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PGM 2014, Utrecht Cedric De Boom, Jasper De Bock, Arthur Van Camp, Gert de Cooman
Robustifying the Viterbi Algorithm
?
A Toy Example
Weather estimation
2
Robustifying the Viterbi Algorithm Cedric De Boom, Jasper De Bock, - - PowerPoint PPT Presentation
Robustifying the Viterbi Algorithm Cedric De Boom, Jasper De Bock, - - PowerPoint PPT Presentation
Robustifying the Viterbi Algorithm Cedric De Boom, Jasper De Bock, Arthur Van Camp, Gert de Cooman PGM 2014, Utrecht A Toy Example Weather estimation ? 2 A Toy Example Weather estimation ? 3 A Toy Example Weather estimation H ? L 4
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SLIDE 3
3
?
A Toy Example
Weather estimation
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H L
?
A Toy Example
Weather estimation
4
SLIDE 5
H ?
A Toy Example
Weather estimation
5
SLIDE 6
H L L
O1 O2 O3
? ? ?
X1 X2 X3
A Toy Example
Weather estimation
6
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H L L
O1 O2 O3 X1 X2 X3
? ? ?
p1(X1) p2(X2|X1) p3(X3|X2) q1(O1|X1) q2(O2|X2) q3(O3|X3)
Hidden Markov Model
Local models
7
SLIDE 8
p1(X1) p2(X2|X1) p3(X3|X2) q1(O1|X1) q2(O2|X2) q3(O3|X3)
Local models
Hidden Markov Model
Local models
8
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p1(X1) p2(X2|X1) p3(X3|X2) q1(O1|X1) q2(O2|X2) q3(O3|X3)
Local models Global model
p(X1:3, O1:3) = p1(X1)q1(O1|X1)
3
Y
i=2
pi(Xi|Xi−1)qi(Oi|Xi)
Hidden Markov Model
Global model
9
SLIDE 10
10
Global model
p(X1:3, O1:3) = p1(X1)q1(O1|X1)
3
Y
i=2
pi(Xi|Xi−1)qi(Oi|Xi) arg max
x1:3 p(x1:3|o1:3)
Hidden Markov Model
Estimating the hidden sequence
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10
Global model
p(X1:3, O1:3) = p1(X1)q1(O1|X1)
3
Y
i=2
pi(Xi|Xi−1)qi(Oi|Xi) arg max
x1:3 p(x1:3|o1:3) = arg max x1:3
p(x1:3, o1:3) p(o1:3)
Hidden Markov Model
Estimating the hidden sequence
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10
Global model
p(X1:3, O1:3) = p1(X1)q1(O1|X1)
3
Y
i=2
pi(Xi|Xi−1)qi(Oi|Xi) arg max
x1:3 p(x1:3|o1:3)
= arg max
x1:3 p(x1:3, o1:3)
= arg max
x1:3
p(x1:3, o1:3) p(o1:3)
Hidden Markov Model
Estimating the hidden sequence
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10
Global model
p(X1:3, O1:3) = p1(X1)q1(O1|X1)
3
Y
i=2
pi(Xi|Xi−1)qi(Oi|Xi) arg max
x1:3 p(x1:3|o1:3)
= arg max
x1:3 p(x1:3, o1:3)
Viterbi algorithm (1967)
= arg max
x1:3
p(x1:3, o1:3) p(o1:3)
Hidden Markov Model
Estimating the hidden sequence
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Hidden Markov Model
Viterbi algorithm
11
Viterbi algorithm (1967)
Recursive
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Hidden Markov Model
Viterbi algorithm
11
Viterbi algorithm (1967)
Recursive Complexity O(nm2) n: length of the sequence m: size of state space
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Hidden Markov Model
Viterbi algorithm
11
Viterbi algorithm (1967)
Recursive Complexity O(nm2) n: length of the sequence m: size of state space Extendible to k-best Viterbi
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Imprecise Hidden Markov Model
Local models
12
H L L ? ? ?
