Bayesian Wavelet Analysis Using Nonlocal Priors with an Application to fMRI Analysis

We propose a Bayesian hierarchical wavelet methodology for nonparametric regression based on nonlocal priors. Our methodology assumes for the wavelet coefficients a prior that is a mixture of a point mass at zero and a Johnson-Rossell nonlocal prior. We consider two choices of Johnson-Rossell nonlocal priors: the moment prior and the inverse moment prior. To borrow strength across wavelet coefficients, in addition to more traditional specifications from the wavelet literature, we consider a logit specification for the mixture probability and a polynomial decay specification for the scale parameter. To estimate these specifications’ hyperparameters, we propose an empirical Bayes methodology based on Laplace approximation that allows fast analytic posterior inference for the wavelet coefficients. To assess performance, we perform a simulation study to compare our methodology to several other wavelet-based methods available in the literature. In terms of mean squared error, our methodology with the inverse moment prior yields superior results for cases of low signal-to-noise ratio, as well as for moderate to large sample sizes. Finally, we illustrate the flexibility of our novel methodology with an application to a functional magnetic resonance imaging (fMRI) dataset from a study of brain activations associated with working memory.