A Logical Approach to Context-Specific Independence

Bayesian networks constitute a qualitative representation for conditional independence (CI) properties of a probability distribution. It is known that every CI statement implied by the topology of a Bayesian network G is witnessed over G under a graph-theoretic criterion called d-separation. Alternatively, all such implied CI statements have been shown to be derivable using the so-called semi-graphoid axioms. In this article we consider Labeled Directed Acyclic Graphs (LDAG) the purpose of which is to graphically model situations exhibiting context-specific independence (CSI). We define an analogue of dependence logic suitable to express context-specific independence and study its basic properties. We also consider the problem of finding inference rules for deriving non-local CSI and CI statements that logically follow from the structure of a LDAG but are not explicitly encoded by it.