IBM-Mode1 Q: A: cash money It is fine today 2
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1 8. IBM 1
2 IBM-Mode1 Q: A: cash money It is fine today 2
3 e f a P (f, a e) â : â = arg max a P (f, a e) â P (f, a e) 3
4 θ P (f e, θ) θ f d = { f, e } L(θ d) = log f,e d P (f e, θ) θ f ˆθ = arg max θ L(θ d) ˆθ 4
5 ( ) 5
6 IBM model-1 P (f, a e) = P (m e) m P (a j a j 1 1, f1 j 1, m, e) j=1 P (f j a j 1, f j 1 1, m, e) (1) P (m e) = ( ) m = ɛ P (a j a j 1 1, f j 1 1, m, e) = j a j = = 1 l + 1 P (f j a j 1, f1 j 1, m, e) = j f j = f j e aj = t(f j e aj ) P (f, a e) = ɛ (l + 1) m m j=1 t(f j e aj ) (2) 6
7 a P (f e) = a = a = = P (f, a e) ɛ m (l + 1) t(f j e m aj ) j=1 ɛ m (l + 1) t(f j e m aj ) a j=1 ɛ (l + 1) m a 1 =0 a m =0 m j=1 t(f j e aj ) (3) a = a 1...a m a j f j e aj e = e 1...e l e 0 = NULL 0 a j l 1 j m 7
8 3 t(f e) max P (f e) t( ) e f t(f e) = 1 e f 1 8
9 3 h(t, λ) = ɛ (l + 1) m t(f j e m aj ) a 1 =0 a m =0 j=1 λ e( t(f e) 1) (4) e f 2 λ e h = ( t(f e) 1) = 0 (5) λ e f λ e 0 t(f e) 4 2 t(f e) λ e( t(f e) 1) = λ e (6) e f 4 1 m j=1 t(f j e aj ) 9
10 m t(f e) t(f j e aj ) j=1 = = 1 k m f k f e ak e 1 k m f k f e ak e = n(e, f)t(f e) 1 t(f k e ak ) t(f k e ak ) = n(e, f)t(f e) 1 m t(f e) t(f e)m j=1 δ(f,f j )δ(e,e aj ) n(e, f)t(f e) n(e,f) 1 1 k m f k f e ak e 1 k m t(f k e ak ) t(f e) n(e,f) t(f k e ak ) (7) n(e, f) = δ(f, f j)δ(e, e aj ) j=1 = f j a j 1 t(f e) 2 t(f e) 3,4 t(f e) 1 1 k m t(f k e ak ) 10
11 h(t, λ) t(f e) ɛ = (l + 1) m m δ(f, f j)δ(e, e aj ) t(f e) 1 a 1 =0 a m =0 j=1 t(f k e ak ) λ e (8) 1 k m 0 t(f e) = λ 1 e l a 1 =0 ɛ (l + 1) m a m =0 m j=1 δ(f, f j)δ(e, e aj ) 1 k m t(f k e ak ) λ e 4 f t(f e) = 1 t(f e) = λe 1 A(f e) λ e = f A(f e) f t(f e) = 1 λ e (9) 11
12 9 t(f e) t(f e) t( ) EM t( ) 1. t( ) 2. 9 t( ) 3. 2 t(e f) 9 12
13 9 t(f e) = λ 1 ɛ e (l + 1) m l a 1 =0 a m =0 3 P (f e) = a P (f, a e) = ɛ m j=1 δ(f, f j)δ(e, e aj ) (l + 1) m a 1 =0 a m =0 1 k m m t(f k e ak ) j=1 t(f j e aj ) t(f e) = λ 1 e P (f, a e) m δ(f, f j)δ(e, e aj ) (10) a j=1 C(f e; f, e) = a P (a f, e) m j=1 δ(f, f j)δ(e, e aj ) (11) m j=1 δ(f, f j )δ(e, e aj ) a f e P (a f, e) a C(f e; f, e) f e t(f e) f e C(f e, f, e) 13
