The Laplace-P-spline methodology for fast approximate Bayesian inference in additive partial linear models

Multiple linear regression is among the cornerstones of statistical model building. Whether from a descriptive or inferential perspective, it is certainly the most widespread approach to analyze the inuence of a collection of explanatory variables on a response. The straightforward interpretability in conjunction with the simple and elegant mathematics of least squares created room for a well appreciated toolbox with an ubiquitous presence in various scientific fields. In this article, the linear dependence assumption of the response variable with respect to the covariates is relaxed and replaced by an additive architecture of univariate smooth functions of predictor variables. An approximate Bayesian approach combining Laplace approximations and P-splines is used for inference in this additive partial linear model class. The analytical availability of the gradient and Hessian of the posterior penalty vector allows for a fast and efficient exploration of the penalty space, which in turn yields accurate point and set estimates of latent field variables. Different simulation settings confirm the statistical performance of the Laplace-P-spline approach and the methodology is applied on mortality data.