MX1 MX2|X1 MX3|X2 MO1|X1 MO2|X2 MO3|X3
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Imprecise Hidden Markov Model
Local models
13
Local models
MX1 MX2|X1 MX3|X2 MO1|X1 MO2|X2 MO3|X3
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M = ( 3 Y
i=1
pi(Xi|Xi−1)qi(Oi|Xi) :
Imprecise Hidden Markov Model
Global model
14
Local models Global model
MX1 MX2|X1 MX3|X2 MO1|X1 MO2|X2 MO3|X3 (∀k ∈ {1, 2, 3}) pk(·|Xk−1) ∈ MXk|Xk−1, qk(·|Xk) ∈ MOk|Xk
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M = ( 3 Y
i=1
pi(Xi|Xi−1)qi(Oi|Xi) :
Imprecise Hidden Markov Model
Global model
14
Local models Global model
MX1 MX2|X1 MX3|X2 MO1|X1 MO2|X2 MO3|X3 (∀k ∈ {1, 2, 3}) pk(·|Xk−1) ∈ MXk|Xk−1, qk(·|Xk) ∈ MOk|Xk
May contain infinitely many precise models!
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Imprecise Hidden Markov Model
Estimating the hidden sequence
15
Partial order
x1:3 ˆ x1:3 , (8p 2 M) p(x1:3|o1:3) > p(ˆ x1:3|o1:3)
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Imprecise Hidden Markov Model
Estimating the hidden sequence
15
Set of maximal solutions
- ptmax(X1:3) , {ˆ
x1:3 ∈ X1:3 : (∀x1:3 ∈ X1:3) x1:3 ⌥ ˆ x1:3}
Partial order
x1:3 ˆ x1:3 , (8p 2 M) p(x1:3|o1:3) > p(ˆ x1:3|o1:3)
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Imprecise Hidden Markov Model
Estimating the hidden sequence
15
Set of maximal solutions
- ptmax(X1:3) , {ˆ
x1:3 ∈ X1:3 : (∀x1:3 ∈ X1:3) x1:3 ⌥ ˆ x1:3}
Indecision
There may be multiple maximal solutions.
Partial order
x1:3 ˆ x1:3 , (8p 2 M) p(x1:3|o1:3) > p(ˆ x1:3|o1:3)
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Imprecise Hidden Markov Model
Rewriting the solution set
16
n , (8p 2 M) p(x1:n|o1:n) > p(ˆ
x1:n|o1:n) x1:n ˆ x1:n ,
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
16
n , (8p 2 M) p(x1:n|o1:n) > p(ˆ
x1:n|o1:n) ⇔ (∀p ∈ M) p(x1:n, o1:n) > p(ˆ x1:n, o1:n) x1:n ˆ x1:n ,
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
16
⇔ (∀p ∈ M) p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1
n , (8p 2 M) p(x1:n|o1:n) > p(ˆ
x1:n|o1:n) ⇔ (∀p ∈ M) p(x1:n, o1:n) > p(ˆ x1:n, o1:n) x1:n ˆ x1:n ,
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
16
⇔ (∀p ∈ M) p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 ⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1
n , (8p 2 M) p(x1:n|o1:n) > p(ˆ
x1:n|o1:n) ⇔ (∀p ∈ M) p(x1:n, o1:n) > p(ˆ x1:n, o1:n) x1:n ˆ x1:n ,
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
16
⇔ (∀p ∈ M) p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 ⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 What if becomes zero? p(ˆ x1:n, o1:n)
n , (8p 2 M) p(x1:n|o1:n) > p(ˆ
x1:n|o1:n) ⇔ (∀p ∈ M) p(x1:n, o1:n) > p(ˆ x1:n, o1:n) x1:n ˆ x1:n ,
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
17
⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 x1:n ˆ x1:n ,
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
17
⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 x1:n ˆ x1:n , ⇔ min
p∈M n
Y
k=1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) qk(ok|xk) qk(ok|ˆ xk) > 1
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
17
⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 x1:n ˆ x1:n , ⇔ min
p∈M n
Y
k=1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) qk(ok|xk) qk(ok|ˆ xk) > 1
