Biology Reference
In-Depth Information
Appendix 2: Causal Effect Cannot Be Extrapolated in Fig.
10.4
The selection diagram in Fig.
10.4
adds a selection variable pointing directly into
E
and thereby prevents deriving a transport formula by summing over
A
. Since
P
ð
c
j
do
ð
u
ÞÞ
cannot be directly extrapolated, Pearl and Bareinboim's Theorem 3 tells us
that
P
can be extrapolated given the selection diagram in Fig.
10.4
only if
one of the following can be reduced by means of the
do
-calculus to a formula in
which a
do
-operator and an
s
never occur in the same probability:
1.
P
e
P
ð
c
j
do
ð
u
ÞÞ
ð
c
j
do
ð
u
Þ
;
e
s
Þ
P
ð
e
j
do
ð
u
Þ
;
s
Þ
;
2.
e
;
a
P
ð
c
j
do
ð
u
Þ
;
e
a
s
Þ
P
ð
e
a
j
do
ð
u
Þ
;
s
Þ
;
;
;
in this manner due to the selection
variable pointing directly into
E
and the confounding arc linking
U
and
E
. The same
reasons preclude reducing
P
In 1, it is not possible to reduce
P
ð
e
j
do
ð
u
Þ
;
s
Þ
ð
e
a
j
do
ð
u
Þ
;
s
Þ
in 2.
;
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