Fast Posterior Approximation

Here we propose a fast but approximate approach for estimating the distribution parameters, $\mu_{\beta_g},\sigma_{\beta_g}^2\Sigma_{\beta_g},\nu_{\beta_g}$, in equation 16. First, we assume high degrees of freedom at the first-level, i.e. $\tau_k=1$ for all $k$. We then obtain a point estimate of $\sigma _g^2$, and use this point estimate to compute a point estimate of $\beta _g$. For the details of how we obtain these point estimates $\widehat{\sigma_g}^2$ and $\mu_{\beta_g}$, see appendix 10.7.

We then make the assumption that the effect of uncertainty in $\sigma^2_g$ is the same as the effect of uncertainty in $\sigma^2_k$ in a first-level model. This means that $p(\beta_g\vert Y)$ is a multivariate non-central t-distribution:

$$\mathcal{N}(\beta_g; \widehat{\beta}_g,(X_{G}^T U^{-1} X_{G})^{-1},\nu).$$ (18)

where $U$ is a diagonal matrix with the $k^{th}$ diagonal element given by $S_k=(\sigma_{\beta_k}^2\Sigma_{\beta_k} / \tau_k)+\sigma_g^2I$. However, we do not know the degrees of freedom (DOF), $\nu$. We might expect the DOF to be within the range, $N_K-P_G\leq \nu \leq \infty$. In the validation section we will look at using $\nu = N_K-P_G$ (lower esimate) and $\nu=\infty$ (upper estimate). The accuracy of these assumptions are examined with simulations in section 6.