... across languages, we can simplify the above formula and write: le+l 1 P(a,e,,f,)= ]-I p(ei,f~ilal )i-1 (5) i=1 where a~ -1 is a shorthand for (al, ,ai-1). p(ei, f~,) is a shorthand for ... bandwidth to largeur de bande., and one from bandwidth to either largeur or hap.de (type 2), or even through the two edges from bandwidth to largeur and bande (type 3), 2. secondly, ... probability of ob- serving the alignment a together with the English and French sentences, es and f~, and (a,e,f)~ -1 is a shorthand for (al, , ai-1, el, , el-l, fal,.', fa,-i ). Since we...