Bayesian Scalar-on-Image Regression with a Spatially Varying Single-layer Neural Network Prior

Deep neural networks (DNN) have been widely used in scalar-on-image regression to predict an outcome variable from imaging predictors. However, training DNN typically requires large sample sizes for accurate prediction, and the resulting models often lack interpretability. In this work, we propose a novel Bayesian nonlinear scalar-on-image regression framework with a spatially varying single-layer neural network (SV-NN) prior. The SV-NN is constructed using a single hidden layer neural network with its weights generated by the soft-thresholded Gaussian process. Our framework enables the selection of interpretable image regions while achieving high prediction accuracy with limited training samples. The SV-NN offers large prior support for the imaging effect function, facilitating efficient posterior inference on image region selection and automatic network structures determination. We establish the posterior consistency for model parameters and selection consistency for image regions when the number of voxels/pixels grows much faster than the sample size. To ensure computational efficiency, we develop a stochastic gradient Langevin dynamics (SGLD) algorithm for posterior inference. We evaluate our method through extensive comparisons with state-of-the-art deep learning approaches, analyzing multiple real datasets, including task fMRI data from the Adolescent Brain Cognitive Development (ABCD) study.