Supplementary to Learning Discriminative Bayesian Networks from High-dimensional Continuous Neuroimaging Data

Proof. As is known, a Bayesian network is equivalent to a topological ordering (Chapter 8, Section 8.1 on Page 362 in [1]). Therefore, we prove Proposition 1 by showing that i) Eqn. (1a ∼ 1d) lead to a topological ordering (the necessary condition), and ii) a topological ordering from a DAG can meet the requirements in Eqn. (1a ∼ 1d) (sufficient condition). First, we prove the necessary condition by contradiction (Fig. 1). We consider three cases for two nodes j and i. Case 1) the nodes j and i are directly connected. If there is an edge from node i to node j, the parameter Θij is then non-zero, and thus Υij must be zero. According to Eqn. (1a), we have oj > oi. If at the same time, there is an edge from node j to node i, similarly we have oi > oj, which contradicts with oj > oi, and therefore is impossible. Case 2: the nodes j and i are not directly linked but connected