3. Bayesian Reasoning We consider a test which detects if a person has a disease. Let R denote the outcome of the test on a person, and let D denote whether the person actually has the disease. We let B denote the true positive rate, that is the likelihood that the test detects the disease among those that have it. Further, we let y denote the false positive rate, that is the probability that test is positive among healthy patients. Formally: p(R=1|D= 1) = 8 and p(R=1|D=0) = 7 Finally, we assume that an a-fraction of the population that gets tested actually has this disease. That is, the prior probability of a person that is being tested having this disease is p(D) = a. (a) A patient has the test performed and it comes back positive. Derive a general formula for the posterior probability that the person actually has the disease, and simplify it in terms of a, 8, and y. Which value do you get for a = 0.1, B = 0.y and y = 0.2, where x,y,z are the three last non-zero digits of your student ID. (Report the digits you use). (b) After the results of the first test comes back positive, the doctor runs it a second time. Again, it comes back positive. Derive the posterior probability that the person actually has the disease after this second round of testing assuming the two test results are independent and simplify in terms of a, b, and 7. Again, in addition to the general expression, report the values you get for a, b, and from the previous question. (c) Analyze under which conditions the posterior probability of having the disease after two positive tests is larger than after only one positive test. How does it depend on a, B, and y? That is, under which conditions on a, B, and y will running multiple tests boost the confidence in the outcome? (d) Let's think of these type of conclusions in the context of the current pandemic. One of the major unknowns is what fraction of the population has already been 2 in touch with the virus and may therefore have developed anti-bodies and be immune. Assuming we a test, which we knew 8 and the true positive and false positive rate). How should we go about estimating the the fraction of a population that is immune? Describe a selection process for whom we should test and how we would calculate the immunity level, given the test results. (e) In the context of a new disease such as COVID-19, is it safe to assume we know B and y? What are the challenges here? 3. Bayesian Reasoning We consider a test which detects if a person has a disease. Let R denote the outcome of the test on a person, and let D denote whether the person actually has the disease. We let B denote the true positive rate, that is the likelihood that the test detects the disease among those that have it. Further, we let y denote the false positive rate, that is the probability that test is positive among healthy patients. Formally: p(R=1|D= 1) = 8 and p(R=1|D=0) = 7 Finally, we assume that an a-fraction of the population that gets tested actually has this disease. That is, the prior probability of a person that is being tested having this disease is p(D) = a. (a) A patient has the test performed and it comes back positive. Derive a general formula for the posterior probability that the person actually has the disease, and simplify it in terms of a, 8, and y. Which value do you get for a = 0.1, B = 0.y and y = 0.2, where x,y,z are the three last non-zero digits of your student ID. (Report the digits you use). (b) After the results of the first test comes back positive, the doctor runs it a second time. Again, it comes back positive. Derive the posterior probability that the person actually has the disease after this second round of testing assuming the two test results are independent and simplify in terms of a, b, and 7. Again, in addition to the general expression, report the values you get for a, b, and from the previous question. (c) Analyze under which conditions the posterior probability of having the disease after two positive tests is larger than after only one positive test. How does it depend on a, B, and y? That is, under which conditions on a, B, and y will running multiple tests boost the confidence in the outcome? (d) Let's think of these type of conclusions in the context of the current pandemic. One of the major unknowns is what fraction of the population has already been 2 in touch with the virus and may therefore have developed anti-bodies and be immune. Assuming we a test, which we knew 8 and the true positive and false positive rate). How should we go about estimating the the fraction of a population that is immune? Describe a selection process for whom we should test and how we would calculate the immunity level, given the test results. (e) In the context of a new disease such as COVID-19, is it safe to assume we know B and y? What are the challenges here