Spatial 3D Matérn priors for fast whole-brain fMRI analysis

Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors have been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the proposed spatial priors have several less appealing properties, such as being improper and having infinite spatial range. Our paper proposes a statistical inference framework for functional magnetic resonance imaging (fMRI) analysis based on the class of Mat\'ern covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of Mat\'ern fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditional on the inferred hyperparameters, we make a fully Bayesian treatment of the main parameters of interest, that is, the brain activity coefficients. We apply the Mat\'ern prior to both experimental and simulated task-fMRI data and clearly demonstrate that this is a more reasonable choice than the previously used priors, by using prior simulation, cross validation and visual inspection of the resulting activation maps. Additionally, to illustrate the potential of the SPDE formulation, we derive an anisotropic version of the Mat\'ern 3D prior.