times for rule-based systems employ- ing forward-chaining; and arguably material implication is still suitable when information about prior or conditional probabilities is not available at all. We investigate a generalization of probabilistic material implication and Bayesian conditionals that combines the advantages of both formalisms in a systematic way and prove basic properties of the generalized implication, in particular, various bounds as well as properties of inference chains in graphs. And most importantly we provide a novel and natural interpretation of the generalized implication’s main parameter.

  title={Generalizing Probabilistic Material Implication and Bayesian Conditionals},
  author={Michael Jahn and Matthias Scheutz},
  journal={International Journal of Approximate Reasoning},