Model

Here we describe the GLM in the fully Bayesian framework. It will be via the priors on the regression parameters that we propose to constrain the possible linear combinations that are allowed. Consider that the preprocessed FMRI data at voxel $i$ and at scan $t$ is $y_{it}$ ( $i=1\ldots N$, $t=1\ldots T$), the $t^{th}$ $1 \times K$ row of the design matrix, $x$, is $x_t$, and $\beta_i$ is a $K\times 1$ vector of parameter estimates. The preprocessed FMRI data, $y$, is taken to have been motion corrected and high-pass filtered. The standard general linear model (GLM) is then:

$$y_{it} = x_t \beta_i + \eta_{it}$$ (1)

We model the error, $\eta_{it}$, as a voxel-wise temporal autoregressive process of order P (AR(P)). We can represent this as:

$$\eta_{it} = \sum_{p=1}^P a_{pi} \eta_{i(t-p)} + \epsilon_{it}$$

$$\epsilon_{it} \sim N(0,\phi_{\epsilon_i}^{-1})$$ (2)

where $a_{pi}$ is the $p^{th}$ AR coefficient ( $p=1,\ldots,P$).