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the probability distributions satisfy the Markov condition with their respective
causal structures. So, Definition 10.1 could be used for cases involving cyclic
causal structures. However, for the purposes of this essay, I will assume that causal
structures are acyclic and that the Markov condition is in force. Finally, R (
Π
*) is
identifiable given R (
), P , P *, and D means in effect that it is a logical consequence
of these premises together with the Markov condition (see Pearl 2000 , p. 77). It may
seem surprising that extrapolation, which seems to be a type of inductive inference,
would be defined in terms of logical consequence (of R (
Π
), P , P *, and
D ). However, inductive inferences would be inevitably be involved in learning
R ( Π ), P , P *, and D , so Definition 10.1 is not a covert expression of a deductivist
perspective on scientific methodology. Moreover, stating the definition in this
manner has the advantage of allowing for proofs about conditions in which extrap-
olation is and is not possible.
Definition 10.1 is similar to Pearl and Bareinboim's ( 2011 , p. 8) definition of
transportability. The main difference is that in Definition 10.1 here, extrapolation is
premised on R (
*) from R (
Π
Π
), P , P *, and D , while in Pearl and Bareinboim's definition of
transportability, it is premised on R (
Π
), P , P *, G , and G *, where G and G * are the
Π
causal graphs for populations
*, respectively. That is, Pearl and
Bareinboim's definition is designed for cases in which the both causal structure
and probability distribution are known for the target prior to the extrapolation. This
is a rather restrictive assumption, as it entails that Pearl and Bareinboim's definition
would not be useful for cases in which the causal structure in the target is not fully
known. For example, animal extrapolation in toxicology often occurs in a back-
ground in which there is substantial uncertainty as to what adverse effect, if any, the
chemical has in humans. In such cases, causal structure is part of what one wishes to
learn by the extrapolation. In contrast, premising extrapolation on the selection
diagram does not presume that the causal structure in the target is fully known,
since a selection diagram indicates uncertainties about causal relations in the target
population. Moreover, the proofs of the main theorems in Pearl and Bareinboim
( 2011 ) depend on knowing the selection diagram, not the causal structure of the
target.
and
Π
Π
3.2 Making Adjustments
Since the model and target typically differ in some causally relevant respects,
extrapolation usually requires making some adjustments. In other words, when
extrapolation is not direct, some adjustment must be made to R (
) in order to
Π
infer R (
*). Pearl and Bareinboim ( 2011 ) prove several theorems about how this
can be done. The most general of these is their Theorem 3, which I restate below
(but with “can be extrapolated” substituted for “is transportable” for consistency
with the foregoing section):
Π
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