A Global Bayes Factor for Observations on an Infinite-Dimensional Hilbert Space, Applied to Signal Detection in fMRI

Functional Magnetic Resonance Imaging (fMRI) is a fundamental tool in advancing our understanding of the brain's functionality. Recently, a series of Bayesian approaches have been suggested to test for the voxel activation in different brain regions. In this paper, we propose a novel definition for the global Bayes factor to test for activation using the Radon-Nikodym derivative. Our proposed method extends the definition of Bayes factor to an infinite dimensional Hilbert space. Using this extended definition, a Bayesian testing procedure is introduced for signal detection in noisy images when both signal and noise are considered as an element of an infinite dimensional Hilbert space. This new approach is illustrated through a real data analysis to find activated areas of Brain in an fMRI data.