2. Bayesian Reasoning II Consider a test which detects if a person has a disease. Let R denote the outcome of the test on a person. D denote whether the person actually has the disease and be the likelihood that the test gives the correct result. That is, the probability that it reports that someone has the disease (R 1) when they actually do (D-1), is , and the probability that it reports that someone doesnt have the disease when they don't is also . Formally: Finally, an a-fraction of the population actually has this disease, that is, the prior probability of a person having this disease is p(D)-a. (a) A patient goes to the doctor, 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 0 and a. Which value do you get for = 0.001 and .95? (b) After the results of the first test come 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 ndependent andl simplify in terms of and . Again, in addition to the general expression. report the values you for = 0.001 and .95 (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 and ? (d) Now we would like to use the above insights for employment of intelligent systems Say security at an airport would like to use a machine learning based system to identify travelers smuggling illegal substances based on expressions of their face while going through security. Say the system was trained to 95% accuracy and we can expect 0.1% of travelers to be smuggling illegal substances, would installing multiple cameras have the same effect as multiple blood tests? Explain you reasoning