Wordbased SMT

1
Word-based SMT

• Ling 580
• Fei Xia
• Week 1 1/3/06

2
Outline

• General concepts
• Source channel model
• Notations
• Word alignment
• Model 1-2
• Model 3-4
• Model 5

3
IBM Model Basics

• Classic paper Brown et. al. (1993)
• Translation F ? E (or Fr ? Eng)
• Resource required
• Parallel data (a set of sentence pairs)
• Main concepts
• Source channel model
• Hidden word alignment
• EM training

4
Intuition

• Sentence pairs word mapping is one-to-one.
• (1) S a b c d e
• T l m n o p
• (2) S c a e
• T p n m
• (3) S d a c
• T n p l
• ? (b, o), (d, l), (e, m), and
• (a, p), (c, n), or
• (a, n), (c, p)

5
Source channel model

• Task S ? T
• Source channel (a.k.a. noisy channel, noisy
  source channel) use the Bayes Rule.
• Two types of parameters
• P(T) language model
• P(S T) its meaning varies.

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Source channel model for MT
P(T)
P(S T)
Tgt sent
Src sent
Noisy channel

• Two types of parameters
• Language model P(T)
• Translation model P(S T)

7
Source channel model for MT
P(E)
P(F E)
Fr sent
Eng sent
Noisy channel

• Two types of parameters
• Language model P(E)
• Translation model P(F E)

8
Source channel for MT

• People think in English.
• English thoughts can be characterized by a
  plausibility filter P(E).
• Sentences are corrupted into a different
  language by a translation model P(F E).
• Our goal is to find the original, uncorrupted
  English sentence e. To achieve this goal, we
  efficiently evaluate P(E) P(F E) over many
  candidate Eng sentences.

9
Source channel vs. direct model

• Source channel demand plausible Eng and strong
  correlation between e and f.
• Direct model demand strong correlation between
  e and f.
• Question
• Formally, they are the same.
• In practice, they are not due to different
  approximations.

10
Word alignment

• a(j)i ? aj i
• a (a1, , am)
• Ex
• F f1 f2 f3 f4 f5
• E e1 e2 e3 e4
• a43
• a (0, 1, 1, 3, 2)

11
The constraint on word alignment

• The constraint each fr word is generated by
  exactly one Eng word (including e0) l is Eng
  sent length, m is Fr sent length
• Without the constraint 2lm.
• With the constraint (l1)m.
• Why the models use the constraint?
• We want to use P(fj ei) to estimate P(F E).
• How to handle the exceptional cases?
• Various methods target word grouping,
  phrase-based SMT, etc.

12
Modeling p(F E) with alignment
13
Notation

• E the Eng sentence E e1 el
• ei the i-th Eng word.
• F the Fr sentence f1 fm
• fj the j-th Fr word.
• e0 the Eng NULL word
• F0 the Fr NULL word.
• aj the position of Eng word that generates
  fj.

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Word alignment

• An alignment, a, is a function from Fr word
  position to Eng word position a(j)i means that
  the fj is generated by ei.
• The constraint each fr word is generated by
  exactly one Eng word (including e0)

15
Notation (cont)

• l Eng sent leng
• m Fr sent leng
• i Eng word position
• j Fr word position
• e an Eng word
• f a Fr word

16
Outline

• General concepts
• Source channel model
• Word alignment
• Notations
• Model 1-2
• Model 3-4

17
Model 1 and 2
18
Model 1 and 2

• Modeling
• Generative process
• Decomposition
• Formula and types of parameters
• Training
• Finding the best alignment
• Decoding

19
Generative process

• To generate F from E
• Pick a length m for F, with prob P(m l)
• Choose an alignment a, with prob P(a E, m)
• Generate Fr sent given the Eng sent and the
  alignment, with prob P(F E, a, m).
• Another way to look at it
• Pick a length m for F, with prob P(m l).
• For j1 to m
• Pick an Eng word index aj, with prob P(aj j, m,
  l).
• Pick a Fr word fj according to the Eng word ei,
  where ajI, with prob P(fj ei ).

20
Decomposition
21
Approximation

• Fr sent length depends only on Eng sent length
• Fr word depends only on the Eng word that
  generates it

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Approximation (cont)

• Estimating P(a E, m)
• Model 1 All alignments are equally likely
• Model 2 alignments have different prob
• Model 1 can be seen as a special case of Model 2,
• where

23
Decomposition for Model 1
24
The magic (for Model 1)
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Final formula and parameters for Model 1

• Two types of parameters
• Length prob P(m l)
• Translation prob P(fj ei), or t(fj ei),

26
Decomposition for Model 2

• Same as Model 1 except that Model 2 does not
  assume all alignments are equally likely.

