Queries Outside a Clique
February 8, 2008
Consider a query
P
(
Y

e
) where the variables
Y
are not present together in
a single clique. One naive approach is to construct a clique tree where we force
one of the cliques to contain
Y
. However, this approach forces us to tailor our
clique tree to different queries, negating many of its advantages. An alternative
approach is to perform variable elimination over a calibrated clique tree.
Example 0.1:
Consider a clique tree obtained from a chain structured network
A—B—C—D, with an appropriate set of factors
F
. Our clique tree in this case
would have three cliques
C
1
=
{
A, B
}
, C
2
=
{
B, C
}
, and
C
3
=
{
C, D
}
. Assume
that we have calibrated the clique tree, so that the clique potentials are equal
to the marginals of
P
F
as in Corollary 11.2.7.
Assume that we now want to
compute the probability
P
F
(
B, D
). If the entire clique tree is calibrated, so is
any (connected) subtree
T
. Letting
T
consist of the two cliques
C
2
and
C
3
,
it follows from Theorem 11.2.11 that:
P
F
(
B, C, D
) =
π
T
.
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 Winter '09
 Conditional Probability, Probability, The Clique series, clique tree

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