. No other edges are needed. (It should be noted that if we had chosen to direct the edge
between
J
and
M
as , then we should added the edge
instead of
(
3.2
.
First, order the variables according to their topological order in
G.
Without loss of generality, we
rename the variables according to their topological order: ,where is the variable to be marginalized.
Note that using the above order, running the minimal Imap algorithm will exactly result in
G
due to
the local Markov assumption. Second, add these variables to using the order above, while skipping ,
and use the independencies that can be read from
G
about
P
to select a parent set for each variable.
For the variables , adding each of them to
G0
results in the same parent set as in
G
: that is, . Where
and
is the parent set of variable
X
in
G
and
respectively. In fact, we can generalize the above
relationship for any variable
such that . That is, . To see why this is true for , if was not selected as
a parent for
then , moreover . Since skipping
will not change the previous assertions, the parent set
of
will remain the same in
as it still satisfies the local Markov assumption. Therefore, the only
variables that will be affected by skipping
are those variables that has
as a parent in
G
, i.e., the
children on
in
G
.
For e ach child of, we ne e d to find a n ew p arent set. Let’s consider one of thes e children a
Must retain its old parent set in
G
, other than, since from
G
, we know that. Therefore, we just need
to replace
with a set of variables that acts as a surrogate for
in
parenting
. Using the local Markov
assumption,
blocks some information flow to , therefore these surrogate variables must block the
same set of paths while themselves cannot be dseparated from
when
is not observed. Moreover,
these variables must appear in the topological order before. For each parent of
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '13
 Dr.ZAre
 Probability theory, Pearson productmoment correlation coefficient, active trail, G. Therefore, G. Moreover, Soheila Molaei

Click to edit the document details