We consider Bayesian optimization of objective functions whose evaluations require evaluating a series of expensive-to-evaluate functions arranged in a network so that each function takes as input the output of its parent nodes. While the standard Bayesian optimization approach observes only the objective value, our approach delivers greater sample efficiency by observing information that the standard approach ignores: intermediate output within the network. Our approach models the nodes of the network using independent Gaussian processes and chooses the points to evaluate using as its acquisition function the expected improvement computed with respect to the implied posterior on the objective function. Although this acquisition function cannot be computed in closed form, we maximize it using a sample average approximation approach. Numerical experiments show that our approach substantially outperforms standard Bayesian optimization benchmarks