Generalized additive models (GAMs) are a well-established statistical tool for modeling complex nonlinear relationships between covariates and a response assumed to have a conditional distribution in the exponential family. To make inference in this model class, a fast and flexible approach is considered based on Bayesian P-splines and the Laplace approximation. The proposed Laplace-P-spline model contributes to the development of a new methodology to explore the posterior penalty space by considering a deterministic grid-based strategy or a Markov chain sampler, depending on the number of smooth additive terms in the predictor. The approach has the merit of relying on a simple Gaussian approximation to the conditional posterior of latent variables with closed form analytical expressions available for the gradient and Hessian of the approximate posterior penalty vector. This enables to construct accurate posterior pointwise and credible set estimators for (functions of) regression and spline parameters at a relatively low computational budget even for a large number of smooth additive components. The performance of the Laplace-P-spline model is confirmed through different simulation scenarios and the method is illustrated on two real datasets.

中文翻译：

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基于 P 样条的广义加性模型中快速贝叶斯推理的拉普拉斯近似

广义加性模型 (GAM) 是一种完善的统计工具，用于对协变量和假设具有指数族条件分布的响应之间的复杂非线性关系进行建模。为了在这个模型类中进行推理，我们考虑了一种基于贝叶斯 P 样条和拉普拉斯近似的快速灵活的方法。所提出的拉普拉斯-P 样条模型有助于开发一种新的方法来探索后验惩罚空间，方法是考虑基于确定性网格的策略或马尔可夫链采样器，这取决于预测器中平滑加性项的数量。该方法的优点是依赖于潜在变量的条件后验的简单高斯近似，具有可用于近似后验惩罚向量的梯度和 Hessian 的封闭形式解析表达式。这使得能够以相对较低的计算预算为回归和样条参数（的函数）构建准确的后验点和可信集估计器，即使对于大量平滑的加性分量也是如此。Laplace-P-spline 模型的性能通过不同的模拟场景得到证实，该方法在两个真实数据集上进行了说明。这使得能够以相对较低的计算预算为回归和样条参数（的函数）构建准确的后验点和可信集估计器，即使对于大量平滑的加性分量也是如此。Laplace-P-spline 模型的性能通过不同的模拟场景得到证实，该方法在两个真实数据集上进行了说明。这使得能够以相对较低的计算预算为回归和样条参数（的函数）构建准确的后验点和可信集估计器，即使对于大量平滑的加性分量也是如此。Laplace-P-spline 模型的性能通过不同的模拟场景得到证实，该方法在两个真实数据集上进行了说明。