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5. Bayes' reasoning
Example 1 in previous section can also be solved by Bayes' theorem. One thing to remember
in understanding Bayes' theorem is the following statistical formula:
P(A & B) = P(A | B) * P(B)
(Formula 3)
Or equivalently,
P(A & B) = P(B | A) * P(A)
In the following, we will explain how Bayes' reasoning works and the meaning of its
subparts.
5.1 Bayes' reasoning explained
Bayes' theorem can be viewed as the bridge that connects the reasoning to physical
evidences: on the left of Formula 1 is the inference/reasoning, and on the right of Formula 1
is the physical evidence that supports the reasoning on the left. When estimating the prior
probabilities (prior probability includes: the baseline probability P(A), the two conditional
probabilities: P(positive x-ray | cancer) and P(positive x-ray | healthy)) on the right, we are
constructing a reasoning model; when applying the Bayes' theorem, we are extracting
information using the constructed model. The process of applying the theorem is the process
of combining the raw data, the knowledge (context) to yield information. We can use the
Bayes' theorem to solve Example 1 as follows:
1.
Start with what we want to achieve: P(cancer | positive x-ray)
2.
Rewrite it as following with the help of Formula 3:
P(cancer | positive x-ray) = P(cancer & positive x-ray) / P(positive x-ray)
3.
P(positive x-ray) can be expanded to P(positive x-ray & cancer) + P(positive x-ray &
~cancer).
Note:
this expansion captures the causality of the reasoning model. It says that the total
number of people with positive x-ray in the annual check group is coming from two groups:
the people with cancer and show the positive x-ray and the people with no cancer and show
the positive x-ray.
4.
Plug in the result from step 3 to the equation in step 2, we get
(
&
)
(
&
)
(
&~
)
(|−)=
5.
The above equation can be rewritten as following with the help of Formula 3:
(|−)
=
P
(
−|
)
∗()
P(−|)∗()+(−|~)∗(~)
This is exactly the same formula as in the Bayes' theorem (Formula 1). If we use the
following data implied by the problem statement in Example 1:
