Multigrid priors for a Bayesian approach to fMRI

We introduce multigrid priors to construct a Bayesian-inspired method to asses brain activity in functional magnetic resonance imaging (fMRI). A sequence of different scale grids is constructed over the image. Starting from the finest scale, coarse grain data variables are sequentially defined for each scale. Then we move back to finer scales, determining for each coarse scale a set of posterior probabilities. The posterior on a coarse scale is used as the prior for activity at the next finer scale. To test the method, we use a linear model with a given hemodynamic response function to construct the likelihood. We apply the method both to real and simulated data of a boxcar experiment. To measure the number of errors, we impose a decision to determine activity by setting a threshold on the posterior. Receiver operating characteristic (ROC) curves are used to study the dependence on threshold and on a few hyperparameters in the relation between specificity and sensitivity. We also study the deterioration of the results for real data, under information loss. This is done by decreasing the number of images in each period and also by decreasing the signal to noise ratio and compare the robustness to other methods.