Neural MT systems generate translations one word at a time. They can still generate fluid translations because they choose each word based on all of the words generated so far. Typically, these systems are just trained to generate the next word correctly, based on all previous words. One systematic problem with this word-by-word approach to training and translating is that the translations are often too short and omit important content. In the paper Neural Machine Translation with Reconstruction, the authors describe a clever new way to train and translate. During training, their system is encouraged not only to generate each next word correctly but also to correctly generate the original source sentence based on the translation that was generated. In this way, the model is rewarded for generating a translation that is sufficient to describe all of the content in the original source.
When doing beam search in sequence to sequence models, one explores next words in order of their likelihood. However, during decoding, there may be other constraints we have or objectives we wish to maximize. For example, sequence length, BLEU score, or mutual information between the target and source sentences. In order to accommodate these additional desiderata, the authors add an additional term Q onto the likelihood capturing the appropriate criterion and then choose words based on this combined objective.