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(85% of people with lung cancer have positive CT scan)
P(positive CT scan | ~cancer) = 0.1%
(0.1% of people without lung cancer have positive CT scan)
And plug in the above data into the above Bayes' theorem, we will get:
P(cancer | positive CT scan) = 85%* 0.2% / (85%*0.2%+0.1%*99.8%)
= 0.0017 / (0.0017+0.001)
= 0.0017 / 0.0027
= 0.63
This result tells use that the CT scan test will shift our belief in positive direction, The
percentage increase is 62.8%. These results support our claim 1.
Note that the x-ray test and CT scan test have the same positive cause-effect probability rate
but different false alarm rate. In x-ray test, the false alarm probability P(positive x-ray |
~cancer) is 6%, while in CT scan test, the false alarm probability P(positive CT scan |
~cancer) is only 0.1%. Here is an example that the
low false alarm probability is the
dominate factor
in deciding the strength of evidence.
6.4 The logarithmical representation of evidence degrees
In the previous section, we used the ratio of two conditional probabilities as the strength
measurement. Under our abstract view of reasoning model in Figure 4, evidences are used
to distort the world space. As indicated in that figure, reasoning is the process of make a
judgment using the knowledge (embedded in the conditionals) based on the evidence (the
right side of “|” on the left side of Formula 1) presented. One thing to point out is that our
abstract reasoning model can be applied to multiple evidences.
To capture the essence of the low-level reasoning in situations with multiple evidences, we
can use a tool in mathematics called ratio and the concept in statistics called odds. Also, the
use of these tools will make reasoning in situations that have multiple evidences easier.
Odds can capture the same information as probability. In statistics, odds are defined as the
ratio of the probability of an event's occurring to the probability of its not occurring. The
reasoning of solving the problem in Example 2 using the odds concept will be like this: in
the initial world, the lung cancer rate is 0.2%. Thus, 2 out of 1000 people have lung cancer,
and 998 people out of 1000 people do not have lung cancer. Using odds, we define the event
of interest as a person has lung cancer vs. a person has no cancer. So the 0.2% cancer rate can
be expressed as the following odds:
2:998
And the evidence strengths of the two tests x-ray, and CT scan can be expressed in odds
notation as:
14.17:1
(get from 0.85/0.06)
