Spatial Bayesian variable selection with application to functional magnetic resonance imaging

Michael S. Smith, Melbourne Business School
Ludwig Fahrmeir, University of Munich

Abstract

In this paper we propose a procedure to undertake Bayesian variable selection and model averaging for a series of regressions that are located on a lattice. For those regressors which are in common in the regressions, we consider using an Ising prior to smooth spatially the indicator variables representing whether or not the variable is zero or non-zero in each regression. This smooths spatially the probabilities that each independent variable is non-zero in each regression, and indirectly smooths spatially the regression coefficients. It is discussed how single site sampling schemes can be used to evaluate the joint posterior distribution. The approach is applied to the problem of functional magnetic resonance imaging in medical statistics, where massive datasets arise that need prompt processing. Here, the Ising prior with a three dimensional neighborhood structure is used to smooth spatially activation maps from regression models of blood oxygenation. The Ising prior also has the advantage of allowing incorporation of anatomical prior information through the external field. It is shown using a visual experiment how a single site sampling scheme can provide fast evaluation of the posterior activation maps and activation amplitudes. The approach is shown to result in maps that are superior to those produced by a recent Bayesian approach using a continuous Markov random field for the activation amplitude.