Abstract: In many practical problems that simultaneously investigate the joint effect of covariates, we first need to identify the subset of significant covariates, and then estimate their joint effect. An example is an epidemiological study that analyzes the effects of exposure variables on a health response. In order to make inferences on the covariate effects, we propose a Bayesian additive nonparametric regression model with a multivariate continuous shrinkage prior to address the model uncertainty and to identify important covariates. Our general approach is to decompose the response function into the sum of the nonlinear main effects and the two-way interaction terms. Then we apply the computationally advantageous Bayesian variable selection method to identify the important effects. The proposed Bayesian method is a multivariate Dirichlet-Laplace prior that aggressively shrinks many terms toward zero, thus mitigating the noise of including unimportant exposures and isolating the effects of the important covariates. Our theoretical studies demonstrate asymptotic prediction and variable selection consistency properties. In addition, we use numerical simulations to evaluate the model performance in terms of prediction and variable selection under practical scenarios. The method is applied to a neurobehavioral data set from the Agricultural Health Study that investigates the association between pesticide usage and neurobehavioral outcomes in farmers. The proposed method shows improved accuracy in predicting the joint effects on the neurobehavioral responses, while restricting the number of covariates included in the model through variable selection.

Key words and phrases: Additive nonparametric regression, Bayesian variable selection, continuous shrinkage prior, environmental epidemiology, posterior consistency.