We consider the estimation of generalized additive models using basis expansions coupled with Bayesian model selection. Although Bayesian model selection is an intuitively appealing tool for regression splines, its use has traditionally been limited to Gaussian additive regression because of the availability of a tractable form of the marginal model likelihood. We extend the method to encompass the exponential family of distributions using the Laplace approximation to the likelihood. Although the approach exhibits success with any Gaussian-type prior distribution, there remains a lack of consensus regarding the best prior distribution for nonparametric regression through model selection. We observe that the classical unit information prior distribution for variable selection may not be well-suited for nonparametric regression using basis expansions. Instead, our investigation reveals that mixtures of g-priors are more suitable. We consider various mixtures of g-priors to evaluate the performance in estimating generalized additive models. Furthermore, we conduct a comparative analysis of several priors for knots to identify the most practically effective strategy. Our extensive simulation studies demonstrate the superiority of model selection-based approaches over other Bayesian methods.

翻译：我们考虑使用基展开结合贝叶斯模型选择来估计广义加性模型。尽管贝叶斯模型选择是回归样条中直观吸引人的工具，但由于边际模型似然具有易处理的形式，其使用传统上仅限于高斯加性回归。我们利用拉普拉斯近似似然，将方法推广至涵盖指数族分布。虽然该方法在使用任何高斯型先验分布时均能成功，但关于通过模型选择进行非参数回归的最佳先验分布仍缺乏共识。我们观察到，经典的变量选择单位信息先验可能不适用于基于基展开的非参数回归。相反，我们的研究表明混合g先验更为合适。我们考虑多种混合g先验，以评估其在估计广义加性模型中的性能。此外，我们对几种关于节点的先验进行比较分析，以确定最实用的有效策略。大量模拟研究表明，基于模型选择的方法优于其他贝叶斯方法。