ISYE 6420A/MSA Homework 3 solution

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1. Consider the Bayesian model
y|θ1, θ2 ∼ N(θ1 + θ2, 1),
θi ∼iid N(0, 1),
for i = 1, 2. Suppose y = 1 is observed. Then, find the marginal posterior distributions
of θ1 and θ2. (Hint: (i) regression. (ii) If In is an n × n identity matrix and Jn is an
n × n matrix of 1’s, then (In + bJn)
−1 = In −
b
1+nbJn. (iii) If θ follows a multivariate
normal distribution, then the marginal distribution of θi
is a normal distribution with
the corresponding mean and variance).

2. Consider the coin example discussed in the class and perform the following simulation.
Simulate the weights of 10 coins from θi ∼ N(5.67, .012
) for i = 1, · · · , 10. Simulate
10 measurements from yi
|θi ∼ N(θi
, .022
). Compute the total error sum of squares
SSEEB =
P10
i=1(θi − ˆθ
EB
i
)
2 and SSEMLE =
P10
i=1(θi − yi)
2
. Repeat this 1000 times
and plot the densities of the two quantities SSEEB and SSEMLE, and make comments.
(Include your R code with the solutions).

3. Let
yi
|θi ∼ind P oisson(θi)
θi ∼iid Exp(λ)
for i = 1, . . . , n. (p(θ) = λe−λθ). Find the empirical Bayes estimator of θi
, i = 1, . . . , n.

4. Consider the Bayesian model:
xi
|φi ∼ind N(0, φi)
1
φi
∼iid Exp(λ),
for i = 1, · · · , n. Find the empirical Bayes estimator of φi
, i = 1, · · · , n. Evaluate the
expressions as far as possible.