HMM, MEMM, CRF メモ

P(X, Y ) = P(Y )P(X|Y ) =
Y
t
P(Yt|Yt 1)P(Xt|Yt)
Yt 1 Yt Yt+1
Xt+1XtXt 1
HMM
X Y
1
HMM
Viterbi
P(X, Y )

P(X) =
X
Y
P(X, Y )
arg max
Y
P(Y |X) = arg max
Y
P(X, Y )
P(X)
= arg max
Y
P(X, Y )
HMM
X
→
X Y
→ Viterbi
EM
→ Baum-Welch
※

t(x, s2) = P(x1, x2, Y2 = s2)
|S| = 3 4
sS
s1
s2
s3
sE
YS Y1 Y2 Y3 Y4 YE
X1 X2 X3 X4
s1
s2
s3
s1
s2
s3
s1
s2
s3
s2
x1 x2

3(x, s1) = P(x4, Y3 = s1)
|S| = 3 4
sS
s1
s2
s3
sE
YS Y1 Y2 Y3 Y4 YE
X1 X2 X3 X4
s1
s2
s3
s1
s2
s3
s1
s2
s3
x4
s1

ˆy = arg max
y
P(y|x)
Viterbi
t
1. sS
2. t = 1, ..., T – 1
3. sE
=
4. t = T – 1, ..., 1
Viterbi
←
x = x1 · · · xT x1 · · · xt si
O(|S|2
T)
t(x, si) = max
y1···yt 1
P(x1, · · · , xt, y1, · · · , yt 1, Yt = si)
1(x, si) = P(si|sS)P(x1|si)
t+1(x, si) = max
sj
[ t(x, sj)P(si|sj)] P(xt+1|si)
⇥t(x, si) = arg max
sj
[ t(x, sj)P(si|sj)]
max
y
P(x, y) = max
sj
[ T (x, sj)] P(sE|sj)
ˆyT = arg max
sj
T (x, sj)
ˆyt = t(x, ˆyt+1)
1

arg max
sj
4(x, sj) = s1
t(x, sj)
Viterbi
|S| = 3 4
1 2 3
s1 s1 s3 s1
s2 s1 s2 s3
s3 s2 s1 s1
sj
t
sS
s1
s2
s3
sE
YS Y1 Y2 Y3 Y4 YE
X1 X2 X3 X4
s1
s2
s3
s1
s2
s3
s1
s2
s3
x1 x2 x3 x4
sS s2 s3 s1 sEs1

✓ ¯✓ = (· · · , ¯i, · · · , ¯aij, · · · ,¯bik, · · · )
※O
✓ = ( 1, · · · , |S|, a11, · · · , a|S||S|, b11, · · · , b|S||O|)
t(si|✓) =
X
sj 2S
t(si, sj|✓) t = 1, · · · , T 1
¯⇥i = ¯P(si|sS, ✓) = 1(si|✓)
¯aij = ¯P(sj|si, ✓) =
PT 1
t=1 t(si, sj|✓)
PT 1
t=1 t(si|✓)
¯bik = ¯P(ok|si, ✓) =
P
t:xt=ok
t(si|✓)
PT
t=1 t(si|✓)

HMM
(features)
→
→
X Y
,
‘er’
HMM ‘er’
‘er’
, ‘er’ …

[[x = y]] =
(
1 if x = y
0 if x 6= y
MEMM
MEMM
X Y
※
( )
HMM
※
Yt 1 Yt Yt+1
Xt+1XtXt 1
Z(Xt, Yt 1)
Ps(Yt|Xt) =
1
Z(Xt, s)
exp
X
a
afa(Xt, Yt)
!
P(Y |X) =
Y
t
Ps(Yt|Xt)[[Yt 1 = s]]
s ME

f<begins-with-number,question>
= 1
features 2
Usenet FAQ
begins-with-number
begins-with-ordinal
begins-with-punctuation
begins-with-question-word
begins-with-subject
blank
contains-alphanum
contains-bracketed-number
contains-http
contains-non-space
contains-number
contains-pipe
contains-question-mark
contains-question-word
ends-with-question-mark
first-alpha-is-capitalized
indented
indented-1-to-4
indented-5-to-10
more-than-one-third-space
only-punctuation
prev-is-blank
prev-begins-with-ordinal
shorter-than-30
Xt 1
head, question,
answer, tail
t
question
f<b,s>(Xt, Yt) =
(
1 if b(Xt) is true and Yt = s
0 otherwise
※ 1
t question

(x(1)
, y(1)
), · · · , (x(n)
, y(n)
)
MEMM
Generalized Iterative Scaling
1. o, s C
※
2.
3. x
4.
5. 3, 4 s ME
fc(o, s) 0 8
o, sfc(o, s) = C
X
a
fa(o, s)
C =
X
a
fa(o, s)
˜E[fa] =
1
n
nX
i=1
1
m
(i)
s
X
t:yt 1=s
fa(x
(i)
t , y
(i)
t )
E[fa] =
1
n
nX
i=1
1
m
(i)
s
X
t:yt 1=s
X
y2S
Ps(y|xt, )fa(x
(i)
t , y)
new
a = a +
1
C
log
˜E[fa]
E[fa]
!

