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:我们考虑利用Bayesian模型选择基础扩展对通用添加模型进行估计。虽然Bayesian模型选择由于灵活的结结安排和模型平均函数估计,对于回归样板是一个直觉的吸引工具,但由于灵活的结结置和模型平均函数估计,因此其使用传统上限于Gausian叠加回归,因为模型空间的后遗迹搜索要求采用边际模型可能性的可移植形式。我们采用拉比近似差推向分布指数式组合的方法。虽然拉普尔近似是所有Gausian型前分销在提供边缘可能性的封闭式表达方面取得成功的,但对于通过模型选择将先前的最佳分布用于非参数回归的最佳分配没有广泛的共识。我们发现,用于变量选择的经典单位信息在分配之前可能不适合使用基础扩展的非参数回归。相反,我们的研究显示,使用拉比差近近近似差的混合物混合物的混合物更为合适。一个大型基基基点模型用于详细审查各种混合物在估计通用添加模型时是如何表现的。此外,我们比较了以前用于非参数回归回归回归的最佳分配方法。我们比较了用于选择模式选择方法的其他方法。我们比较了选择方法的模型的模拟方法。比较了其他选择方法。