Dynamic Programming Viterbi

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Dynamic ProgrammingViterbi
2
Today

• Take a look at dynamic programming algorithms
• Walk through some examples
• Simple chart parse
• Viterbi example (from speech)
• Viterbi example (from POS tagging)

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Dynamic Programming

• Definition Algorithmic approach for solving
  optimization problems by caching sub-problem
  solutions rather than recomputing them.
• Typical device used is a chart to hold the
  various sub-problem solutions.
• Common implementation Charts used in parsing
  (based on the Earley algorithm)

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BU Chart Parser Walk Through
Sample Sentence The large can can hold the
water.
Sample Grammar
1. S ? NP VP 2. NP ? D ADJ N 3. NP ? D N 4. NP ?
ADJ N 5. VP ? AUX VP 6. VP ? V NP
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Chart Parser

• Allows one to preserve the results of the parse
  so far in a chart
• Partial results and all previous results are kept
  so that work is not repeated

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1. S ? NP VP 2. NP ? D ADJ N 3. NP ? D N 4. NP ?
ADJ N 5. VP ? AUX VP 6. VP ? V NP
The large can can hold the water.
D1
ADJ1
1
2
3
NP ? D ADJ N
NP ? D ADJ N
NP ? D N
NP ? ADJ N
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1. S ? NP VP 2. NP ? D ADJ N 3. NP ? D N 4. NP ?
ADJ N 5. VP ? AUX VP 6. VP ? V NP
The large can can hold the water.
NP2
NP1
N1
D1
ADJ1
AUX1 - V1
1
2
3
4
NP ? D ADJ N
NP ? D ADJ N
NP ? D ADJ N
NP ? D N
NP ? ADJ N
NP ? ADJ N
S ? NP VP
S ? NP VP
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Viterbi Example
See word models, p. 245
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Viterbi Example
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Viterbi example

• Input is aa n iy dh ax
• I need the
• Actually dialectal/fast speech variant, Ah nee
  the
• a ni ð?

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aa n iy dh ax
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Viterbi Example
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A smaller example
very
quick
very
quick
0.1
0.9
0.8
0.2
0.7
end
start
JJ
RB
1
1
0.1
0.3
0.9

• What is the best sequence of states for the input
  string very very quick?
• Computing all possible paths and finding the one
  with maximum prob. is exponential

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quick
very
very
quick
0.1
0.9
0.8
0.2
0.7
end
start
JJ
RB
1
1
0.1
0.3
0.9
Input sequence / time Input sequence / time Input sequence / time Input sequence / time
e --gt very very --gt very very very --gt quick
e --gt RB 0.8 (1.0x0.8)

e --gt JJ 0.0 (0x0.8)

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quick
very
very
quick
0.1
0.9
0.8
0.2
0.7
end
start
JJ
RB
1
1
0.1
0.3
0.9
Input sequence / time Input sequence / time Input sequence / time Input sequence / time
e --gt very very --gt very very very --gt quick
e --gt RB 0.6 (1.0x0.8) RB --gt RB 0.192 (0.8x0.3x0.8)

e --gt JJ 0.0 (0x0.8) RB --gt JJ 0.056 (0.8x0.7x0.1)

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quick
very
very
quick
0.1
0.9
0.8
0.2
0.7
end
start
JJ
RB
1
1
0.1
0.3
0.9
Input sequence Input sequence Input sequence Input sequence
e --gt very very --gt very very very --gt quick
e --gt RB 0.6 (1.0x0.8) RB --gt RB 0.192 (0.8x0.3x0.8) RB RB --gt RB 0.01152 (0.192x0.3x0.2)
RB RB --gt JJ 0.12096 (0.192x0.7x 0.9)
e --gt JJ 0.0 (0x0.8) RB --gt JJ 0.056 (0.8x0.7x0.8) RB JJ --gt RB 0.001122 (0.056x0.1x0.2)
RB JJ --gt JJ 0.04536 (0.056x0.9x0.9)
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Implementation
// Initialize viterbi1,PERIOD 1.0 at the
start, assuming dummy period for i1 to n step 1
do // all input words (presumably in a sentence)
for all tags tj do // for all possible
tags // max probability of being in state j (tag
j) at word i1 (path probability
matrix) viterbii1,tj max1kT(viterbii,tk
x P(wi1tj) x P(tj tk)) // most likely
state (tag) at word i given that were in state j
at word i1 // In other words, statei1 is
keeping a pointer to the state that got us here
statei1,tj argmax1kT ( viterbii,tk x
P(wi1tj) x P(tj tk)) end end //Termination
and path-readout bestPathn1 argmax1jT
viterbin1,j for jn to 1 step -1 do // for
all input words bestPathj statei1,
bestPathj1 end P(bestPath1,, bestPathn )
max1jT viterbin1,j
Emission probability
State transition probability