We describe a Bayesian estimation and inference procedure for fMRI time series based on the use of General Linear Models (GLMs). Importantly, we use a spatial prior on regression coefficients which embodies our prior knowledge that evoked responses are spatially contiguous and locally homogeneous. Further, using a computationally efficient Variational Bayes framework, we are able to let the data determine the optimal amount of smoothing. We assume an arbitrary order Auto-Regressive (AR) model for the errors. Our model generalizes earlier work on voxel-wise estimation of GLM-AR models and inference in GLMs using Posterior Probability Maps (PPMs). Results are shown on simulated data and on data from an event-related fMRI experiment.

This paper provides a method for the analysis of the spatial and temporal diffusion of shocks in a dynamic system. We use changes in real house prices within the UK economy at the level of regions to illustrate its use. Adjustment to shocks involves both a region specific and a spatial effect. Shocks to a dominant region-London- are propagated contemporaneously and spatially to other regions. They in turn impact on other regions with a delay. We allow for lagged effects to echo back to the dominant region. London in turn is influenced by international developments through its link to New York and other financial centers. It is shown that New York house prices have a direct effect on London house prices. We analyse the effect of shocks using generalised spatio-temporal impulse responses. These highlight the diffusion of shocks both over time (as with the conventional impulse responses) and over space.

Abstract. In recent years there has been explosive growth in the number of neuroimaging studies performed using functional Magnetic Resonance Imaging (fMRI). The field that has grown around the acquisition and analysis of fMRI data is intrinsically interdisciplinary in nature and involves contributions from researchers in neuroscience, psychology, physics and statistics, among others. A standard fMRI study gives rise to massive amounts of noisy data with a complicated spatio-temporal correlation structure. Statistics plays a crucial role in understanding the nature of the data and obtaining relevant results that can be used and interpreted by neuroscientists. In this paper we discuss the analysis of fMRI data, from the initial acquisition of the raw data to its use in locating brain activity, making inference about brain connectivity and predictions about psychological or disease states. Along the way, we illustrate interesting and important issues where statistics already plays a crucial role. We also seek to illustrate areas where statistics has perhaps been underutilized and will have an increased role in the future. Key words and phrases: fMRI, brain imaging, statistical analysis, challenges. 1.