Homework 4

Due Thursday October 12th at 5:00pm

Exercise 1

Let

\[ \begin{aligned} Y | \theta, \sigma^2 &\sim N(\theta, 1/\gamma)\\ \theta | \sigma^2 &\sim N(\mu_0, 1/\gamma \kappa_0)\\ \gamma &\sim \text{gamma}(a, b) \end{aligned} \]

so \(\gamma\) is the precision (inverse-variance) of the normal distribution.

  1. Derive and simplify the joint pdf \(p(y_1, \ldots y_n | \theta, \gamma)\)

  2. Derive the posterior of the precision, \(p(\gamma| y_1, \ldots y_n)\).

  3. Derive the posterior of \(\theta\), \(p(\theta | y_1, \ldots y_n)\)

Exercise 2

Exercise 5.1 from Hoff. You can read in the data from the three schools with the R code below. Hint: the problem specification is the same as exercise 1, except \(a = \nu_0/2\) and \(b = \nu_0 \sigma_0^2/2\).

library(tidyverse)
school1 = read_csv("https://sta360-fa23.github.io/data/school1.csv")
school2 = read_csv("https://sta360-fa23.github.io/data/school2.csv")
school3 = read_csv("https://sta360-fa23.github.io/data/school3.csv")

Exercise 3

3.12 from Hoff.