MIDTERM EXAM ISyE6420 Fall 2021 Released October 21, 12:00am – due October 24, 11:59pm. This exam is not proctored and not time limited except the due date. Late submissions will not be accepted. Use of all course materials is allowed. Internet search and direct communication with others that violate Georgia Tech Academic Integrity Rules are not permit- ted. Please show necessary work to get full credit. The exam must be typed in word/latex/RMarkdown and submitted as a pdf file. Please include the Win- Bugs/R/Python/Matlab codes as separate files. Name Problem Chad Normal-uniform Genetic study Total Score /35 /25 /40 /100 1. Chad, Bayes, Car, and Vacation. Chad is taking a Bayesian Analysis course. He believes he will get an A with probability 0.7. At the end of semester he will get a car as a present form his rich uncle depending on his class performance. For getting an A in the course he will get a car with probability 0.8, for anything less than A, he will get a car with probability of 0.1. If Chad gets a car, he would travel to Cocoa Beach with probability 0.7, or stay on campus with probability 0.3. If he does not get a car, these two probabilities are 0.2 and 0.8, respectively. Figure 1: Chad on the road After the semester was over you learn that Chad is in Cocoa Beach. What is the proba- bility that he got a car? Hint: You can solve this problem by any of the 3 ways: (i) use of WinBUGS or Open- BUGS, (ii) direct simulation using Octave/MATLAB, R, or Python, and (iii) exact calcula- tion. 2. Normal-uniform. Consider the Bayesian model yi|θ ∼iid N(θ, σ2), θ ∼ Uniform(0, 1), for i = 1, · · · , n, where σ2 is known. Find the posterior distribution of θ. 3. Genetic study. A genetic study has divided n = 197 animals into four categories: y = (125, 18, 20, 34). A genetic model for the population cell probabilities is given by( 1 2 + θ 4 , 1− θ 4 , 1− θ 4 , θ 4 ) and thus, the sampling model is a multinomial distribution: p(y|θ) = n! y1!y2!y3!y4! ( 1 2 + θ 4 )y1 (1− θ 4 )y2 (1− θ 4 )y3 (θ 4 )y4 , 2 where n = y1+y2+y3+y4. Assume the prior distribution for θ to be Uniform(0, 1). To find the posterior distribution of θ, a Gibbs sampling algorithm can be implemented by splitting the first category into two (y0, y1 − y0) with probabilities (12 , θ4). Here y0 can be viewed as another parameter (or a latent variable). Thus, p(θ, y0|y) ∝ n! y0!(y1 − y0)!y2!y3!y4! ( 1 2 )y0 (θ 4 )y1−y0 (1− θ 4 )y2 (1− θ 4 )y3 (θ 4 )y4 . 1. Derive the full conditional distributions of θ and y0. 2. Implement Gibbs sampling in R, Matlab, Python, or Winbugs and obtain the posterior distribution of θ (plot the density). 3. Find the estimate and 95% credible interval of θ. Hint: 1 2 1 2 + θ 4 + θ 4 1 2 + θ 4 = 1 . 3