Autoregressive Parameters Spatial Prior

In Penny et al. (2003) non-spatial non-informative priors were used on the autoregressive parameters. Previous work has shown that neighbouring voxels have similar temporal autocorrelation (Woolrich et al., 2001; Worsley et al., 2002). Therefore, we want to model the assumption that a priori we expect neighbouring voxels to have similar temporal autocorrelation. To do this we use a Gaussian conditionally specified auto-regressive (CAR) or continuous Markov Random Field (MRF) prior (Cressie, 1993) on each of the $P$ autoregressive parameter maps, i.e. $p(a)=\prod^P_{p=1} p(a_p)$ with:

$$p(a_p\vert\phi_{a_p})\sim MVN(\mathbf{0},{\phi_{a_p}^{-1}D^{-1}})$$ (7)

where $MVN$ denotes a multivariate Normal distribution, ${D}$ is an ${N}\times {N}$ matrix whose (i,j)th element is $d_{ij}$, and $\phi_{a_p}$ is the MRF control parameter which controls the amount of spatial regularisation. We set $d_{ii}=1$, $d_{ij}=-1/N_{ij}$ if $i$ and $j$ are spatial neighbours and $d_{ij}=0$ otherwise (where $N_{ij}$ is the geometric mean of the number of neighbours for voxels $i$ and $j$). We also require a hyperprior on $\phi_{a_p}$, for which we use a standard noninformative conjugate gamma prior:

$$\phi_{a_p} \sim Ga({b}_{{a_0}}, {c}_{{a_0}} )$$ (8)

Before we specify the prior for the regression parameters $\beta$, we consider how we can use basis functions within this framework.