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Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysiswith three-dimensional spatial smoothing priors has been shown to producestate-of-the-art activity maps without pre-smoothing the data. The proposedinference algorithms are computationally demanding however, and the proposedspatial priors have several less appealing properties, such as being improperand having infinite spatial range. We propose a statistical inference frameworkfor whole-brain fMRI analysis based on the class of Mat\'ern covariancefunctions. The framework uses the Gaussian Markov random field (GMRF)representation of possibly anisotropic spatial Mat\'ern fields via thestochastic partial differential equation (SPDE) approach of Lindgren et al.(2011). This allows for more flexible and interpretable spatial priors, whilemaintaining the sparsity required for fast inference in the high-dimensionalwhole-brain setting. We develop an accelerated stochastic gradient descent(SGD) optimization algorithm for empirical Bayes (EB) inference of the spatialhyperparameters. Conditionally on the inferred hyperparameters, we make a fullyBayesian treatment of the brain activity. The Mat\'ern prior is applied to bothsimulated and experimental task-fMRI data and clearly demonstrates that it is amore reasonable choice than the previously used priors, using comparisons ofactivity maps, prior simulation and cross-validation.

Spatial 3D Mat\'ern priors for fast whole-brain fMRI analysis