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Given a fixed dependency graph $G$ that describes a Bayesian network of binary variables $X_1,\cdots ,X_n$, our main result is a tight bound on the mutual information $I_c(Y_1,\cdots ,Y_k)={\sum }_{j=1}^{k}H\left(Y_j\right)/c-H(Y_1,\cdots ,Y_k)$ of an observed subset $Y_1,\cdots ,Y_k$ of the variables $X_1,\cdots ,X_n$. 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.