Given a fixed dependency graph G that describes a Bayesian network of binary variables X1,…,Xn, our main result is a tight bound on the mutual information Ic(Y1,…,Yk)=∑kj=1H(Yj)/c−H(Y1,…,Yk) of an observed subset Y1,…,Yk of the variables X1,…,Xn. Our bound depends on certain quantities that can be computed from the connective structure of the nodes in G. Thus it allows to discriminate between different dependency graphs for a probability distribution, as we show from numerical experiments.