Entailment in Probability of Thresholded Generalizations

TL;DR

This paper introduces a nonmonotonic logic based on thresholded generalizations, defining probabilistic entailment that ensures conclusions are trustworthy given true premises, and relates it to existing systems like System-Z^+.

Contribution

It formalizes a new nonmonotonic probabilistic entailment framework and connects it to established reasoning systems, ensuring probabilistic trustworthiness of conclusions.

Findings

Entailment in probability is closely related to System-Z^+.

The framework guarantees probabilistic trustworthiness of conclusions.

A procedure for checking entailment is provided.

Abstract

A nonmonotonic logic of thresholded generalizations is presented. Given propositions A and B from a language L and a positive integer k, the thresholded generalization A=>B{k} means that the conditional probability P(B|A) falls short of one by no more than c*d^k. A two-level probability structure is defined. At the lower level, a model is defined to be a probability function on L. At the upper level, there is a probability distribution over models. A definition is given of what it means for a collection of thresholded generalizations to entail another thresholded generalization. This nonmonotonic entailment relation, called "entailment in probability", has the feature that its conclusions are "probabilistically trustworthy" meaning that, given true premises, it is improbable that an entailed conclusion would be false. A procedure is presented for ascertaining whether any given collection of premises entails any given conclusion. It is shown that entailment in probability is closely related to Goldszmidt and Pearl's System-Z^+, thereby demonstrating that the conclusions of System-Z^+ are probabilistically trustworthy.