⇔
n
Y
k=1
min
pk(·|Xk−1)∈MXk|Xk−1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) min
qk(·|Xk)∈MOk|Xk
qk(ok|xk) qk(ok|ˆ xk) > 1 Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
17
⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 x1:n ˆ x1:n , ⇔ min
p∈M n
Y
k=1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) qk(ok|xk) qk(ok|ˆ xk) > 1
⇔
n
Y
k=1
min
pk(·|Xk−1)∈MXk|Xk−1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) min
qk(·|Xk)∈MOk|Xk
qk(ok|xk) qk(ok|ˆ xk) > 1
⇔
n
Y
k=1
χk(xk, xk−1, ˆ xk, ˆ xk−1)ωk(xk, ˆ xk, ok) > 1
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
17
⇔ min
p∈M
p(x1:n, o1:n) p(ˆ x1:n, o1:n) > 1 x1:n ˆ x1:n , ⇔ min
p∈M n
Y
k=1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) qk(ok|xk) qk(ok|ˆ xk) > 1
⇔
n
Y
k=1
min
pk(·|Xk−1)∈MXk|Xk−1
pk(xk|xk−1) pk(ˆ xk|ˆ xk−1) min
qk(·|Xk)∈MOk|Xk
qk(ok|xk) qk(ok|ˆ xk) > 1
⇔
n
Y
k=1
χk(xk, xk−1, ˆ xk, ˆ xk−1)ωk(xk, ˆ xk, ok) > 1 Can be calculated in advance
Partial order
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Imprecise Hidden Markov Model
Rewriting the solution set
18
Set of maximal solutions
⇔ max
x1:n∈X1:n n
Y
k=1
χk(xk, xk−1, ˆ xk, ˆ xk−1)ωk(xk, ˆ xk, ok) ≤ 1
- ptmax(X1:n) , {ˆ
x1:n 2 X1:n : (8x1:n 2 X1:n) x1:n 6 ˆ x1:n}
⇔
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Imprecise Hidden Markov Model
Rewriting the solution set
18
How do we calculate this set?
Set of maximal solutions
⇔ max
x1:n∈X1:n n
Y
k=1
χk(xk, xk−1, ˆ xk, ˆ xk−1)ωk(xk, ˆ xk, ok) ≤ 1
- ptmax(X1:n) , {ˆ
x1:n 2 X1:n : (8x1:n 2 X1:n) x1:n 6 ˆ x1:n}
⇔
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Imprecise Hidden Markov Model
Rewriting the solution set
18
How do we calculate this set?
MaxiHMM algorithm Set of maximal solutions
⇔ max
x1:n∈X1:n n
Y
k=1
χk(xk, xk−1, ˆ xk, ˆ xk−1)ωk(xk, ˆ xk, ok) ≤ 1
- ptmax(X1:n) , {ˆ
x1:n 2 X1:n : (8x1:n 2 X1:n) x1:n 6 ˆ x1:n}
⇔
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
19
MaxiHMM algorithm
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MaxiHMM algorithm
General overview
20
We recursively define two parameters:
SLIDE 50
MaxiHMM algorithm
General overview
20
γk(xk, ˆ xk) , min
ˆ xk+1∈Xk+1
max
xk+1∈Xk+1 χk+1(xk+1, xk, ˆ
xk+1, ˆ xk) ωk+1(xk+1, ˆ xk+1, ok+1)γk+1(xk+1, ˆ xk+1) γn(xn, ˆ xn) , 1 We recursively define two parameters:
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MaxiHMM algorithm
General overview
20
γk(xk, ˆ xk) , min
ˆ xk+1∈Xk+1
max
xk+1∈Xk+1 χk+1(xk+1, xk, ˆ
xk+1, ˆ xk) ωk+1(xk+1, ˆ xk+1, ok+1)γk+1(xk+1, ˆ xk+1) γn(xn, ˆ xn) , 1 … and … δk(xk, ˆ x1:k) , max
xk−1∈Xk−1 χk(xk, xk−1, ˆ
xk, ˆ xk−1) ωk(xk, ˆ xk, ok)δk−1(xk−1, ˆ x1:k−1) δ1(x1, ˆ x1) , χ1(x1, ˆ x1)ω(x1, ˆ x1, o1) We recursively define two parameters:
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MaxiHMM algorithm
General overview
21
We want to be able to check whether: (8ˆ x1:n 2 optmax(X1:n)) ˆ x1:k 6= ˆ x∗
1:k
some initial segment
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MaxiHMM algorithm
General overview
21
We want to be able to check whether: (8ˆ x1:n 2 optmax(X1:n)) ˆ x1:k 6= ˆ x∗
1:k
some initial segment max
xk∈Xk δk(xk, ˆ
x∗
1:k)γk(xk, ˆ
x∗
k) > 1
⇐
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MaxiHMM algorithm
