  A fully Bayesian approach is proposed for ultrahigh-dimensional nonparametric
additive models in which the number of additive components may be larger than
the sample size, though ideally the true model is believed to include only a
small number of components. Bayesian approaches can conduct stochastic model
search and fulfill flexible parameter estimation by stochastic draws. The
theory shows that the proposed model selection method has satisfactory
properties. For instance, when the hyperparameter associated with the model
prior is correctly specified, the true model has posterior probability
approaching one as the sample size goes to infinity; when this hyperparameter
is incorrectly specified, the selected model is still acceptable since
asymptotically it is proved to be nested in the true model. To enhance model
flexibility, two new $g$-priors are proposed and their theoretical performance
is examined. We also propose an efficient MCMC algorithm to handle the
computational issues. Several simulation examples are provided to demonstrate
the computational advantages of our method.
