Information Technology Reference
In-Depth Information
cancer is reduced only from 20 to 17. Thus, the proportion of 17 within 616 (the total number
of people with positive x-ray) is much larger than the proportion of 20 within 10,000.
5.2 Conditional probabilities play the role as shifters
From Example 2, you may have already seen the role played by the two conditional
probabilities: P(positive x-ray | cancer) and P(positive x-ray | healthy). They are the
shifters: P(positive x-ray | cancer) shifts our view positively and P(positive x-ray | healthy)
shifts our view negatively. In other words, large value of P(positive x-ray | cancer) will
increase our confidence in predicting a person has cancer given that he has a positive test.
On the other hand, small value of P(positive x-ray | healthy) will increase our confidence in
predicting a person has cancer given that he has a positive test. The quality measurement of
a test in altering our view to the world is the inter-play of these two conditional
probabilities. They map the number of cancerous people and the number of healthy people
in one world into another world. Their ratio can be used as a measurement of effectiveness
for a test to be evidence.
We will show later that for a test to be effective, its positive conditional probability cannot
have the same value as its negative conditional probability. Otherwise, the test will shift our
view to the same amount and the net effect is nil.
The second application of the Bayes' theorem alters the ratio of number of healthy people to
the number of cancer people in the universe even further. In the second mapped world, the
number of people who have cancer to be included is 14, and the number of people who are
healthy to be included is 0.6. In the second new world, seeing both positive evidences (a
positive x-ray and a positive CT scan) is convincing evidence that the person has lung
cancer (96% probability).
Bayes' theorem is important in understanding the basic statistical reasoning mechanism. In
its original form, it is not easy to use, especially in the face of multiple evidences. In the next
section, we will introduce a computer reasoning theory: evidence theory that is based on the
Bayes' theory.
6. The evidence theory of computer reasoning
In this section, we are going to present a computer reasoning method called evidence theory
that is more convenient and easier to use than the Bayes' theorem. To help our presentation,
we will define some terms and use some mathematical formulas along the way.
If we take an abstract view, the computer reasoning can be summarized as: capture the
causality relationships from raw data, build a knowledge database using these relationships,
and make a judgment (or inference) on pieces of evidence based on existing knowledge
database. The essence of the summary is shown in Figure 4.
The computer reasoning mechanism shown in Figure 4 can be explained as having two
stages: the knowledge/pattern building and the application of knowledge to the new
evidence. In the first stage (indicated by arrows from the Raw data to the Knowledge
database), knowledge is produced either by data mining from raw data or by direct human
insertion; in the second stage (indicated by arrows from the Knowledge database to the
Search WWH ::




Custom Search