Page 42 - Data Science Algorithms in a Week

Naive Bayes

Medical test - basic application of Bayes'

theorem

A patient takes a special cancer test which has the accuracy test_accuracy=99.9%: if the result
is positive, then 99.9% of the patients tested will suffer from the special type of cancer.
99.9% of the patients with a negative result do not suffer from the cancer.

Suppose that a patient is tested and scores positive on the test. What is the probability that a
patient suffers from the special type of cancer?

Analysis:

We will use Bayes' theorem to find out the probability of the patient having the cancer:

P(cancer|test_positive)=(P(test_positive|cancer) * P(cancer))/P(test_positive)

To know the prior probability that a patient has the cancer, we have to find out how
frequently the cancer occurs among people. Say that we find out that 1 person in 100,000
suffers from this kind of cancer. Then P(cancer)=1/100,000. So, P(test_positive|cancer) =
test_accuracy=99.9%=0.999 given by the accuracy of the test.

P(test_positive) has to be computed:
P(test_positive)=P(test_positive|cancer)*P(cancer)+P(test_positive|no_cancer)*P(no_cancer)

= test_accuracy*P(cancer)+(1-test_accuracy)*(1-P(cancer))

= 2*test_accuracy*P(cancer)+1-test_accuracy-P(cancer)
Therefore, we can compute the following:

P(cancer|test_positive) = (test_accuracy * P(cancer))/(2 * test_accuracy * P(cancer)+1-
test_accuracy-P(cancer))

= 0.999 * 0.00001 / (2 * 0.999 * 0.00001 + 1 - 0.999-0.00001)
= 0.00989128497 which is approximately 1%

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