Biomedical Engineering Reference
In-Depth Information
C
0
= C
1
= ::: = C
m
= 1. As before, the target parameter of interest can
then be expressed as
(P
O;U
) = E
Y
m+1
Y
m+1
where P
O;U
is the joint distribution of O and the vector of exogenous errors
U = (U
M
0
;:::;U
M
m+1
;U
A
;U
Y
0
;:::;U
Y
m+1
;U
0
;:::;U
m
;U
C
0
;:::;U
C
m
) :
For given a 2f0; 1g, set L(0) = M
0
, A(0) = a, L(1) = (Y
0
;
0
), A(1) = C
0
,
L(k) = (M
k1
;Y
k1
;
k1
)
and A(k) = C
k1
; k = 2; 3;:::;m;
L(m + 1) = (M
m
;Y
m
), A(m + 1) = (
m
;C
m
), L(m + 2) = M
m+1
, and
Y = Y
m+1
. Denote by L(k) the vector (L(0);L(1);:::;L(k)), and similarly,
dene
A(k) = (A(0);A(1);:::;A(k)) and v(k) = (v(0);v(1);:::;v(k)) for any
xed vector v = (v(0);v(1);:::;v(m + 1)).
As in Section 8.3, to obtain an identity relating the distribution of the
observed data to the causal effect of interest, we define the following iterative
sequence of conditional expectations:
E
YjL(m + 1); A(m + 1) = a
0;a
(m + 1)
Q
a
m+2
=
E
Q
a
m+2
jL(m); A(m) = a
0;a
(m)
Q
a
m+1
=
E
Q
a
m+1
jL(m 1); A(m 1) = a
0;a
(m 1)
Q
a
m
=
E
Q
2
jL(0);A(0) = a
0;a
(0)
Q
1
=
E
Q
1
;
Q
0
=
where a
0;a
(m + 1) = (a; 1; 1;:::; 1; (1; 1)). Suppose that all intervention nodes
are sequentially randomized in the sense that (i) for each k 2 f1;:::;m + 1g
and each a 2f0; 1g A(k) and Y
a
are independent given L(k) and A(k1) =
a
0;a
(k 1), and (ii) for each a 2 f0; 1g A(0) and Y
a
are independent given
L(0). Suppose further that the interventions considered satisfy the positivity
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