Homework 1 Introduction & Single Parameter Models


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Homework 1
Introduction & Single Parameter Models

Practice of Bayes Formula (3 × 10 points). Suppose that if θ = 1, then y has a
normal distribution with mean 1 and standard deviation σ, and if θ = 2, then y has
a normal distribution with mean 2 and standard deviation σ. Also, suppose Pr(θ =
1) = 0.5 and Pr(θ = 2) = 0.5.
1. For σ = 2, derive the formula for the marginal probability density for y, p(y),
and sketch/visualize it in R.
2. What is P r(θ = 1|y = 1), again supposing σ = 2?
3. Describe how the posterior density of θ changes in shape as σ is increased and
as it is decreased.
Guideline for Submission: submit a hard copy (handwritten or printed).
Normal distribution with unknown mean (3 × 10 points). A random sample
of n students is drawn from a large population, and their weights are measured. The
average weight of the n sampled students is ¯y = 150 pounds. Assume the weights in
the population are normally distributed with unknown mean θ and known standard
deviation 20 pounds. Suppose your prior distribution for θ is normal with mean 180
and standard deviation 40.
1. Give your posterior distribution for θ. (Your answer will be a function of n.)
2. A new student is sampled at random from the same population and has a weight
of ˜y pounds. Give a posterior predictive distribution for ˜y. (Your answer will
still be a function of n.)
STATS 551, Unversity of Michigan Instructor: Yang Chen
3. For n = 10, give a 95% posterior interval for θ and a 95% posterior predictive
interval for ˜y. Do the same for n = 100.
Guideline for Submission: submit a hard copy (handwritten or printed).
Nonconjugate single parameter model (2 × 20 points). Suppose you observe
y = 285 from the model Binomial(500, θ), where θ is an unknown parameter. Assume
the prior on θ has the following form
p(θ) =

8θ, 0 ≤ θ < 0.25
3 −
θ, 0.25 ≤ θ ≤ 1
0, otherwise.
1. Compute the unnormalized posterior density function on a grid of m points
for some large integer m. Using the grid approximation, compute and plot the
normalized posterior density function p(θ|y), as a function of θ.
2. Sample 10000 draws of θ from the posterior density and plot a histogram of
the draws.
Guideline for Submission: submit R markdown (or jupyter notebook) with annotated
code followed by results. Discussions about the results should follow the results.
Optional Reading. Read one of the following papers and post your summary and
thoughts on Canvas.
1. Efron, B. (2005). Bayesians, frequentists, and scientists. Journal of the American Statistical Association, 100(469), 1-5.
2. Biostatistics and Bayes, Norman Breslow, Statist. Sci., Volume 5, Number 3
(1990), 269-284.
3. Bayesian Methods in Practice: Experiences in the Pharmaceutical Industry,
A. Racine, A. P. Grieve, H. Fluhler and A. F. M. Smith, Journal of the Royal
Statistical Society. Series C (Applied Statistics), Vol. 35, No. 2 (1986), pp.

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