Integrated nested Laplace approximation method for hierarchical Bayesian inference of spatial model with application to functional magnetic resonance imaging data

Analysis of functional magnetic resonance imaging (fMRI) data plays a crucial role in medical statistics. Many developed approaches have relied on the voxel-wise general linear model (GLM). Despite the popularity of classical GLM, this procedure has some drawbacks in modeling complex structure of fMRI data. Bayesian approaches have largely dealt with this issue. Due to the computational cost of Bayesian approach such as Monte Carlo Markov Chain (MCMC), approximate Bayesian methods including variational Bayes (VB) and Integrated Nested Laplacian Approximation (INLA) have been proposed. In this study, we use INLA for inferences of spatial hierarchical model. Moreover, temporal correlations of time series is addressed by pre-whitening approach. To detect regions of activation, we apply joint posterior distributions as opposed to marginal distributions. The joint posterior probability map (PPM) is computed based on excursions set theory, and multiple comparisons problem is properly corrected. The performance of the proposed Bayesian approach is evaluated via simulation studies and a real task fMRI dataset. The results are compared with VB and classical GLM. The Bayesian approach based on joint PPM leads to more precise results by controlling false positive rate. The posterior mean from INLA fairly agree with the posterior mean from VB, however VB underestimates posterior variance.