Adaptive Gaussian Markov random fields with applications in human brain mapping

Summary. Functional magnetic resonance imaging has become a standard technology in human brain mapping. Analyses of the massive spatiotemporal functional magnetic resonance imaging data sets often focus on parametric or non‐parametric modelling of the temporal component, whereas spatial smoothing is based on Gaussian kernels or random fields. A weakness of Gaussian spatial smoothing is underestimation of activation peaks or blurring of high curvature transitions between activated and non‐activated regions of the brain. To improve spatial adaptivity, we introduce a class of inhomogeneous Markov random fields with stochastic interaction weights in a space‐varying coefficient model. For given weights, the random field is conditionally Gaussian, but marginally it is non‐Gaussian. Fully Bayesian inference, including estimation of weights and variance parameters, can be carried out through efficient Markov chain Monte Carlo simulation. Although motivated by the analysis of functional magnetic resonance imaging data, the methodological development is general and can also be used for spatial smoothing and regression analysis of areal data on irregular lattices. An application to stylized artificial data and to real functional magnetic resonance imaging data from a visual stimulation experiment demonstrates the performance of our approach in comparison with Gaussian and robustified non‐Gaussian Markov random‐field models.