  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. In this
article, P-splines and the Laplace approximation are coupled for flexible and
fast approximate Bayesian inference in GAMs. 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. Our approach has the merit of relying on closed form analytical
expressions for the gradient and Hessian of the approximate posterior penalty
vector, which enables to construct accurate posterior pointwise and credible
set estimators for latent field variables at a relatively low computational
budget even for a large number of smooth additive components. Based upon simple
Gaussian approximations of the conditional latent field posterior, the
suggested methodology enjoys excellent statistical properties. The performance
of the Laplace-P-spline model is confirmed through different simulation
scenarios and the method is illustrated on two real datasets.
