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Inference in Bayesian network refers to answering probabilistic queries regard-
ing the (random) variables present in the model. For instance, when a subset
of the variables in the model are observed - the so-called
evidence
variables -
then inference is used in order to update the probabilistic information about the
remaining variables. In other words, once the state of some variables became
known, the probabilities of the other variables (might) change and distributions
need to be coherently revised in order to maintain the integrity of the knowledge
encoded by the model. This process of computing the posterior distribution of
variables given evidence is called
probabilistic inference
in which essentially three
computation rules from probability theory are used: marginalisation, condition-
ing, and Bayes' theorem.
Next we present two specific Bayesian network architectures where the notion
of independence is used to represent compactly the joint probability distribution.
4.2 Naive Bayes
A naive Bayesian (NB) classifier is a very popular special case of a Bayesian
network. These network models have a central node, called the
class
node
Cl
,
and nodes representing feature variables
F
i
,
i
=1
,...,n
,
n
≥
1; the arcs are
going from the class node
Cl
to each of the feature nodes
F
i
[26]. A schematic
representation of a naive Bayes model is shown in Figure 2.
Cl
F
1
F
2
···
F
n
Fig. 2.
A schematic representation of a naive Bayes model
The conditional independencies that explain the adjective 'naive' determine
that each feature
F
i
is conditionally independent of every other feature
F
j
,for
j
=
i
, given the class variable
Cl
.Inotherwords,
P
(
F
i
|
Cl,F
j
)=
P
(
F
i
|
Cl
)
.
The joint probability distribution underlying the model is:
n
P
(
Cl,F
1
,...,F
n
)=
P
(
Cl
)
P
(
F
i
|
Cl
)
,
i
=1
and the conditional distribution over the class variable
Cl
canbeexpressedas
n
P
(
Cl
|
F
1
,...,F
n
)
∝
P
(
Cl
)
P
(
F
i
|
Cl
)
.
i
=1
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