Results

Figure 7 shows the posterior means of the AR parameters from the single-event pain dataset for two different models: with non-spatial and with spatial MRF AR parameter priors. For the model with spatial MRF priors, the Variational Bayesian inference adaptively determines the amount of spatial regularisation to impose for each of the autoregressive parameter MRFs separately. The spatial maps for $p=1$ are very similar, but as we increase $p$ the MRF spatial regularisation increases. This shows how the adaptive determination of the amount of spatial regularisation automatically adjusts to avoid overfitting of the high order autoregressive parameters. In terms of inference on the normalised power of the f-contrasts, figure 8(a) suggests that avoiding overfitting and spatially regularising the AR parameters does make a significant difference to the pseudo-z-statistics.

Figure 7: Posterior means of parameters from an autoregressive model of order 4 from the single-event pain dataset. From left to right we have increasing $p$ from $1$ to $4$. We show this for two different priors on the AR parameters: [top] non-informative non-spatial, and [bottom] spatial MRF.

Figure 8: Histogram of the voxelwise difference in pseudo-z-statistics between the model with non-spatial non-informative AR priors and the model with adaptive spatial MRF AR priors for the single-event pain dataset.

Figure 9 shows the histogram of pseudo-z-statistics obtained for the two different models with and without HRF constraints. For both conditions we can see how the right hand tail, i.e. those voxels which are strongly activating, is relatively unaffected. Whereas, the main body of the histogram, i.e. the background non-activating voxels, is shifted to the left in the same way that it was for the null artificial data.

Figure 9: Histogram of pseudo-z-statistics obtained for two different models (with and without HRF constraints) on (a) the visual boxcar dataset and (b) the pain single-event dataset.

Figures 10 and 12 show the results of applying the spatial mixture modelling on the unconstrained HRF model. Figures 11 and 13 show the results for the constrained HRF model. Figure 14 shows the difference in voxel classification between the constrained HRF model and the unconstrained HRF model. This difference highlights the increased sensitivity. With the constrained HRF model smaller strength activating voxels have increased probability of being in the activation class. This is because the non-activating class distribution is shifted to lower pseudo-z-statistics when we use the constrained HRF model. Figures 15 and 16 show samples from the marginal posterior of the HRF at a voxel which is not activating and a voxel which is strongly activating respectively in the pain experiment. In particular, figure 15 highlights the difference between the unconstrained and constrained HRF models for a non-activating voxel. Whereas, in figure 16 the HRF shape conforms to prior expectations, hence there is little different between the unconstrained and constrained HRF models for a strongly activating voxel.

Figure 10: Visual condition. Results of applying the spatial mixture modelling on the unconstrained HRF model. [top] Pseudo-z-statistic spatial maps. [bottom] Probability of being in the activation class. [right] Mixture model fit to histogram of pseudo-z-statistics.

Figure 11: Visual condition. Results of applying the spatial mixture modelling on the constrained HRF model. [top] Pseudo-z-statistic spatial maps. [bottom] Probability of being in the activation class. [right] Mixture model fit to histogram of pseudo-z-statistics ('A' and 'B' mark the pseudo-z-statistics for which voxels with greater pseudo-z-statistics have higher probability of being active than non-active for the constrained HRF model and unconstrained HRF model respectively).

Figure 12: Pain single-event condition. Results of applying the spatial mixture modelling on the unconstrained HRF model. [top] Pseudo-z-statistic spatial maps. [bottom] Probability of being in the activation class. [right] Mixture model fit to histogram of pseudo-z-statistics.

Figure 13: Pain single-event condition. Results of applying the spatial mixture modelling on the constrained HRF model. [top] Pseudo-z-statistic spatial maps. [bottom] Probability of being in the activation class. [right] Mixture model fit to histogram of pseudo-z-statistics ('A' and 'B' mark the pseudo-z-statistics for which voxels with greater pseudo-z-statistics have higher probability of being active than non-active for the constrained HRF model and unconstrained HRF model respectively).

Figure 14: Difference in voxel classification between the constrained HRF model and the unconstrained HRF model. [red] voxels are active for both models, [blue] voxels are active for just the unconstrained HRF model, and [yellow] voxels are active for just the constrained HRF model. Voxels are classified as activating with probability of being in the activation class greater then 0.5. [top] Visual boxcar dataset [Bottom] Pain single-event dataset.

Figure 15: Samples from the marginal posterior of the HRF at a single voxel which is not activating in the pain experiment. (a) Unconstrained with $m=0$ and $C=I$ (b) Constrained with $m=\tilde{m}$ and $C=\tilde{C}$

Figure 16: Samples from the marginal posterior of the HRF at a single voxel which is strongly activating in the pain experiment. (a) Unconstrained with $m=0$ and $C=I$ (b) Constrained with $m=\tilde{m}$ and $C=\tilde{C}$