We are given the posterior distribution:
\[ \beta \mid y \sim t_p\left( \beta^*,\ \frac{2b^*}{n+2a} (M^*)^{-1},\ n+2a \right) \]
We want the variance (covariance matrix) of \(\beta \mid y\).
A \(p\)-dimensional random vector \(X\) follows a multivariate \(t\) distribution with parameters \(\mu\) (location vector), \(V\) (scale matrix), and \(\nu\) (degrees of freedom) if:
\[ X = \mu + \frac{Z}{\sqrt{W}} \]
where: - \(Z \sim N_p(0, V)\) independent of \(W\) - \(W \sim \frac{\chi^2_\nu}{\nu}\) (so \(W\) has mean 1)
Then: \[ \text{Var}(X) = \frac{\nu}{\nu-2} V \quad \text{for } \nu > 2 \]
Reason: \[ \text{Var}(X) = E\left[ \frac{V}{W} \right] = V \cdot E\left[ \frac{1}{W} \right] \] and for \(W \sim \text{Gamma}(\nu/2, \nu/2)\) (i.e., \(\chi^2_\nu / \nu\)), \[ E\left[ \frac{1}{W} \right] = \frac{\nu}{\nu-2}, \quad \nu > 2. \]
From the problem: - Location: \(\mu = \beta^*\) - Scale matrix: \[ V = \frac{2b^*}{n+2a} (M^*)^{-1} \] - Degrees of freedom: \[ \nu = n + 2a \]
For \(\nu > 2\) (i.e., \(n+2a > 2\)):
\[ \text{Var}(\beta \mid y) = \frac{\nu}{\nu - 2} \cdot V \]
Substitute \(V\) and \(\nu\):
\[ \text{Var}(\beta \mid y) = \frac{n+2a}{n+2a - 2} \cdot \frac{2b^*}{n+2a} (M^*)^{-1} \]
Cancel \(n+2a\) in numerator and denominator:
\[ \text{Var}(\beta \mid y) = \frac{2b^*}{n+2a - 2} (M^*)^{-1} \]
\[ \boxed{\frac{2b^*}{n+2a - 2} (M^*)^{-1}} \]
This is the posterior variance (covariance matrix) of \(\beta\) given \(y\), valid for \(n + 2a > 2\).
The original parameterization used \(\frac{2b^*}{n+2a} (M^*)^{-1}\) as the scale matrix, not the variance matrix. The variance is obtained by scaling by \(\frac{\nu}{\nu-2}\).