Bayesian Multilevel Network Recovery Selection

Variable selection and network estimation have been popular tools for identifying key variables associated with a response variable of interest in settings involving non-negligible dependency structures among variables. However, the ability to identify relevant variables in a high-dimensional setting while accounting for conditional dependencies within a multilevel structure under a nonadditive model is still limited. Hence, in this article, we examine multilevel network recovery selection under a two-level structure in which higher-level variables contain lower-level variables nested within them. Due to the dependency structure, the variables work together to accomplish certain tasks at both levels. Our main interest is to simultaneously explore variable selection and identify dependency structures between higher- and lower-level variables under a nonadditive model framework. We develop a multi-level nonparametric kernel machine approach with a newly proposed multilevel Ising spike-slab prior, using Markov-chain Monte Carlo and variational Bayes inference to identify multi-level variables and jointly build the network. The variational inference approach is novel in using the sampled dependency structure as the observed variable rather than the response. In addition to the variable selection and network recovery capabilities, our approach can produce both mean and quantile estimations of the original response variable of interest. We demonstrate the advantages of our approach using simulation studies and a genetic pathway-based analysis. Supplementary materials for this article are available online.