Databases Reference
In-Depth Information
of
Y
.
Each variable is conditionally independent of its nondescendants in the graph, given
its parents.
Figure 9.1 is a simple belief network, adapted from Russell, Binder, Koller, and
Kanazawa [RBKK95] for six Boolean variables. The arcs in Figure 9.1(a) allow a rep-
resentation of causal knowledge. For example, having lung cancer is influenced by a
person's family history of lung cancer, as well as whether or not the person is a smoker.
Note that the variable
PositiveXRay
is independent of whether the patient has a family
history of lung cancer or is a smoker, given that we know the patient has lung cancer. In
other words, once we know the outcome of the variable
LungCancer
, then the variables
FamilyHistory
and
Smoker
do not provide any additional information regarding
Posi-
tiveXRay
. The arcs also show that the variable
LungCancer
is conditionally independent
of
Emphysema
, given its parents,
FamilyHistory
and
Smoker
.
A belief network has one
conditional probability table (CPT)
for each variable.
The CPT for a variable
Y
specifies the conditional distribution
P
.
Y
j
Parents
.
Y
//
, where
Parents
are the parents of
Y
. Figure 9.1(b) shows a CPT for the variable
LungCancer
.
The conditional probability for each known value of
LungCancer
is given for each pos-
sible combination of the values of its parents. For instance, from the upper leftmost and
bottom rightmost entries, respectively, we see that
.
Y
/
P
.
LungCancer
D
yes
j
FamilyHistory
D
yes
,
Smoker
D
yes
/D 0.8
P
.
LungCancer
D
no
j
FamilyHistory
D
no
,
Smoker
D
no
/D 0.9.
,
Y
n
,
respectively. Recall that each variable is conditionally independent of its nondescen-
dants in the network graph, given its parents. This allows the network to provide a
complete representation of the existing joint probability distribution with the following
equation:
Let
X
D.
x
1
,
:::
,
x
n
/
be a data tuple described by the variables or attributes
Y
1
,
:::
n
Y
P
.
x
1
,
:::
,
x
n
/D
P
.
x
i
j
Parents
.
Y
i
//
,
(9.1)
i
D1
where
P
.
x
1
,
:::
,
x
n
/
is the probability of a particular combination of values of
X
, and the
values for
P
correspond to the entries in the CPT for
Y
i
.
A node within the network can be selected as an “output” node, representing a class
label attribute. There may be more than one output node. Various algorithms for infer-
ence and learning can be applied to the network. Rather than returning a single class
label, the classification process can return a probability distribution that gives the prob-
ability of each class. Belief networks can be used to answer probability of evidence
queries (e.g., what is the probability that an individual will have
LungCancer
, given that
they have both
PositiveXRay
and
Dyspnea
) and most probable explanation queries (e.g.,
which group of the population is most likely to have both
PositiveXRay
and
Dyspnea
).
Belief networks have been used to model a number of well-known problems. One
example is genetic linkage analysis (e.g., the mapping of genes onto a chromosome). By
casting the gene linkage problem in terms of inference on Bayesian networks, and using
.
x
i
j
Parents
.
Y
i
//