27
The magic for Model 2
28
Final formula and parameters for Model 2

• Three types of parameters
• Length prob P(m l)
• Translation prob t(fj ei)
• Distortion prob d(i j, m, l)

29
Summary of Modeling
Model 1
Model 2

• Parameters
• Length prob P(m
  l)
• Translation prob t(fj
  ei)
• Distortion prob (for Model 2) d(i j, m, l)

30
Model 1 and 2

• Modeling
• Generative process
• Decomposition
• Formula and types of parameters
• Training
• Finding the best alignment
• Decoding

31
Training

• Mathematically motivated
• Having an objective function to optimize
• Using several clever tricks
• The resulting formulae
• are intuitively expected
• can be calculated efficiently
• EM algorithm
• Hill climbing, and each iteration guarantees to
  improve objective function
• It does not guaranteed to reach global optimal.

32
Length prob P(j i)

• Let Ct (j, i) be the number of sentence pairs
  where the Fr leng is j, and Eng leng is i.
• Length prob
• No need for iterations

33
Estimating t(fe) a naïve approach

• A naïve approach
• Count the times that f appears in F and e appears
  in E.
• Count the times that e appears in E
• Divide the 1st number by the 2nd number.
• Problem
• It cannot distinguish true translations from pure
  coincidence.
• Ex t(el white) t(blanco white)
• Solution count the times that f aligns to e.

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Estimating t(fe) in Model 1

• When each sent pair has a unique word alignment
• When each sent pair has several word alignments
  with prob
• When there are no word alignments

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When there is a single word alignment

• We can simply count.
• Training data
• Eng b c b
• Fr x y y
• Prob
• ct(x,b)0, ct(y,b)2, ct(x,c)1, ct(y,c)0
• t(xb)0, t(yb)1.0, t(xc)1.0, t(yc)0

36
When there are several word alignments

• If a sent pair has several word alignments, use
  fractional counts.
• Training data
• P(aE,F)0.3 0.2 0.4 0.1
  1.0
• b c b c b c
  b c b
• x y x y x y
  x y y
• Prob
• Ct(x,b)0.7, Ct(y,b)1.5, Ct(x,c)0.3,
  Ct(y,c)0.5
• P(xb)7/22, P(yb)15/22, P(xc)3/8, P(yc)5/8

37
Fractional counts

• Let Ct(f, e) be the fractional count of (f, e)
  pair in the training data, given alignment prob
  P.

Alignment prob
Actual count of times e and f are linked in
(E,F) by alignment a
38
When there are no word alignments

• We could list all the alignments, and estimate
  P(a E, F).

39
Formulae so far
? New estimate for t(fe)
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The algorithm

• Start with an initial estimate of t(f e) e.g.,
  uniform distribution
• Calculate P(a F, E)
• Calculate Ct (f, e), Normalize to get t(fe)
• Repeat Steps 2-3 until the improvement is too
  small.

41
So far, we estimate t(f e) by enumerating all
possible alignments

• This process is very expensive, as the number of
  all possible alignments is (l1)m.

Prev iterations Estimate of Alignment prob
Actual count of times e and f are linked in
(E,F) by alignment a
42
No need to enumerate all word alignments

• Luckily, for Model 1, there is a way to calculate
  Ct(f, e) efficiently.

43
The algorithm

• Start with an initial estimate of t(f e) e.g.,
  uniform distribution
• Calculate P(a F, E)
• Calculate Ct (f, e), Normalize to get t(fe)
• Repeat Steps 2-3 until the improvement is too
  small.

44
Estimating t(f e) in Model 2

• Ct(f, e) is slightly different from the one in
  Model 1

45
Estimating d(i j, m,l) in Model 2

• Let Ct(i, j, m, l) be the fractional count that
  Fr position j is linked to the Eng position i.

46
The algorithm

• Start with an initial estimate of t(f e) e.g.,
  uniform distribution
• Calculate P(a F, E)
• Calculate Ct (f, e), Normalize to get t(fe)
• Repeat Steps 2-3 until the improvement is too
  small.

47
Training Summary

• EM algorithm
• Hill climbing, and each iteration guarantees to
  improve objective function
• It does not guaranteed to reach global optimal.
• The resulting formulae
• are intuitively expected
• can be calculated efficiently

48
Model 1 and 2

• Modeling
• Generative process
• Decomposition
• Formula and types of parameters
• Training
• Finding the best alignment

49
The best alignment in Model 1-5
Given E and F, we are looking for the best
alignment a
50
The best alignment in Model 1
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The best alignment in Model 2
52
Summary of Model 1 and 2

• Modeling
• Pick the length of F with prob P(m l).
• For each position j
• Pick an English word position aj, with prob P(aj
  j, m, l).
• Pick a Fr word fj according to the Eng word ei,
  with t(fj ei), where iaj
• The resulting formula can be calculated
  efficiently.
• Training EM algorithm. The update can be done
  efficiently.
• Finding the best alignment can be easily done.

53
Limitations of Model 1 and 2

• There could be some relations among the Fr words
  generated by the same Eng word (w.r.t. positions
  and fertility).
• The relations are not captured by Model 1 and 2.
• They are captured by Model 3 and 4.

