Bayesian analysis of functional magnetic resonance imaging data with spatially varying auto‐regressive orders

Statistical modelling of functional magnetic resonance imaging data is challenging as the data are both spatially and temporally correlated. Spatially, measurements are taken at thousands of contiguous regions, called voxels, and temporally measurements are taken at hundreds of time points at each voxel. Recent advances in Bayesian hierarchical modelling have addressed the challenges of spatiotemporal structure in functional magnetic resonance imaging data with models incorporating both spatial and temporal priors for signal and noise. Whereas there has been extensive research on modelling the functional magnetic resonance imaging signal (i.e. the convolution of the experimental design with the functional choice for the haemodynamic response function) and its spatial variability, less attention has been paid to realistic modelling of the temporal dependence that typically exists within the functional magnetic resonance imaging noise, where a low order auto‐regressive process is typically adopted. Furthermore, the auto‐regressive order is held constant across voxels (e.g. AR(1) at each voxel). Motivated by an event‐related functional magnetic resonance imaging experiment, we propose a novel hierarchical Bayesian model with automatic selection of the auto‐regressive orders of the noise process that vary spatially over the brain. With simulation studies we show that our model is more statistically efficient and we apply it to our motivating example.