Homework 6

Due Friday March 21 at 5:00pm

Exercise 1

6.2 from Hoff. Note the typo: \(1/\sigma_j^2\) is gamma, not \(1/\sigma_j\). Use the code below to load the data.

glucose = readr::read_csv("https://sta602-sp25.github.io/data/glucose.csv")

Exercise 2

Recall that if \(W \sim \text{Wishart}_p(m, S)\) then \(W = \sum_{i=1}^m z_i z_i^T\) where \(z_1, \ldots, z_m \sim \text{i.i.d. } N_p(0, S)\)

  1. Show that \(E[W] = mS\).

  2. Show that \(W\) is positive definite if \(m \geq p\).

Exercise 3

Suppose \(Y\) is a random normal vector \(Y \sim N_p(\theta, \Sigma)\). Let \(Y_A\) be the first \(p_1\) elements of \(Y\) and \(Y_B\) be the last \(p_2 = p - p_1\) elements, so that \(Y = (Y_A, Y_B)\). Similarly, write \(\theta = (\theta_A, \theta_B)\). Finally, let

\[ \Sigma^{-} \equiv \Psi = \left[ {\begin{array}{cc} \Psi_{AA} & \Psi_{AB} \\ \Psi_{BA} & \Psi_{BB} \\ \end{array} } \right] \] and note that \(\Psi_{AB} = \Psi_{BA}^T\). Find the conditional distribution of \(Y_B\) given \(Y_A\) in terms of \(\theta_A\), \(\theta_B\) and components of \(\Psi\). Try to interpret how \(E[Y_B|Y_A]\) differs from \(E[Y_B]\) and how \(V[Y_B|Y_A]\) differs from \(V[Y_B]\).

Identities for exercise 3

Some of the following identities will be helpful for interpretation.

Let

\[ \Sigma = \left[ {\begin{array}{cc} \Sigma_{AA} & \Sigma_{AB} \\ \Sigma_{BA} & \Sigma_{BB} \\ \end{array} } \right] \]

and

\[ \Psi = \left[ {\begin{array}{cc} \Psi_{AA} & \Psi_{AB} \\ \Psi_{BA} & \Psi_{BB} \\ \end{array} } \right]. \]

Then

\[ \begin{aligned} \Psi_{AA}^- &= \Sigma_{AA} - \Sigma_{AB} \Sigma_{BB}^- \Sigma_{BA}\\ \Psi_{BB}^- &= \Sigma_{BB} - \Sigma_{BA} \Sigma_{AA}^- \Sigma_{AB}\\ \Psi_{AB} &= -\Psi_{AA} \Sigma_{AB} \Sigma_{BB}^-\\ \Psi_{BA} &= -\Psi_{BB} \Sigma_{BA} \Sigma_{AA}^-, \end{aligned} \]

and note that \(\Sigma_{AB} = \Sigma_{BA}^T\) and \(\Psi_{AB} = \Psi_{BA}^T\).