Generating Graphoids from Generalised Conditional Probability
This work addresses foundational issues in uncertainty reasoning for AI and statistics, offering incremental theoretical insights into independence structures.
The paper tackles the problem of understanding when uncertainty measures on product spaces satisfy graphoid properties, providing more intuitive sufficient conditions that explain why probability and other formalisms generate graphoids, including a condition for the Intersection property that works even with strong logical relationships between variables.
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.