Spatial Bayesian Variable Selection With Application to Functional Magnetic Resonance Imaging

Abstract

We propose a procedure to undertake Bayesian variable selection and model averaging for a series of regressions located on a lattice. For those regressors that 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 nonzero in each regression. This smooths spatially the probabilities that each independent variable is nonzero in each regression and indirectly smooths spatially the regression coefficients. We discuss 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 require 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 anatomic prior information through the external field. Using a visual experiment, we show how a single-site sampling scheme can provide rapid 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.KEY WORDS: Binary Markov random field; Human brain mapping; Ising prior; Markov chain Monte Carlo; Model averaging.

Bayesian Variable Selection in Structured High-Dimensional Covariate Spaces With Applications in Genomics

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“…In a regression context, the idea of adding such indicator variables is usually referred to as variable selection (see e.g. [5]) and has been used in [6,2] to capture evoked brain activity in a sparse manner. In the JDE framework, this activity detection task is already handled within the model in a more general way.…”

“…In the JDE framework, this activity detection task is already handled within the model in a more general way. Activated and non-activated voxels are modelled using a two-class Gaussian mixture instead of a BernoulliGaussian prior [6]. Our use of binary variables is then rather oriented toward the selection of stimulus types which has to be done across the whole set of voxelwise regressions (Section 2).…”

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“…Following recent advances in convex optimization theory, these methods can be made more sophisticated by adding sparse priors on the underlying signal and solve maximum a posteriori estimation rather than naive maximum likelihood estimation. They have been exploited as an extension to standard GLM analysis by defining spatial priors on spatial activation maps (Flandin and Penny, 2007;Harrison et al, 2008;Smith and Fahrmeir, 2007;Vincent et al, 2010). Within a temporal fMRI deconvolution framework, while the linear system assumption on the hemodynamic model is retained, regularization terms use ' 1 -norm to favor sparse solutions in time; i.e., a limited number of spike-like activations.…”