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Initialisation

We do not need to initialise the approximate distribution parameters of $ q(\beta_i,\bar{\beta}_i\vert y)$. This is because we can initialise the other approximate distribution parameters and then update $ q(\beta_i,\bar{\beta}_i\vert y)$ first. To allow us to provide sensible initialisation of the other approximate distributions, $ q(\phi_{\epsilon_i}\vert Y)$ and $ q(a_p)$, we use the model with the autoregression parameters set to zero ($ a_p=0$) and with the HRF constraints removed ($ m=0$ and $ C=I$). This means we can use the standard ordinary least squares (OLS) voxelwise frequentist solution to the GLM to get:
$\displaystyle \hat{\beta_i}$ $\displaystyle =$ $\displaystyle (x^Tx)^{-1}x^Ty_i$ (28)
$\displaystyle \hat{\eta}_{it}$ $\displaystyle =$ $\displaystyle y_{it}-x_t\hat{\beta_i}$  

and
$\displaystyle S_i$ $\displaystyle =$ $\displaystyle \sum_t{(\hat{\eta}_{it})^2}$ (29)

The approximate distribution, $ q(\phi_{\epsilon_i}\vert y)$, is set using frequentist results. The mean is set to $ (T-K)/S$ and the variance to $ 2(T-K)/S^2$. This mean and variance gives the approximate distribution parameters $ b_{\epsilon_i}$ and $ c_{\epsilon_i}$ (see appendix A for the conversion). We initialise $ q(a_p\vert y)$ by using the frequentist solution to equation 2, where $ \eta_i$ is set to the residuals, $ \hat{\eta_i}$. This gives:
$\displaystyle \mu_{ai}$ $\displaystyle =$ $\displaystyle (\tilde{\eta}_i\tilde{\eta}_i^T)^{-1}\tilde{\eta}_i\hat{\eta_0}$ (30)
$\displaystyle \Lambda_{ai}$ $\displaystyle =$ $\displaystyle \sigma_\epsilon^2(\tilde{\eta}\tilde{\eta}^T)^{-1}$  

where $ \tilde{\eta}_i=[\hat{\eta_{i1}}\ldots \hat{\eta_{iP}}]$, $ \hat{\eta}_{ip}=[\hat{\eta}_{i(p+1)} \ldots \hat{\eta}_{iT}]^T$ and where:
$\displaystyle \sigma_{\epsilon_i}^2$ $\displaystyle =$ $\displaystyle \sum_t{(\eta_{it}-\eta_{i(t-p)}\mu_{ai})^2}$ (31)

We also initialise $ b_{a_p}=1$, $ c_{a_p}=1$ and $ \phi_{\bar{\beta}_k}=1$.
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Next: f-contrasts Up: Inference Previous: Inference