Bayesian Regression with Undirected Network Predictors with an Application to Brain Connectome Data

Abstract

This article proposes a Bayesian approach to regression with a continuous scalar response and an undirected network predictor. Undirected network predictors are often expressed in terms of symmetric adjacency matrices, with rows and columns of the matrix representing the nodes, and zero entries signifying no association between two corresponding nodes. Network predictor matrices are typically vectorized prior to any analysis, thus failing to account for the important structural information in the network. This results in poor inferential and predictive performance in presence of small sample sizes. We propose a novel class of network shrinkage priors for the coefficient corresponding to the undirected network predictor. The proposed framework is devised to detect both nodes and edges in the network predictive of the response. Our framework is implemented using an efficient Markov Chain Monte Carlo algorithm. Empirical results in simulation studies illustrate strikingly superior inferential and predictive gains of the proposed framework in comparison with the ordinary high dimensional Bayesian shrinkage priors and penalized optimization schemes. We apply our method to a brain connectome dataset that contains information on brain networks along with a measure of creativity for multiple individuals. Here, interest lies in building a regression model of the creativity measure on the network predictor to identify important regions and connections in the brain strongly associated with creativity. To the best of our knowledge, our approach is the first principled Bayesian method that is able to detect scientifically interpretable regions and connections in the brain actively impacting the continuous response (creativity) in the presence of a small sample size.