Biology Reference
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Fig. 10.1 A DAG
W
X
Y
Z
likely to be similar (Steel
2008
, pp. 88-89). Typically, the model does not resemble
the target in all relevant respects, and hence extrapolation can involve some
adjustment for to account for differences. A first step towards a conceptual frame-
work for representing extrapolation, then, is a means for compactly and
perspicuously representing causally relevant similarities and differences between
the model and target. Steel (
2008
) and Pearl and Bareinboim (
2011
) develop similar
representational frameworks for this purpose through an extension of directed
acyclic graphs (DAGs), which are often used to represent causal relationships. A
DAG consists of a set of nodes linked by arrows, for example, as in Fig.
10.1
.A
DAG is
directed
in that every line (or “edge”) has an arrowhead attached at one end,
and it is
acyclic
in that it does not contain loops, that is, a sequence of arrows
aligned head to tail that begin and end with the same node. If there is an arrow
pointing directly from node
X
to node
Y
, then
X
is said to be a
parent
of
Y
. For
example,
X
is a parent of
Y
in Fig.
10.1
, and
Y
is a parent of
Z
.If
Y
can be reached
from
X
by following a chain of arrows aligned head to tail, then
Y
is said to be a
descendant
of
X
.
6
Thus, in Fig.
10.1
,
X
,
Y
, and
Z
are all descendants of
W
. The nodes
in a DAG are normally taken to represent variables. When DAGs are interpreted
causally, parents are direct causes and descendants are effects. A Bayesian network
consists of a DAG together with a probability distribution that satisfies something
known as the Markov condition with respect to that DAG. The Markov condition
asserts that every variable in the DAG is probabilistically independent of its non-
descendants conditional on its parents. For example, in Fig.
10.1
,
Z
is probabilisti-
cally independent of
W
and
X
conditional on
Y
. Intuitively, this means that once the
value of the variable
Y
is known, learning the values of
W
and
X
provides no further
information concerning the value of
Z
. The graphical concept of d-separation
allows one to read off all of the independence relationships entailed by the Markov
condition for a DAG (see the Appendix
1
for the “Definition of d-Separation”). The
Markov condition will play an important role in the account of extrapolation
described below.
A simple extension of DAGs can be used to represent similarities and differences
between causal relationships in model and target populations. That extension
consists of adding additional variables to a DAG to represent differences between
model and target populations that may alter the relationships represented in the
DAG. For example, consider the diagram in Fig.
10.2
. This diagram represents a
6
This definition should be understood to entail that every node is descendant of itself (as any node
is trivially reachable from itself). This seemingly odd feature of the definition simplifies the
statement of the Markov condition.
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