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As shown in Figure 3, the application of Bayes' theorem is the mapping from one space to
another space. In the initial world, the probability of a person in the sample is healthy is
99.8% while the probability of having the lung cancer is 0.2%. The first application of the
Bayes' theorem has two distortions: one distorts the probability of having cancer, P(cancer),
to the probability of both having cancer and being positive for x-ray test, P(positive x-ray &
cancer), (the distorting leverage/filter is the conditional probability P(positive x-ray |
cancer)), the other distorts the probability of being healthy, P(healthy), to the probability of
being positive for x-ray and being healthy, P(positive x-ray & healthy), (the distorting
leverage/filter is the conditional probability P(positive x-ray | healthy)). In the new
alternate universe, though the number of people who have cancer to be included is almost
the same as in the initial world (from 20 in the initial world to 17 in the first mapped world),
the number of people who are healthy to be included is greatly reduced (from 9980 in the
initial world to 599 in the first mapped world). Thus, when we try to answer the question of
“the probability of a person having lung cancer given that he has a positive x-ray” by
dividing the number of people with cancer by the total number of people with positive x-
ray, we will get a much higher probability. In other words, the mapping altered our
assessment. The mapping reflects the effect of the evidence “positive x-ray” in shifting our
judgment of deciding whether a person has lung cancer.
Fig. 3. We apply Bayes' theorem twice for two tests; each application of Bayes' theorem can
be viewed as a mapping
One thing to point out, the x-ray test will not affect the actual probability of a person has
cancer (otherwise no one will take the test). However, the test will affect our beliefs. A
positive x-ray is a membership test. If the test is positive, it will eliminate many more people
without lung cancer than people with the cancer. The number of people without cancer is
reduced by a factor of more than 16, from 9980 to 599, while the number of people with
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