|Bayesianism: perceives probability as the degree of a belief. See also subjective probability, objective probability, chance, likelihood.|
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P(A I X) = P(X I A) * P(A)
Says to what extent we should update our knowledge about A with respect to the new information.
P(A): Prior knowledge: 1% of women over 40 get breast cancer.
New information: good test proving breast cancer.
P(XI A): 80% of all women with breast cancer are tested positive in the test.
P(X ~ A): 9.6% of women without breast cancer are tested positive in the test.
P(A I X): what is the proportion of those with a positive test result who actually have breast cancer?
Common mistake: most consider the proportion of women who actually have breast cancer to be very high.
Solution: E.g. 10 000 women are examined
Group 1: 100 with cancer
Group 2: 9,900 without cancer.
After the examination one has 4 groups:
A: 80 women with cancer and with a positive test result
B: 20 with cancer but with a false negative result
In A, the 80% with a correct-positive result are: p(X I A)
C: 950 without cancer but with a false-positive result
D: 8950 without cancer and with a negative result.
Question: What is the percentage of women with a positive outcome who actually have cancer?
Solution: Group A is divided by the sum of A and C. This is 7.8%
N.B.: more than 90% of positively tested women have no cancer.
Although the test is a good test, the Bayesian theorem says that the new information is not particularly helpful.
Bayesian Theorem/Frith: the Bayesian theorem tells us precisely how much new information should influence our ideas about the world.
Definition ideal Bayesian observer/Frith: the ideal Bayesian observer always uses information in an optimal way.
Problem: we can classify information badly when it comes to rare events and large numbers.
Brain: although we as persons are not ideal observers, there is much evidence that our brain is an ideal observer.
For example, if an object is very rare, an observer needs more information to believe that it is actually there. Therefore, in the case of bombs, we are not an ideal observer.
Brain: has the task of combining the information from the different sensory channels.
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