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Problem 1 - Medical test analysis¶
Disclaimer: This example is a modified version of the one found in a 2013 lecture on Bayesian Scientific Computing taught by Prof. Nicholas Zabaras. I am not sure where the original problem is coming from.
We are tasked with assessing the usefulness of a tuberculosis test. The prior information I is:
The percentage of the population infected by tuberculosis is 0.4%. We have run several experiments and determined that:
If a tested patient has the disease, then 80% of the time the test comes out positive.
If a tested patient does not have the disease, then 90% of the time the test comes out negative.
To facilitate your analysis, consider the following logical sentences concerning a patient:
A: The patient is tested and the test is positive.
B: The patient has tuberculosis.
A. Find the probability that the patient has tuberculosis (before looking at the result of the test), i.e., \(p(B|I)\). This is known as the base rate or the prior probability.
B. Find the probability that the test is positive given that the patient has tuberculosis, i.e., \(p(A|B,I)\).
C. Find the probability that the test is positive given that the patient does not have tuberculosis, i.e., \(p(A|\neg B, I)\).
D. Find the probability that a patient that tested positive has tuberculosis, i.e., \(p(B|A,I)\).
E. Find the probability that a patient that tested negative has tuberculosis, i.e., \(p(B|\neg A, I)\). Does the test change our prior state of knowledge about about the patient? Is the test useful?
F. What would a good test look like? Find values for
There are more than one solutions. How would you pick a good one? Thinking in this way can help you set goals if you work in R&D. If you have time, try to figure out whether or not there exists such an accurate test for tuberculosis