The end-to-end performance of natural language processing systems for compound tasks, such as question answering and textual entailment, is often hampered by use of a greedy 1-best pipeline architecture, which causes errors to propagate and compound at each stage. We present a novel architecture, which models these pipelines as Bayesian networks, with each low level task corresponding to a variable in the network, and then we perform approximate inference to find the best labeling. Our approach is extremely simple to apply but gains the benefits of sampling the entire distribution over labels at each stage in the pipeline. We apply our method to two tasks – semantic role labeling and recognizing textual entailment – and achieve useful performance gains from the superior pipeline architecture.