1. Let \(S\sim \text{Wishart}_n(\Sigma)\) for some \(\Sigma\in \mathcal D_p^+\), \(S' \sim \text{Wishart}_n(I_p)\), and let \(R\) and \(R'\) be the corresponding correlation matrices. Show that \(R \stackrel{d}{=} R'\).

  2. Analyze the pendigit data [pendigits.rds] in the [Code] directory for partial isotropy. Specifically, for each digit, use BIC to identify a plausible dimension of a subspace with isotropic errors. What do the results suggest about compressibility of the data?

  3. Consider “Bayesian probabilistic PCA” using the model \(Y \sim N_{n\times p}(0,(AA^\top +\sigma^2 I_p )\otimes I_n)\).

    1. Describe the steps of a Gibbs sampler for estimation of \(A\) and \(\sigma^2\) using an inverse-gamma prior distribution for \(\sigma^2\) and \(A \sim N_{p\times r}( 0, \tau^2 I)\).
    2. Find the posterior mean \(\hat A\) of \(A\), and discuss the possibility of estimating \(AA^\top +\sigma^2I\) with \(\hat A \hat A^\top + \hat\sigma^2 I\), where \(\hat\sigma^2\) is the posterior mean of \(A\). Can you come up with a better Bayesian estimator of \(AA^\top + \sigma^2 I\)?