  We present a probabilistic framework for multilingual neural machine
translation that encompasses supervised and unsupervised setups, focusing on
unsupervised translation. In addition to studying the vanilla case where there
is only monolingual data available, we propose a novel setup where one language
in the (source, target) pair is not associated with any parallel data, but
there may exist auxiliary parallel data that contains the other. This auxiliary
data can naturally be utilized in our probabilistic framework via a novel
cross-translation loss term. Empirically, we show that our approach results in
higher BLEU scores over state-of-the-art unsupervised models on the WMT'14
English-French, WMT'16 English-German, and WMT'16 English-Romanian datasets in
most directions. In particular, we obtain a +1.65 BLEU advantage over the
best-performing unsupervised model in the Romanian-English direction.
