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observational data using information about their probabilistic correlations and
assumptions about their causal structure. These assumptions are: 1. Causal
Markov Condition (CMC), 2. Faithfulness Condition (FC) and 3. Causal Suf-
ficiency Condition (CSC). According to CMC, a variable X is independent of
every other variable (except X's effects) conditional on all of its direct causes. A
is a direct cause of X if A exerts a causal influence on X that is not mediated
by any other variables in a given graph. The FC says that all the conditional
independencies in the graph are only implied by CMC, while CSC states that
all common causes of measured variables are explicitly included in the model.
Since these theorists are interested in teasing out reliable causal relationships
from data, they would like to make sure that those probability distributions are
faithful in representing causal relations in them.
One reason for SP being causal, according to this account, is that (for the
example given in Table 1) applying to the school has a causal dimension in-
volving causal dependencies between “gender” and “acceptance rate”. More fe-
male students chose to apply to the departments where rates of acceptance are
significantly lower,
causing
their overall rates of acceptance to be lower in the
combined population. Similarly, with regard to Simpson's own example in the
literature, Spirtes et al. write, “[t]he question is what
causal dependencies
can
produce such a table, and that question is properly known as “Simpson's para-
dox”.” [15, p. 40].
4 Counter-Example to the Causal Account
It is not easy to come up with an example which precludes invoking some sort of
appeal to “causal intuitions” with regard to SP. But what follows is, we think,
such a case. It tests in a crucial way the persuasiveness of the causal accounts.
8
Tabl e 5.
Simpson's Paradox (Marble Example)
Rates of red
Marbles
Marbles
of two
sizes
Bag 1
Bag 2
Overall rates for
red marbles
Red
Blue
Red
Blue
Bag 1
Bag 2
Big
marbles
180
20
100
200
90%
33%
56%
Small
Marbles
480
120
10
90
80%
10%
70%
Suppose, as in Table 5, we have two bags of marbles, all of which are either big or
small, and red or blue. Suppose in each bag, the proportion of big marbles that
are red is greater than the portion of small marbles that are red (Bag 1: 90%
>
80% and Bag 2: 33%
>
10%). Now suppose we pour all the marbles from both
bags into a box. Would we expect the portion of big marbles in the box that
are red to be greater than the portion of small marbles in the box that are red?
Most of us would be surprised to find that our usual expectation is incorrect.
8
This counter-example is due to John G. Bennett.
