# Homework 8¶

• Type your name and email in the “Student details” section below.

• Develop the code and generate the figures you need to solve the problems using this notebook.

• For the answers that require a mathematical proof or derivation you can either:

• Type the answer using the built-in latex capabilities. In this case, simply export the notebook as a pdf and upload it on gradescope; or

• You can print the notebook (after you are done with all the code), write your answers by hand, scan, turn your response to a single pdf, and upload on gradescope.

• The total homework points are 100. Please note that the problems are not weighed equally.

Note

• This is due before the beginning of the next lecture.

• Please match all the pages corresponding to each of the questions when you submit on gradescope.

• First Name:

• Last Name:

• Email:

## 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

$p(A|B,I)= p(\text{test is positive} |\text{has tuberculosis},I),$

and

$p(A| \neg B, I) = p(\text{test is positive}|\text{does not have tuberculosis}, I),$

so that

$p(B|A, I) = p(\text{has tuberculosis}|\text{test is positive}, I) = 0.99.$

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