Download Bayes Theorem example

Examples of Bayes Theorem
Law of total probability: Let B i events be mutually exclusive and exhaustive.
n
P(A ) =
∑ P ( A B i )P ( Bi )
(1)
i=1
Bayes Theorem (Devore)
Let A 1, A 2, …, A k be a collection of k mutually exclusive and exhaustive events with prior probability P ( A i ) , where i = 1, 2, …, k . Then for any other event B for which P ( B ) > 0 , the posteriori probability of A j given that B has occurred is
P ( Aj∩ B )
P ( B A j )P ( A j )
- j = 1, 2, … , k
P ( A j B ) = ------------------------- = ------------------------------------------k
P(B)
∑ P ( B Ai )P ( Ai )
i=1
Drug testing example: wikipedia
Suppose a drug test is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of
people are users of the drug. If a randomly selected individual tests positive, what is the probability he or she is a user?
A 1 : Event that person is a drug user, and A 2 event that the person is not a drug user
B : Positive drug test result
P ( A 1 B ) : Required output
Prior probabilities P ( A 1 ) = 0.005 and P ( A 2 ) = 0.995
P ( B A 1 ) = 0.99 True positive for drug users
P ( B A 2 ) = 0.01 True negative for non-drug user is 99% and hence there is only 1% chance that
the drug test result will be positive if the person is a non-drug user.
Posterior probability:
P ( B A 1 )P ( A 1 )
0.99 × 0.005
- = ------------------------------------------------------------------------------P ( A 1 B ) = ------------------------------------------k
P ( B A 1 )P ( A 1 ) + P ( B A 2 )P ( A 2 )
∑ P ( B Ai )P ( Ai )
i=1
0.99 × 0.005
P ( A 1 B ) = ------------------------------------------------------------------- = 33.2 %
0.99 × 0.005 + 0.01 × 0.995
From wikipedia entry for Bayes Theorem taken Sept. 2, 2013.
Despite the apparent accuracy of the test (99%), if an individual tests positive, it is more likely
that they do not use the drug than that they do.
This surprising result arises because the number of non-users is very large compared to the number of users. To use concrete numbers, if 1000 individuals are tested, there are expected to be 995
non-users and 5 users. From the 995 non-users, 0.01 × 995 ≈ 10 false positives are expected.
From the 5 users, 0.99 × 5 ≈ 5 true positives are expected. Out of 15 positive results, only 5,
about 33%, are genuine.
Nate Silver’s book had examples of Bayes theorem and emphasized its power; also Comscore
wanted expertise with applying this theorem. Let’s internalize it.