Laplace approximations for fast Bayesian inference in generalized additive models based on P-splines

• Published in: Computational statistics & data analysis Vol. 154; p. 107088
• Main Authors: Gressani, Oswaldo, Lambert, Philippe
• Format: Journal Article Web Resource
• Language: English
• Published: Elsevier B.V 01.02.2021; Elsevier
• Subjects: Fast Bayesian computation; Fast Bayesian inference; Generalized additive models; Laplace approximation; Mathematics; Mathématiques; P-splines; Physical, chemical, mathematical & earth Sciences; Physique, chimie, mathématiques & sciences de la terre; P-splines; Fast Bayesian computation; Generalized additive models; Laplace approximation
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/j.csda.2020.107088

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. •Laplace-P-spline(s) (LPS) is a methodology for fast approximate Bayesian inference.•Bayesian P-splines permit a flexible modeling of smooth terms in additive models.•LPS is generally much faster than existing Bayesian methods fully relying on MCMC.•Analytical gradient/Hessian of the posterior penalty vector permits efficient exploration.