Mechanisms and Extrapolation in the Abortion-Crime Controversy - Mechanism and Causality in Biology and Economics - page 189

Fig. 10.3 The U -manipulated
version of the selection
diagram in Fig. 10.2

I

U E C

S

A

Theorem 3. Let D be the selection diagram characterizing two populations, and

Π

P ð *, and S the set of selection variables in D. The causal effect R ¼ y j do ð x ÞÞ

Π

can be extrapolated from * if and only if the expression P ( y | do ( x ) ,S ) is

reducible, using the rules of do-calculus, to an expression in which no do-operator
is conjoined with S. (Pearl and Bareinboim 2011 , p. 32)

to

Π Π

As noted above, the causal effect of X upon Y , represented by the formula P ,

is the probability distribution of Y conditional on an intervention on X .Aninterven-
tion or manipulation on X is an exogenous cause that targets X alone and eliminates
all other causes of X . To illustrate, consider the U -manipulated version of the
selection diagram from Fig. 10.2 (i.e., the version of that diagram resulting from an
intervention on the variable U ). As shown in Fig. 10.3 , this modified selection
diagram eliminates the confounding arcs and adds a new variable labeled I ,
representing the intervention, with an arrow pointing directly into U . The do-calculus
consists of three basic rules, derived from the Markov condition, for manipulating
probabilities that contain do -operators, such as do ( x )(Pearl 2000 , pp. 85-86).
To understand Theorem 3, first recall that the selection variables are assumed
to account for all differences between the probability distributions in the model
and target, so that P ðÞ¼

ð y j do ð x ÞÞ

Þ . Thus, any P -formula from which S is eliminable
can be directly extrapolated from the model to the target, since in that case
P ðÞ¼

P ðj s

P ðÞ

, any P -formula
without a do -operator can be derived from the probability distribution of the target.
In contrast, since no experimental manipulation was performed in the target popu-
lation and selection variables are unmeasured, P probabilities with both an irreduc-
ible do ( u ) and s are unidentifiable. Consider, then, how Theorem 3 applies to the
selection diagram in Fig. 10.2 , when P

P ðj s Þ¼ P ðÞ . On the other hand, since P ðj s Þ¼

ð c j do ð u ÞÞ is the causal effect to be

extrapolated. This proceeds as follows:

P ð c j do ð u ÞÞ¼ P ð c j do ð u Þ ; s Þ

X

¼ P ð c j do ð u Þ ; a s Þ P ð a j do ð u Þ ; s Þ

;

a

X

¼ P ð c j do ð u Þ ; a Þ P ð a j do ð u Þ ; s Þ

a

X

¼ P ð c j do ð u Þ ; a Þ P ð a j s Þ

a

X

P ð

¼ P ð c j do ð u Þ ; a Þ a Þ

a