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Generalized additive models (GAMs) are a well-established statistical toolfor modeling complex nonlinear relationships between covariates and a responseassumed to have a conditional distribution in the exponential family. In thisarticle, P-splines and the Laplace approximation are coupled for flexible andfast approximate Bayesian inference in GAMs. The proposed Laplace-P-splinemodel contributes to the development of a new methodology to explore theposterior penalty space by considering a deterministic grid-based strategy or aMarkov chain sampler, depending on the number of smooth additive terms in thepredictor. Our approach has the merit of relying on closed form analyticalexpressions for the gradient and Hessian of the approximate posterior penaltyvector, which enables to construct accurate posterior pointwise and credibleset estimators for latent field variables at a relatively low computationalbudget even for a large number of smooth additive components. Based upon simpleGaussian approximations of the conditional latent field posterior, thesuggested methodology enjoys excellent statistical properties. The performanceof the Laplace-P-spline model is confirmed through different simulationscenarios and the method is illustrated on two real datasets.

Laplace approximation for fast Bayesian inference in generalized additive models based on penalized regression splines