Biomedical Engineering Reference
In-Depth Information
A feature
f
may be any real-valued function of the state. The focus of this chapter
is on binary features,
f
j
2f
0; 1
g
. Thus, if we translate from the potential-function
form, the model will have one feature for each possible state x
k
of each clique and
its weight will be log..x
fkg
/. This representation is exponential in the size of the
cliques, but, however, we can specify a much smaller number of features in a more
compact representation than the potential-function form. This is the case when large
cliques are present and MLNs try to take advantage of this.
A first-order knowledge base (KB) can be considered as a set of hard constraints
on a set of possible worlds: if a world violates a single formula, it will have zero
probability. The idea in Markov logic is to soften these constraints: when a world
violates a formula in the KB it will be less probable, but not impossible. The fewer
formulas a world will violate, the more probable it will be. Each formula has an
attached weight that represents how hard a constraint it is. A higher weight of a
formula means there is a greater difference in log probability between a world that
satisfies that formula and one that does not, all other things being equal.
An MLN [
23
] T is a set of pairs .
F
i
I
w
i
/,where
F
i
is a formula in first-
order logic (FOL) and
w
i
is a real number. Together with a finite set of constants
C
Df
c
1
;c
2
;:::;c
p
g
it defines a MN
M
T
I
C
as follows:
1.
There is a binary node in
M
T
I
C
for each possible grounding of each predicate
appearing in T and the value of the node is 1 if the ground predicate is true, and
0otherwise.
2.
There is one feature in
M
T
I
C
for each possible grounding of each formula
F
i
in T
and the value of this feature is 1 if the ground formula is true, and 0 otherwise.
The weight
w
i
of the formula
F
i
in T becomes the weight of this feature. There
is an edge between two nodes of
M
T
I
C
if and only if the corresponding ground
predicates appear together in at least one grounding of a formula in T.
An MLN can be viewed as a template for constructing MNs. The probability
distribution over possible worlds
x
defined by the ground MN
M
T
I
C
is given by:
exp
F
X
i D1
!
1
Z
P.X
D
x/
D
w
i
n
i
.x/
;
(4.4)
where
F
is the number of formulas in T and
n
i
.x/ is the number of true groundings
of
F
i
in
x
. When formula weights increase, an MLN will resemble a purely logical
KB, and in the limit of all infinite weights it becomes equivalent to it.
The focus of this chapter is on MLNs with function-free clauses assuming
domain closure in order to ensure that the MNs generated will be finite. In this case,
the groundings of a formula are formed by replacing the variables with constants in
all possible ways.
A simple example of a first-order KB is given in Fig.
4.1
. Statements in FOL are
always true. The following FOL formulas state that if someone drinks heavily, he
will have an accident, and that if two people are friends, they either both drink or
both don't drink.
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