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
