Generating Graphoids from Generalised Conditional Probability

TL;DR

This paper explores the conditions under which various uncertainty measures, including probability, produce independence structures that satisfy graphoid properties, providing insights into qualitative conditional probability theories.

Contribution

It introduces sufficient conditions for the independence structures of uncertainty measures to satisfy graphoid properties, including the Intersection property, even with strong logical relationships.

Findings

Conditions for graphoid properties in uncertainty measures

Sufficient criteria for the Intersection property

Framework for qualitative conditional probability theories

Abstract

We take a general approach to uncertainty on product spaces, and give sufficient conditions for the independence structures of uncertainty measures to satisfy graphoid properties. Since these conditions are arguably more intuitive than some of the graphoid properties, they can be viewed as explanations why probability and certain other formalisms generate graphoids. The conditions include a sufficient condition for the Intersection property which can still apply even if there is a strong logical relations hip between the variables. We indicate how these results can be used to produce theories of qualitative conditional probability which are semi-graphoids and graphoids.