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structure
G
and parameters
, under one of the distributional assumptions detailed
in Sect.
2.2.4
. Subsequently, we want to investigate the effects of a new piece of
evidence
E
on the distribution of
X
using the knowledge encoded in
B
,thatis,to
investigate the posterior distribution P
Θ
.
The approaches used for this kind of analysis vary depending on the nature of
E
and on the nature of information we are interested in. The two most common kinds
of evidence are as follows:
•
(
X
|
E
,
B
)=
P
(
X
|
E
,
G
,
Θ
)
Hard evidence
, an instantiation of one or more variables in the network. In other
words,
=
X
i
1
=
e
k
,
E
e
1
,
X
i
2
=
e
2
, ...,
X
i
k
=
i
1
,...,
i
k
∈{
1
,...
n
},
(4.1)
which ranges from the value of a single variable
X
i
to a complete specification for
X
. Such an instantiation may come, for instance, from a new (partial or complete)
observation recorded after the Bayesian network was learned.
•
Soft evidence
, a new distribution for one or more variables in the network. Since
both the network structure and the distributional assumptions are treated as fixed,
soft evidence is usually specified as a new set of parameters,
X
i
1
∼
(
Θ
X
i
1
)
,
E
=
X
i
2
∼
(
Θ
X
i
2
)
, ...,
X
i
k
∼
(
Θ
X
i
k
)
.
(4.2)
This new distribution may be, for instance, the null distribution in a hypothesis
testing problem.
As far as queries are concerned, we will focus on
conditional probability queries
(CPQ) and
maximum a posteriori
(MAP) queries, also known as
most probable
explanation
(MPE) queries. Both apply mainly to hard evidence, even though they
can be used in combination with soft evidence.
Conditional probability queries are concerned with the distribution of a subset of
variables
Q
=
{
X
j
1
,...,
X
j
l
}
given some hard evidence
E
on another set
X
i
1
,...,
X
i
k
of variables in
X
. The two sets of variables are usually assumed to be disjoint. In
discrete Bayesian networks, this distribution is computed as the posterior probability
CPQ
(
Q
|
E
,
B
)=
P
(
Q
|
E
,
G
,
Θ
)=
P
(
X
j
1
,...,
X
j
l
|
E
,
G
,
Θ
)
,
(4.3)
which is the marginal posterior probability distribution of
Q
, i.e.,
P
P
(
Q
|
E
,
G
,
Θ
)=
(
X
|
E
,
G
,
Θ
)
d
(
X
\
Q
)
.
(4.4)
In Gaussian Bayesian networks, likewise,
CPQ
(
Q
|
E
,
B
)=
f
(
Q
|
E
,
G
,
Θ
)=
f
(
X
|
E
,
G
,
Θ
)
d
(
X
\
Q
)
.
(4.5)
This class of queries has many useful applications due to their versatility. For in-
stance, conditional probability queries can be used to assess the interplay between
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