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CXRs are a crucial and extraordinarily common diagnostic tool, leading to heavy research for CAD solutions. However, both high classification accuracy and meaningful model predictions that respect and incorporate clinical taxonomies are crucial for CAD usability. To this end, we present a deep HMLC approach for CXR CAD. Different than other hierarchical systems, we show that first training the network to model conditional probability directly and then refining it with unconditional probabilities is key in boosting performance. In addition, we also formulate a numerically stable cross-entropy loss function for unconditional probabilities that provides concrete performance improvements. Finally, we demonstrate that HMLC can be an effective means to manage missing or incomplete labels. To the best of our knowledge, we are the first to apply HMLC to medical imaging CAD. We extensively evaluate our approach on detecting abnormality labels from the CXR arm of the PLCO dataset, which comprises over $198,000$ manually annotated CXRs. When using complete labels, we report a mean AUC of 0.887, the highest yet reported for this dataset. These results are supported by ancillary experiments on the PadChest dataset, where we also report significant improvements, 1.2% and 4.1% in AUC and AP, respectively over strong "flat" classifiers. Finally, we demonstrate that our HMLC approach can much better handle incompletely labelled data. These performance improvements, combined with the inherent usefulness of taxonomic predictions, indicate that our approach represents a useful step forward for CXR CAD.