# Statistical Machine Translation

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1 Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm of finding th bst English translation for a Frnch sntnc 1 as a noisy channl: Th Frnch sntnc f 2 to translat is considrd to b a corruption of th English sntnc as dpictd in Figur 1. Which English sntnc is at th sourc of its Frnch countrpart is unknown, but w can look for sntncs that maximis P ( f). Figur 1: Th noisy channl modl of machin translation As by Bays thorm, P ( f) = P () P (f ) P (f) (1) th bst English translation for a Frnch sntnc is: ê = arg max P ()P (f ) (2) Th translation problm is thus rducd to a languag task, to obtain P (), 1 W follow th convntion that th sourc languag of th translation task is Frnch, and th targt languag English. 2 S th Appndix for a list of notations 1

2 and a translation task, to obtain P (f ) 3. Th IBM modls focus on th translation task only. To mphasiz that th English sntnc rprsnts a collction of concpts, th words of th sntnc ar calld cpts. Each cpt gnrats on or mor words in th Frnch translation. Convrsly, ach word in th Frnch sntnc is said to b alignd to a cpt in th English sntnc. To account for th possibility that Frnch words hav no countrpart in th English translation, on says that ths words ar alignd to th mpty cpt, 0. Figur 2 shows a possibl alignmnt btwn two sntncs. In this cas, th last thr words of th Frnch sntnc ar alignd to th sam word in th English sntnc. Not that th IBM modl of alignmnt dos not allow for th rvrs, that is to say, a Frnch word cannot b alignd with mor than on English word. Gnrally, a group of words cannot b alignd to anothr group of words with th IBM modls. This would in fact rquir a Phras-Basd approach. Figur 2: Alignmnt btwn a Frnch and an English sntnc, aftr [1]. In th IBM notation, this alignmnt is rprsntd as [2, 3, 4, 5, 6, 6, 6]. Each IBM modl builds on th prvious on, ach incrasing th complxity of th alignmnt modl. In all cass, th translation probability P (f ) is sn as th sum on all alignmnts of th conditional probabilitis P (f, a ), whr a is an alignmnt btwn th Frnch and th English sntncs: P (f ) = P (f, a ) (3) a Th conditional probability P (f, a ) can itslf b xprssd as: P (f, a ) = P (m ) m j=1 P (a j a j 1 1, f j 1 1, m, )P (f j a j 1, f j 1 1, m, ) (4) 3 Th translation modl informs us on what sntncs ar good translations, whil th languag modl nsurs that ths sntncs ar wll-formd. By combining ths modls, w thus gt bttr rsults than if w wr to look dirctly for th sntnc that maximiss P ( f). 2

3 Th following sctions prsnt th main assumptions takn by modls 1 to 3, along with th principal quations and th algorithm proposd to comput P (f ). 1.1 Modl 1 Modl 1 taks P (m ) to b a paramtr ɛ indpndnt of and m. Th trm P (a j a j 1 1, f j 1 1, m, ) is assumd to dpnd only on l and to b qual to (l+ 1) 1, and P (f j a j 1, f j 1 1, m, ) is assumd to dpnd only on f j and aj, and can thus b xprssd as th conditional probability t(f j aj ). Equation 4 can thrfor b rwrittn as: P (f, a ) = ɛ (l + 1) m m t(f j aj ) (5) j=1 Importantly, ths assumptions mak it possibl to comput P (f ) fficintly, as Equation 3 can b rwrittn as: ɛ m l P (f ) = (l + 1) m t(f j i ) (6) j=1 i=0 Th translation probabilitis t(f j i ) can b computd using: whr: c(f ; f, ) = t(f ) = λ 1 t(f ) t(f 0 ) t(f l ) and λ is a normalisation paramtr ( f t(f ) = 1): S c(f ; f (s), (s) ) (7) s=1 m l δ(f, f j ) δ(, i ) (8) j=1 i=0 λ = f S c(f ; f (s), (s) ) (9) s=1 Th algorithm usd to comput th translation probabilitis t(f j i ) is th following: 1. Choos initial valus for t(f ). As P (f ) has a uniqu local maximum in this modl, it is sufficint to tak ths initial valus to b qual and diffrnt from zro. 2. Comput th counts c(f ; f (S), (S) ) for ach pair of sntncs using Equation 8. 3

4 3. For ach apparing in th English sntncs, comput λ using Equation 9, and t(f ) using Equation Rpat stps 2 and 3 until th valus of t(f ) hav convrgd to th dsird dgr. 4

5 A f f l m i j 0 a a j Notations English word Frnch word English sntnc Frnch sntnc lngth of English sntnc lngth of Frnch sntnc word indx in English sntnc word indx in Frnch sntnc th mpty cpt alignmnt btwn a Frnch and an English sntnc th indx of th English word that is alignd to th Frnch word at indx j Rfrncs [1] Brown t. Al. (1993): Th mathmatics of statistical machin translation: paramtr stimation. Computational Linguistics Vol. 19, No. 2 [2] Th Europarl paralll corpus: 5

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