Nested MCMC for Uncertainty Propagation in Spatial Modeling of Temporal Long-Memory in Resting-State fMRI

Resting-state functional magnetic resonance imaging (rs-fMRI) signals exhibit long-range temporal dependence, characterized by a slow power-law decay of autocorrelation. This scale-free temporal structure, quantified by the long-memory (LM) parameter, has been associated with brain function and age-related changes in neural dynamics. Investigating spatial variation in cortical LM and its association with demographic and clinical covariates requires methods that account for spatial dependence and propagate uncertainty in LM estimation into population-level inference. We develop a two-stage spatial Bayesian framework that incorporates a conditional autoregressive (CAR) prior to model dependence among neighbouring cortical vertices and a nested Markov chain Monte Carlo (MCMC) strategy to propagate uncertainty from subject-specific LM estimation into regression inference. A composite-basis representation enables computationally feasible population-level inference over the high-dimensional cortical surface. In simulation studies, the CAR prior substantially improved subject-level LM estimation accuracy, while CAR-based approaches reduced population-level estimation error and generally improved detection of localized group effects relative to independent estimation. Nested MCMC produced posterior mean estimates comparable to those of CAR plug-in, while yielding slightly wider credible intervals that account for uncertainty in subject-level LM estimation. Application to HCP-Aging resting-state fMRI data identified significant negative associations between age and LM in localized cortical regions, particularly in frontal and precentral areas. These findings demonstrate the benefits of spatial modeling for cortical LM estimation and uncertainty propagation for population-level Bayesian inference.