Finding interpretable sparse structure in fMRI data with dependent relevance determination priors (Jonathan Pillow)

In many problem settings, parameters are not merely sparse, but dependent in such a way that non-zero coefficients tend to cluster together. We refer to this form of dependency as region sparsity". Classical sparse regression methods, such as the lasso and automatic relevance determination (ARD), which models parameters as independent a priori, and therefore do not exploit such dependencies. Here we introduce a hierarchical model for smooth, region-sparse weight vectors and tensors in a linear regression setting. Our approach represents a hierarchical extension of the relevance determination framework, where we add a transformed Gaussian process to model the dependencies between the prior variances of regression weights. We combine this with a structured model of the prior variances of Fourier coefficients, which eliminates unnecessary high frequencies. The resulting prior encourages weights to be region-sparse in two different bases simultaneously. We develop Laplace approximation and Monte Carlo Markov Chain (MCMC) sampling to provide efficient inference for the posterior, and show substantial improvements over existing methods for both simulated and real fMRI datasets.