Title: Putting Machine Translation in Context with the Noisy Channel Model. Abstract: We show that Bayes' rule provides a compelling mechanism for controlling unconditional document language models, using the long-standing challenge of effectively leveraging document context in machine translation. In our formulation, we estimate the probability of a candidate translation as the product of the unconditional probability of the candidate output document and the ``reverse translation probability'' of translating the candidate output back into the input source language document---the so-called ``noisy channel'' decomposition. A particular advantage of our model is that it requires only parallel sentences to train, rather than parallel documents, which are not always available. Using a new beam search reranking approximation to solve the decoding problem, we find that document language models outperform language models that assume independence between sentences, and that using either a document or sentence language model outperform comparable models that directly estimate the translation probability. We obtain the best-published results on the NIST Chinese--English translation task, a standard task for evaluating document translation. Our model also outperforms the benchmark Transformer model by approximately 2.5 BLEU on the WMT19 Chinese--English translation task.