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We propose a stochastic variational inference algorithm for training large-scale Bayesian networks, where noisy-OR conditional distributions are used to capture higher-order relationships. One application is to the learning of hierarchical topic models for text data. While previous work has focused on two-layer networks popular in applications like medical diagnosis, we develop scalable algorithms for deep networks that capture a multi-level hierarchy of interactions. Our key innovation is a family of constrained variational bounds that only explicitly optimize posterior probabilities for the sub-graph of topics most related to the sparse observations in a given document. These constrained bounds have comparable accuracy but dramatically reduced computational cost. Using stochastic gradient updates based on our variational bounds, we learn noisy-OR Bayesian networks orders of magnitude faster than was possible with prior Monte Carlo learning algorithms, and provide a new tool for understanding large-scale binary data.