In order to extract event information from text, a machine reading model must learn to accurately read and interpret the ways in which that information is expressed. But it must also, as the human reader must, aggregate numerous individual value hypotheses into a single coherent global analysis, applying global constraints which reflect prior knowledge of the domain. In this work we focus on the task of extracting plane crash event information from clusters of related news articles whose labels are derived via distant supervision. Unlike previous machine reading work, we assume that while most target values will occur frequently in most clusters, they may also be missing or incorrect. We introduce a novel neural architecture to explicitly model the noisy nature of the data and to deal with these aforementioned learning issues. Our models are trained end-to-end and achieve an improvement of more than 12.1 F$_1$ over previous work, despite using far less linguistic annotation. We apply factor graph constraints to promote more coherent event analyses, with belief propagation inference formulated within the transitions of a recurrent neural network. We show this technique additionally improves maximum F$_1$ by up to 2.8 points, resulting in a relative improvement of $50\%$ over the previous state-of-the-art.