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Fig. 6.1
The C-CRF model
6.1.2 Continuous Conditional Random Fields
In [ 7 ], Qin et al. also investigate the formalization of relational ranking, but using a
graphical model called continuous conditional random fields (C-CRF). C-CRF is a
graphical model, as depicted in Fig. 6.1 . In the conditioned undirected graph, a white
vertex represents a ranking score, a gray vertex represents a document, an edge
between two white vertexes represents the dependency between ranking scores, and
an edge between a gray vertex and a white vertex represents the dependency of a
ranking score on its document (content). (In principle a ranking score can depend on
all the documents of the query; here for ease of presentation we consider the simple
case in which it only depends on the corresponding document.)
Specifically, let
K 1
k =
1 be a set of real-valued feature functions defined
on document set x and label y j ( j =
{
g k (y j , x )
}
K 2
k
{ g k (y u ,y v , x ) }
1 ,...,m ), and
be a set of
=
1
real-valued feature functions defined on y u , y v , and x ( u, v
v ). Then
C-CRF is a conditional probability distribution with the following density function,
=
1 ,...,m , u
=
Z( x ) exp
j
β k g k (y u ,y v , x ) ,
K 1
K 2
1
Pr ( y
|
x )
=
α k g k (y j , x )
+
(6.7)
k =
1
u,v
k =
1
where α is a K 1 -dimensional parameter vector and β is a K 2 -dimensional parameter
vector, and Z( x ) is a normalization function,
exp
j
β k g k (y u ,y v , x ) d y .
K 1
K 2
Z( x )
=
α k g k (y j , x )
+
(6.8)
y
u,v
k =
1
k =
1
x (i) , y (i)
n
i
Given training data
{
}
1 , the Maximum Likelihood Estimation can be
=
used to estimate the parameters
of C-CRF. Specifically, the conditional log
likelihood of the training data with respect to the C-CRF model can be computed as
follows:
{
α, β
}
n
log Pr y (i)
; α, β .
x (i)
L(α, β) =
|
(6.9)
i
=
1
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