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I have a couple of questions when reading the chapter 7 The Multivariate Normal Model of Peter Hoff's "A First Course in Bayesian Statistical Methods".

First, could anyone give me any resource about "$z^i$ is a gamma random variable given z is a mean-zero univariate normal random variable"? (first paragraph on page 110).

My second question is, on the same page 110, it samples a covariance matrix $\Sigma$ from an inverse-Wishart distribution in the three steps below:

  1. sample $\textbf{z}_1,\dotsc,\textbf{z}_{v0}\sim i.i.d. \text{multi normal}(0,\textbf{S}_0^{-1})$ (According to text, $\textbf{z}_{v0}$ is a $p \times1$ matrix, correct me if I am wrong);
  2. calculate $Z^TZ=\sum_{i=1}^{v0}z_i z_i^T$ (according to text, this should be sum of $v_0$ $z_i z_i^T$, this result will be a $p\times p$ matrix);
  3. set $\Sigma=(Z^TZ)^{-1}$.

Then in the next paragraph, the book points out that the expectation is $\Sigma^{-1}$ and $\Sigma$ are:

$$E[\Sigma^{-1}]=v_0\textbf{S}_0^{-1}$$

$$E[\Sigma]=\frac{1}{v_0-p-1}(\textbf{S}_0^{-1})^{-1}=\frac{1}{v_0-p-1}\textbf{S}_0$$

Here is what I am confused, the books continues comments: choosing $v_0=p+2$ and $\textbf{S}_0=\Sigma_0$ makes $\Sigma$ only loosely centered around $\Sigma_0$. In this way, I think the result of $E[\Sigma]$ will be the same as making $\textbf{S}_0=(v_0-p-1)\Sigma_0$. However, the book comments that the distribution $\Sigma$ concentrated around $\Sigma_0$ by making $\textbf{S}_0=(v_0-p-1)\Sigma_0$. I feel the comments contradicts with each other.

Thank you so much for helping me solve the problems.

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