14 t(f e) = λ 1 e P (f, a e) m δ(f, f j)δ(e, e aj ) a j=1 = λ 1 e a P (f e)p (a f, e) m j=1 δ(f, f j)δ(e, e aj ) = λ 1 e P (f e) P (a f, e) m δ(f, f j)δ(e, e aj ) a j=1 = λ 1 e P (f e)c(f e; f, e) = λ e 1 C(f e; f, e) (12) λ e = λ e P (f e) 1 λ e = f C(f e; f, e) f t(f e) = 1 e f s e (s) f (s) t(f e) = λ 1 e C(f e; f (s), e (s) ) s 14
15 t(f e) = λ 1 e, e (s) ) s (13) λ e =, e (s) ) f s (14) C(f e; f, e) = a P (a f, e) m j=1 δ(f, f j)δ(e, e aj ) (15) 1. f e t(f e) 2. s C(f e; f (s), e (s) ) 3. e λ e 4. f t(f e) 5. goto 2 or exit 15
16 C(f e; f, e) = P (a f, e) m δ(f, f j)δ(e, e aj ) a j=1 = λ e P (f e) 1 t(f e) ( 12 ) (16) 4 h(t, λ) = ɛ (l + 1) m t(f j e m aj ) a 1 =0 a m =0 j=1 λ e( t(f e) 1) e f (l + 1) m m 16
17 a 1 =0 a m =0 m j=1 t(f j e aj ) = m j=1 i=0 t(f j e i ) (l + 1) m m m(l + 1) m = 3,l = 1 t(f j e i ) = t ji = 1 t(f j e aj ) a 1 =0 a 2 =0 a 3 =0 j=1 = t 10 t 20 t 30 + t 11 t 20 t t 11 t 21 t 30 + t 11 t 21 t 31 1 = 3 t(f j e i ) j=1 i=0 = (t 10 + t 11 )(t 20 + t 21 )(t 30 + t 31 ) 17
18 4 ɛ m h(t, λ) = (l + 1) t(f j e m i ) λ e( j=1 i=0 e f 2 t(f e) λ e t(f e) 1) n f = m δ(f, f j) = (f = f j ) f j j=1 n e = l m j=1 = = i=0 t(f j e i ) 1 j m f j f 1 j m f j f δ(e, e i) = (e = e i ) e i i=0 i=0 t(f j e i ) i=0 t(f j e i ) 2 t(f e) = n f n e = n f n e = 0 i l e i e 0 i l n f n e 0 i l t(f e i ) i=0 t(f e i) 0 i l e i e n f t(f e i ) + n e t(f e) t(f e i ) + n e t(f e) t(f e i ) 0 i l 18 n f 1 n f 1 n f (17) f t(f e i ) n (18)
19 t(f e) m j=1 i=0 t(f j e i ) = n f n e 0 i l t(f e i ) m j=1 i=0 t(f j e i ) (19) h(t, λ) t(f e) n f n e ɛ m = l i=0 t(f e i )(l + 1) t(f j e m i ) λ e j=1 i=0 n f n e = l i=0 t(f e i ) P (f e) λ e (20) h(t,λ) t(f e) = 0 λ e P (f e) 1 = n f n e l i=0 t(f e i ) 16 C(f e; f, e) = λ e P (f e) 1 t(f e) t(f e) = l i=0 t(f e i ) n fn e (21) t(f e) l i=0 t(f e i ) f e n f n e 19
20 C(f e; f, e) = t(f e) l i=0 t(f e i ) n fn e (22) λ e =, e (s) ) f s (23) t(f e) = λ 1 e, e (s) ) s (24) t( ) 1. t(f e) 2. s C(f e; f (s), e (s) ) 3. e λ e 4. f t(f e) 5. goto 2 or exit. Peter F. Brown, Stephen A. Della Pietra, Vincent J. Della Pietra, and Robert L. Mercer. (1993) The Mathematics of Statistical Machine Translation: Parameter Estimation. 20
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