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RELEVANT LEMMAS
The following result is used in
Section 5.4:
Lemma 1
If a joint distribution satisfies constraints
,
then it satisfies
constraints
.
To prove this result, take
in the following lemma.
Lemma 2
Consider a joint distribution that satisfies constraints
,
and for every node
Xi,
is a subset
of
that does not overlap
with the parents of
Xi. Then the following constraints are
also satisfied:
|
(6) |
Sketch of proof.
Consider an arbitrary joint distribution satisfying constraints
.
Denote the set
by
.
Obtain by marginalization the distribution of
,
.
Select all constraints that are repetitions of a single original
constraint for fixed
.
These constraints are all
identical, except that values of
and
vary across constraints. Multiply every one of these constraints
by the appropriate value of
,
and
add all constraints that refer to a particular value of
;
constraints (6) are
then obtained after algebraic manipulations.
Fabio Gagliardi Cozman
1998-07-03