MEMM
s0
s5
s4
s6
s3
s1
s2
0.65
0.35
1
0.5
0.5 1
1
1 s0 → s1 → s2 → s3 : 0.325
x1 x2 x3 s0 s1 s2 s3 s0 s1 s4 s3
s0 s5 s6 s3
ME
x1 x2 x3
1
1
s0 → s1 → s4 → s3 : 0.325
s0 → s5 → s6 → s3 : 0.35

P(Y |X) =
1
Z(X)
exp
0
@
X
t,i
ifi(Yt 1, Yt, X, t) +
X
t,j
µjgj(Yt, X, t)
1
A
Z(X)
X
CRF
CRF
MEMM
HMM
Yt 1 Yt Yt+1

log
Y
t
P(Yt|Yt 1)P(Xt|Yt)
!
=
X
t
{log P(Yt|Yt 1) + log P(Xt|Yt)}
=
X
t
8
<
:
X
<s,s0>
[[Yt 1 = s0
]][[Yt = s]] log P(s|s0
) +
X
<o,s>
[[Xt = o]][[Yt = s]] log P(o|s)
9
=
;
=
X
<s,s0>
log P(s|s0
)[[Yt 1 = s0
]][[Yt = s]] +
X
t,<o,s>
log P(o|s)[[Xt = o]][[Yt = s]]
P(Y |X) =
1
P(X)
exp
0
@
X
t,<o,s>
<o,s>f<o,s>(Yt 1, Yt, X, t) +
X
t,<s,s0>
µ<s,s0>g<s,s0>(Yt, X, t)
1
A
X
o
exp( <o,s>) = 1
X
s
exp(µ<s,s0>) = 1
HMM CRF
CRF
P(X, Y ) = P(Y )P(X|Y ) =
Y
t
P(Yt|Yt 1)P(Xt|Xt)
<o,s> f<o,s>(Yt 1, Yt, X, t) g<s,s0>(Yt, X, t)µ<s,s0>

ˆyT = arg max
sm
T (x, sm)
x skt k
1. sS
2. k = 1, ..., T – 1
3.
4. t = T – 1, ..., 1
Viterbi
←
ˆyt = k(x, ˆyt+1)
1(x, sl) = h1(sS, sl, x)
k+1(x, sl) = max
sm
[ k(x, sm) + hk+1(sm, sl, x)]
⇥k(x, sl) = arg max
sm
[ k(x, sm) + hk+1(sm, sl, x)]
k(x, sl) = max
y1···yk 1
"k 1X
t=1
ht(yt 1, yt, x) + hk(yk 1, sl, x)
#
ht(Yt 1, Yt, X) =
X
i
X
j
µjgj(Yt, X, t)
ˆy = arg max
y
P(Y |X) = arg max
y
2
4
X
t,i
X
t,j
µjgj(Yt, X, t)
3
5

(|S| + 2) × (|S| + 2)
|S| + 2
|S| + 2
sS
sm
sE
sl
Mt(X)
Mt(sl, sm|X) = exp ht(sl, sm, X)
↵t(X)
0(Y |X) =
(
1 if Y = sS
0 otherwise
T +1(Y |X) =
(
1 if Y = sE
0 otherwise
t(X)T
= t 1(X)T
Mt(X)
t(X)
t(X) = Mt+1(X) t+1(X)
sS sEMt(X)
↵t(X)
t(X)

C MEMM
(x(1)
, y(1)
), · · · , (x(n)
, y(n)
)
Generalized Iterative Scaling
1. C
※
2.
3. x
4.
5. 3, 4
C =
X
t,i
fi(yt 1, yt, x, t) +
X
t,j
gj(yt, x, t)
c(x, y) = C
X
t,i
fi(yt 1, yt, x, t)
X
t,j
gj(yt, x, t)
new
i = i +
1
C
log
˜E[fi]
E[fi]
!
E[fi] =
1
n
nX
k=1
X
t
X
sl,sm
t 1(sl|x(k)
, , µ)Mt(sl, sm|x(k)
, , µ)⇥t(sm|x(k)
, , µ)
Z(x(k)| , µ)
fi(sl, sm, x(k)
, t)
Z(x) =
Y
t
Mt(x)
!
sS ,sE
P(Yt 1 = sl, Yt = sm|x, , µ)
˜E[fi] =
1
n
nX
k=1
X
t
fi(y
(k)
t 1, y
(k)
t , x(k)
, t)
c(x(k)
, y(k)
) 0 k = 1, · · · , n

• , ( ). . , 1999.
• A. McCallum, D. Freitag, and F. Pereira. Maximum entropy Markov models for
information extraction and segmentation. Proc. ICML, pp. 591-598, 2000.
• J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: probabilistic
models for segmenting and labeling sequence data. Proc. ICML, pp. 282-289
, 2001.
• Charles Elkan. Log-Linear Models and Conditional Random Fields. Notes for a
tutorial at CIKM, 2008.
• Hanna M. Wallach. Conditional Random Fields: An Introduction. Technical Report
MS-CIS-04-21. Department of Computer and Information Science, University of
Pennsylvania, 2004.
• , , . Conditional Random Fields
. , pp. 89-96, 2004.
• http://www.dbl.k.hosei.ac.jp/~miurat/readings/Nov0706b.pdf
• http://www.dbl.k.hosei.ac.jp/~miurat/readings/Nov0706a.pdf

Conditional Random Fields
X
Y1 Y2 Y3
Y4 Y5
Y5 Y4 Y6
X
cf. PRML8

p(y|x) =
1
Z
Y
C
C(yC|x) =
1
Z
exp
X
C
E(yC|x)
!
chain-structed CRFs
Y1 Y2
Y2 Y1 Y3
PRML8
X
Y1 Y2 Y3 Y4 Y5 Y6

i yi C
CRF
E(yC|x) =
X
j
jfj(yi 1, yi, x, i)
X
C
E(yC|x) =
X
C
X
j
jfj(yi 1, yi, x, i)
=
X
j
jFj(y, x)
Fj(y|x) =
X
C
jfj(yi 1, yi, x, i)
p(y|x) =
1
Z
exp
0
@
X
j
jFj(y, x)
1
A

HMM, MEMM, CRF メモ

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