Marginalising over $(\vec{\beta_K},\vec{\sigma^2_K})$ in the two-level model

From the two-level model the full joint posterior distribution is (equation 12):

$$p(\beta_g,\sigma_g^2,\vec{\beta_K},\vec{\sigma_K^2}\vert Y) \prop... ...k^2)\}p(\beta_K\vert\beta_g,\sigma_g^2) p(\beta_g,\sigma_g^2,\vec{\sigma_K^2}),$$ (36)

where the prior is the reference prior for this full two-level model (equation 13):

$$p(\beta_g,\sigma_g^2,\vec{\sigma_K^2}) = \frac{1}{\sigma_g^2} \prod_k \frac{1}{\sigma_k^2}.$$ (37)

If we marginalise out $\vec{\sigma_K^2}$ then we get:

$$p(\beta_g,\sigma_g^2,\vec{\beta_K}\vert Y) \propto \prod_k \left\{ \int p(Y_k\vert\beta_k,\sigma_k^2)/\sigma_k^2 d\sigma_k^2 \right\} \mathcal{N}(\beta_K;X_g \beta_g,\sigma_g^2I)1/\sigma_g^2$$ (38)

and then substitute in the summary result of the first-level model in isolation (equation 10):

$$p(\beta_g,\sigma_g^2,\vec{\beta_K}\vert Y) \propto \prod_k \left\{ \mathcal{T}(\beta_k;\mu_{\beta_k},\sigma_{\beta_k... ...u_{\beta_k}) \right\} \mathcal{N}(\beta_K;X_g \beta_g,\sigma_g^2I)1/\sigma_g^2.$$ (39)

We can represent a multivariate non-central t-distribution using a two-parameter Gamma distribution and a multivariate Normal distribution (see appendix 10.3). This is achieved by introducing a parameter $\tau_k$ for each vector $\beta_k$:

$$p(\beta_g,\sigma_g^2,\vec{\beta_K},\vec{\tau_K}\vert Y) \propto \prod_k \left\{ \mathcal{N}(\beta_k;\mu_{\beta_k},(\sigma_{\beta_... ...\beta_k} / \tau_{k})) f_{Ga}(\tau_{k};\nu_{\beta_k}/2,\nu_{\beta_k}/2) \right\}$$

$$\mathcal{N}(\beta_K;X_g \beta_g,\sigma_g^2I) 1/\sigma_g^2.$$ (40)

Writing $\mathcal{N}(\beta_K;X_g \beta_g,\sigma_g^2I)=\prod_k \mathcal{N}(\beta_k;X_{gk} \beta_g,\sigma_{G}^2I)$, where $X_{gk}$ is the $k^{th}$ row vector of the second-level design matrix $X_{g}$, we can now easily integrate out $\beta_k$ for all $k$ to give:

$$p(\beta_g,\sigma_g^2,\vec{\tau_K}\vert Y) \propto \prod_k \left\{ \mathcal{N}(\mu_{\beta_k};X_{gk} \beta_g,(\sigma_... ...ma_g^2I) \Gamma (\tau_{k};\nu_{\beta_k}/2,\nu_{\beta_k}/2) \right\}1/\sigma_g^2$$ (41)

where $\vec{\tau_K}$ is a $(K\times 1)$ vector of the variables $\tau_k$ for $k=1\ldots K$.