54
Outline

• General concepts
• Source channel model
• Word alignment
• Notations
• Model 1-2
• Model 3-4

55
Model 3 and 4
56
Model 3 and 4

• Modeling
• Generative process
• Decomposition and final formula
• Types of parameters
• Training
• Finding the best alignment
• Decoding

57
Generative process

• For each Eng word ei, choose a fertility
• For each ei, generate Fr words
• Choose the position of each Fr word.

58
An example
NULL the cheapest nonstop flights
59
An example
NULL the cheapest nonstop flights
vols
sans
escale
le
moins
cher
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Decomposition
61
Approximations and types of parameters
Where N is the number of empty slots.
62
Approximations and types of parameters (cont)
63
Modeling summary

• For each Eng word ei, choose a fertility
• which only depends on ei.
• For each ei, generate Fr words, which only
  depends on ei.
• Choose the position of each Fr word
• Model 3 the position depends only on the
  position of the Eng word generating it.
• Model 4 the position depends on more.

64
Training

• Use EM, just like Model 1 and 2
• Translation and distortion probabilities can be
  calculated efficiently, fertility probabilities
  cannot.
• No efficient algorithms to find the best
  alignment.

65
Model 3 and 4

• Modeling
• Generative process
• Decomposition and final formula
• Types of parameters
• Training
• Finding the best alignment
• Decoding

66
Model 1-4 modeling
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Model 1-4 training

• Similarities
• Same objective function
• Same algorithm EM algorithm
• Differences
• Summation over all alignments can be done
  efficiently for Model 1-2, but not for Model 3-4.
• Best alignment can be found efficiently for Model
  1-2, but not for Model 3-4.

68
Summary

• General concepts
• Source channel model P(E) and P(FE)
• Notations
• Word alignment each Fr word comes from exactly
  one Eng word (including e0).
• Model 1-2
• Model 3-4

69
Additional slides
70
An example of Model 1 training

• Training data
• Sent 1 Eng b c, Fr x y
• Sent 2 Eng b, Fr y
• To reduce the number of alignments, assume that
  each Eng word generates exactly one Fr word ? Two
  possible alignments for Sent1, and one for Sent2.
• Step 1 Initial t(fe) t(xb)t(yb)1/2,
  t(xc)t(yc)1/2

71
Step 2 calculating P(aF,E)

• a1 b c a2 b c a3
  b
• x y x y
  y
• Before normalization
• P(a1E1,F1)Z1/21/21/4
• P(a2E1,F1)Z1/21/21/4
• P(a3E2,F2)Z1/2
• After normalization
• P(a1E1,F1)1/4 / (1/41/4) ½
• P(a2E1,F1)1/4 / ½ ½.
• P(a3E2,F2) ½ / ½ 1

72
Step 3 calculating t(f e)

• a1 b c a2 b c a3
  b
• x y x y
  y
• Collecting counts
• Ct(x,b) 1/2
• Ct(y,b) ½ 1 3/2
• Ct(x,c)1/2
• Ct(y,c)1/2
• After normalization
• t(x b) ½ / (1/23/2) ¼, t(y b) 3/4
• t(x c) ½ / 1 ½, t(y c)1/2

73
Repeating step 2 calculating P(aF,E)

• a1 b c a2 b c a3
  b
• x y x y
  y
• Before normalization
• P(a1E1,F1)Z1/41/21/8
• P(a2E1,F1)Z3/41/23/8
• P(a3E2,F2)Z3/4
• After normalization
• P(a1E1,F1)1/8 / (1/83/8) 1/4
• P(a2E1,F2)3/8 / 4/8 3/4.
• P(a3E2,F2) 3/4 / 3/4 1

74
Repeating step 3 calculating t(f e)

• a1 b c a2 b c a3
  b
• x y x y
  y
• Collecting counts
• Ct(x,b) 1/4
• Ct(y,b) 3/4 1 7/4
• Ct(x,c)3/4
• Ct(y,c)1/4
• After normalization
• t(x b) 1/4 / (1/47/4) 1/8, t(y b) 7/8
• t(x c) 3/4 / (3/41/4) 3/4, t(y c)1/4

75
See the trend?
76
Calculating t(f e) with the new formulae

• E1 b c E2 b
• F1 x y F2 y
• Collecting counts
• Ct(x,b) 1/2/(1/21/2)
• Ct(y,b) ½ /(1/21/2) 1/1 3/2
• Ct(x,c)1/2 / (1/21/2) 1/2
• Ct(y,c)1/2 / (1/21/2) 1/2
• After normalization
• t(x b) ½ / (1/23/2) ¼, t(y b) 3/4
• t(x c) ½ / 1 ½, t(y c)1/2

77
EM algorithm

• EM expectation maximization
• In a model with hidden states (e.g., word
  alignment), how can we estimate model parameters?
• EM does the following
• E-step Take an initial model parameterization
  and calculate the expected values of the hidden
  data.
• M-step Use the expected values to maximize the
  likelihood of the training data.

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Objective function