We consider estimation of generalized additive models using basis expansions with Bayesian model selection. Although Bayesian model selection is an intuitively appealing tool for regression splines by virtue of the flexible knot placement and model-averaged function estimates, its use has traditionally been limited to Gaussian additive regression, as posterior search of the model space requires a tractable form of the marginal model likelihood. We introduce an extension of the method to the exponential family of distributions using the Laplace approximation to the likelihood. Although the Laplace approximation is successful with all Gaussian-type prior distributions in providing a closed-form expression of the marginal likelihood, there is no broad consensus on the best prior distribution to be used for nonparametric regression via model selection. We observe that the classical unit information prior distribution for variable selection may not be suitable for nonparametric regression using basis expansions. Instead, our study reveals that mixtures of g-priors are more suitable. A large family of mixtures of g-priors is considered for a detailed examination of how various mixture priors perform in estimating generalized additive models. Furthermore, we compare several priors of knots for model selection-based spline approaches to determine the most practically effective scheme. The model selection-based estimation methods are also compared with other Bayesian approaches to function estimation. Extensive simulation studies demonstrate the validity of the model selection-based approaches. We provide an R package for the proposed method.

翻译：我们考虑使用基展开与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凭借灵活的节点放置和模型平均函数估计，在回归样条中是一种直观吸引人的工具，但其使用传统上仅限于高斯加性回归，因为模型空间的后验搜索需要边缘模型似然具有可处理的形式。我们利用拉普拉斯近似于似然函数，将该方法扩展到指数族分布。尽管拉普拉斯近似在使用所有高斯型先验分布时能成功给出边缘似然的闭式表达式，但对于通过模型选择进行非参数回归时应使用的最佳先验分布，尚未达成广泛共识。我们观察到，经典的变量选择单位信息先验可能不适用于使用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。我们考虑了一大类混合g-先验，以详细考察各种混合先验在估计广义加性模型时的表现。此外，我们比较了几种用于基于模型选择的样条方法的节点先验，以确定最实用的有效方案。还将基于模型选择的估计方法与其他贝叶斯函数估计方法进行了比较。大量的模拟研究证明了基于模型选择的方法的有效性。我们为所提出的方法提供了一个R包。