Variational regression methods are an increasingly popular tool for their efficient estimation of complex. Given the mixed model representation of penalized effects, additive regression models with smoothed effects and scalar-on-function regression models can be fit relatively efficiently in a variational framework. However, inferential procedures for smoothed and functional effects in such a context is limited. We demonstrate that by using the Mean Field Variational Bayesian (MFVB) approximation to the additive model and the subsequent Coordinate Ascent Variational Inference (CAVI) algorithm, we can obtain a form of the estimated effects required of a Frequentist test for semiparametric curves. We establish MFVB approximations and CAVI algorithms for both Gaussian and binary additive models with an arbitrary number of smoothed and functional effects. We then derive a global testing framework for smoothed and functional effects. Our empirical study demonstrates that the test maintains good Frequentist properties in the variational framework and can be used to directly test results from a converged, MFVB approximation and CAVI algorithm. We illustrate the applicability of this approach in a wide range of data illustrations.

翻译：变分回归方法因其对复杂模型的高效估计而日益流行。鉴于惩罚效应的混合模型表示，具有平滑效应的加性回归模型和标量-函数回归模型可以在变分框架中相对高效地拟合。然而，在此类背景下对平滑效应和函数效应的推断程序十分有限。我们证明，通过对加性模型使用平均场变分贝叶斯（MFVB）近似以及随后的坐标上升变分推断（CAVI）算法，我们可以获得半参数曲线频率学派检验所需的估计效应形式。我们为具有任意数量平滑效应和函数效应的高斯加性模型和二元加性模型建立了MFVB近似和CAVI算法。随后，我们推导出针对平滑效应和函数效应的全局检验框架。我们的实证研究表明，该检验在变分框架中保持良好的频率学派性质，并可用于直接检验来自收敛的MFVB近似和CAVI算法的结果。我们通过广泛的数据示例说明了该方法的适用性。