Graphoids and natural conditional functions

Abstract

A natural conditional function (NCF) is a function from possible states of the world to natural numbers, representing the degrees of implausibility of the different states. This paper shows that the relation of conditional independence, whe defined in terms of NCFs, has a natural representation in terms of influence diagrams. First it is shown that the relation of conditional independence relative to an NCF satisfies certain axioms for conditional independence (the graphoid axioms). Then it is proved that the conditional independencies deducible from the graphoid axioms together with a set of conditional independence statements structured in a certain way (those forming a causal input list) are exactly the conditional independencies semantically implied by that input list and are also identical with the set of conditional independencies that can be read off the corresponding influence diagram using the graphical criterion of d-separation. The computational implications of these results are discussed.