In practice, we are not searching in the space of possible translations, but in the space of possible derivations, which are sequences of phrase translations p 1 (s 1 , t 1 ), ..., p n (s n , t n ) that can be applied to s to generate an output t with the score given by P (t) n i=1 P (s i , t i ).
in our metric, we do not have any means for detecting spurious phrase pairs, in fact, spurious phrase pairs are probably kept in the phrase table, since each distinct spurious phrase pair is only extracted once, and thus, they have very few derivations in its support set.