A hierarchical Bayesian approach is proposed for variable selection and function estimation in additive nonparametric Gaussian regression models and additive nonparametric binary regression models. The prior for each component function is an integrated Wiener process resulting in a posterior mean estimate that is a cubic smoothing spline. Each of the explanatory variables is allowed to be in or out of the model, and the regression functions are estimated by model averaging. To allow variable selection and model averaging, data-based priors are used for the smoothing parameter and the slope at 0 of each component function. A two-step Markov chain Monte Carlo method is used to efficiently obtain the data-based prior and to carry out variable selection and function estimation. It is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously. The methodology is illustrated for a binary regression using heart attack data.