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.)

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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

8

3 −

8

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.

93-150.

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