The Mathematics of Statistical Machine Translation: Parameter Estimation

3 Alignments

1. What is a word alignment between a French and an English sentence?
   An object indicating, for each French word, the English word(s) it arose from, drawn as connections. Every alignment is considered correct with some probability.

   Written (Le programme a été mis en application | And the(1) program(2) has(3) been(4) implemented(5,6,7)).
   • Brown et al., 1993, The Mathematics of Statistical Machine Translation (Computational Linguistics), §3
2. What is a cept, and what is the empty cept?
   The set of English words connected to a French word in an alignment (e.g. don’t have any money → sont démunis). French words connected to no English word are generated by the empty cept \(e_0\), placed by convention at position 0 (e.g. à in J’applaudis à la décision).
   • Brown et al., 1993, The Mathematics of Statistical Machine Translation (Computational Linguistics), §3
3. How many alignments are there between an English string of length \(l\) and a French string of length \(m\)?
   There are \(lm\) possible connections, so \(2^{lm}\) alignments in general. Restricting each French word to one English word (or \(e_0\)), as in the models, gives \((l + 1)^m\).
   • Brown et al., 1993, The Mathematics of Statistical Machine Translation (Computational Linguistics), §3