A Hierarchical Prior for Bayesian Variable Selection with Interactions

Selecting subsets of variables has always been a vital and challenging topic in educational and psychological settings. In many cases, the probability that an interaction is active is influenced by whether the related variables are active. In this chapter, we proposed a hierarchical prior for Bayesian variable selection to account for a structural relationship between variables and their interactions. Specifically, an interaction is more likely to be active when all the associated variables are active and is more likely to be inactive when at least one variable is inactive. The proposed hierarchical prior is based upon the deterministic inputs, noisy "and" gate model and is implemented in the stochastic search variable selection approach (George and McCulloch (J Amer Statist Assoc 88(423):881-889, 1993)). A Metropolis-within-Gibbs algorithm is used to uncover the selected variables and to estimate the coefficients. Simulation studies were conducted under different conditions and in a real data example. The performance of the proposed hierarchical prior was compared with the widely adopted independent priors in Bayesian variable selection approaches, including traditional stochastic search variable selection prior, Dirac spike and slab priors (Mitchell and Beauchamp (J Amer Statist Assoc 83(404):1023-1032, 1988)), and hyper g-prior (Liang et al. (J Amer Statist Assoc 103(481):410-423, 2008)).