General overview
22
MaxiHMM algorithm
SLIDE 55
Cedric De Boom
MaxiHMM algorithm
General overview
23
SLIDE 56
MaxiHMM algorithm
Properties
24
MaxiHMM algorithm
Recursive
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MaxiHMM algorithm
Properties
24
MaxiHMM algorithm
Recursive Heuristic complexity O(Snm2) S: number of solutions n: length of the sequence m: size of state space
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MaxiHMM algorithm
Properties
25
MaxiHMM algorithm
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MaxiHMM algorithm
Applications
26
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OCR
Optical Character Recognition
27
Willem, die vele bouke maecte, Daer hi dicken omme waecte, Hem vernoyde so haerde Dat die avonture van Reynaerde In Dietsche onghemaket bleven
- Die Willem niet hevet vulscreven -
Dat hi die vijte van Reynaerde soucken Ende hise na den Walschen boucken In Dietsche dus hevet begonnen.
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OCR
Hidden Markov models
28
Willem, die vele bouke maecte, Daer hi dicken omme waecte, Hem vernoyde so haerde Dat die avonture van Reynaerde In Dietsche onghemaket bleven
- Die Willem niet hevet vulscreven -
Dat hi die vijte van Reynaerde soucken Ende hise na den Walschen boucken In Dietsche dus hevet begonnen.
D ?
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OCR
Hidden Markov models
29
M ? M ? E ? D ?
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OCR
Hidden Markov models
30
M M M M E E D O
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31
Van den Vos Reynaerde
OCR
Textual data
Medieval Dutch (13th century)
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31
Van den Vos Reynaerde
OCR
Textual data
Medieval Dutch (13th century) Often little data on hand
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31
Van den Vos Reynaerde
OCR
Textual data
Medieval Dutch (13th century) Often little data on hand Building very accurate precise models is impossible
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31
Van den Vos Reynaerde
OCR
Textual data
Medieval Dutch (13th century) Often little data on hand Building very accurate precise models is impossible Use imprecise models
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OCR
Results
32
BEWAENT
read as
BEWAEHT 3-best Viterbi solutions
BEWAENT BEWAERT BEWAEHT
MaxiHMM solutions
BEWAENT
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OCR
Results
33
UP
read as
UF 3-best Viterbi solutions
VE OF UP
MaxiHMM solutions
OF UF UP VE
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OCR
Results
33
UP
read as
UF 3-best Viterbi solutions
VE OF UP
MaxiHMM solutions
OF UF UP VE
Multiple solutions are typically returned in cases where the Viterbi algorithm fails to return the correct result
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OCR
Results
34
COMT
read as
COMT 3-best Viterbi solutions
CONT COMI CONI
MaxiHMM solutions
COMI COMT CONT
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OCR
Results
34
COMT
read as
COMT 3-best Viterbi solutions
CONT COMI CONI
MaxiHMM solutions
COMI COMT CONT
Imprecision takes care of problems with small data sets
SLIDE 73
Questions?