For f ( ) = | | 1 / 2 [ 1 + 1 v ( )

For $f(\theta )=|\mathrm{\Sigma }{|}^{-1/2}{[1+\frac{1}{v}(\theta -\mu {)}^{\mathrm{\prime }}{\mathrm{\Sigma }}^{-1}(\theta -\mu )]}^{-(\nu +\alpha )/2}$, consider the $d$-dimensional integral ${\int }_{R^d}f(\theta )d\theta$. The integrand is the kernel of a multivariate- $t$ density, so the correct answer is the inverse of the normalizing constant.

(a) Evaluate this integral as a Monte Carlo average ${\mathcal{S}}^{-1}\sum _{s=1}^{S}f\left({\theta }^{(s)}\right)/h\left({\theta }^{(s)}\right)$, ${\theta }^{(s)}\sim h(\theta )$, where the importance density $h(\theta )$ is multivariate-t with the same location and scale as $f(\theta )$, but with a different degrees-of-freedom parameter.

(b) Explore the stability of this average as you vary the degrees of freedom of $h(\theta )$. Increase the mismatch between $f(\theta )$ and $h(\theta )$ by changing the location and scale of $h(\theta )$ and explore further.