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850:1
(get from 0.85/0.001)
To get the answer for low level reasoning, we calculate the odds for a person with cancer
who score positive on the two tests, versus a person without cancer who score positive on
the two tests. Using the basic principles in algebra, the above odds can be calculated as
following
2*14.17*850:998*1*1 =
24089:998
Once get the final odds, we can get probability of a person having lung cancer given that he
score both tests positive as following:
P(cancer | positive x-ray & positive CT scan) = 24089 / (24089+998)
= 24089 / 25087
= 96%
This is the same answer as we get using Bayes' theorem in section 5.
As you can see, using the ratio and the odds tool is simpler than using the Bayes' theorem
directly. We can simplify our calculation even further by using another tool called logarithm
in mathematics. Before we can take the advantage of logarithm, we need to give a new
definition on evidence called evidence degree.
Definition of evidence degree: we define evidence degree of a test as the as the following
formula:
degree(test) = 10 log 10 strength (test)
(Formula 6)
To get the strength from the degree, we use the following formula:
strength(test) = 10 degree(test) / 10
(Formula 7)
Once represented in logarithmic format (degree of evidence), the aggregated effect of
evidence toward a goal can be obtained by simple adding instead of multiplying.
6.5 The evidence based reasoning
As mentioned before, at low-level reasoning, the logic employed by a human is the same as
the Bayes' theorem. In this section, we will show how to reason using the evidence
expressed in the form of degree. As the topic suggested, the focus of our reasoning method
is on evidence. The reasoning method addresses the question of the following type:
Question Type: Given a set of evidences and prior probability of an event A, we want to
reason about the posterior probability of A (here, the event of interest can be anything, such
as the survival chance of a disease, the goal in a planning problem, etc.). In other words, we
want to figure out the left side of the following equation:
P(A | seen evidences x, y, z, …) = ?
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