An Algorithm for Reading Dependencies from the Minimal Undirected Independence Map of a Graphoid that Satisfies Weak Transitivity

Abstract

We present a sound and complete graphical criterion for reading dependencies from the minimal undirected independence map G of a graphoid M that satisfies weak transitivity. Here, complete means that it is able to read all the dependencies in M that can be derived by applying the graphoid properties and weak transitivity to the dependencies used in the construction of G and the independencies obtained from G by vertex separation. We argue that assuming weak transitivity is not too restrictive. As an intermediate step in the derivation of the graphical criterion, we prove that for any undirected graph G there exists a strictly positive discrete probability distribution with the prescribed sample spaces that is faithful to G . We also report an algorithm that implements the graphical criterion and whose running time is considered to be at most O ( n 2 ( e + n )) for n nodes and e edges. Finally, we illustrate how the graphical criterion can be used within bioinformatics to identify biologically meaningful gene dependencies.