Bayesian variable selection in additive partial linear models

Many studies in recent time include a large number of predictor variables, but typically only a few of the predictors have significant roles. Variable selection techniques have been developed using both non-Bayesian and Bayesian approaches. Additive partial linear models (APLM) provide a flexible yet manageable extension of linear models, where some variables can have non-linear effects. We develop a Bayesian method for variable selection for APLM by expanding the non-linear functions in a polynomial basis and introducing sparsity by allowing point masses in the prior distribution of regression coefficients. We address variable selection for both linear and non-linear parts. The nonsingular part of the prior is given by a Laplace or multivariate Laplace density depending on whether the predictor has only a linear effect or a general effect. However, instead of using Markov Chain Monte Carlo methods, which are extremely slow in high dimensional models, we use Laplace approximation technique around posterior mode, which can be identified with the group lasso solution. We conduct a simulation study and present real data analysis for a nutritional epidemiology study and prostate cancer data.