Title: Fast Automatic Smoothing for Generalized Additive Models,
Abstract: Multiple generalized additive models (GAMs) are a type of distributional
regression wherein parameters of probability distributions depend on predictors
through smooth functions, with selection of the degree of smoothness via $L_2$
regularization. Multiple GAMs allow finer statistical inference by
incorporating explanatory information in any or all of the parameters of the
distribution. Owing to their nonlinearity, flexibility and interpretability,
GAMs are widely used, but reliable and fast methods for automatic smoothing in
large datasets are still lacking, despite recent advances. We develop a general
methodology for automatically learning the optimal degree of $L_2$
regularization for multiple GAMs using an empirical Bayes approach. The smooth
functions are penalized by different amounts, which are learned simultaneously
by maximization of a marginal likelihood through an approximate
expectation-maximization algorithm that involves a double Laplace approximation
at the E-step, and leads to an efficient M-step. Empirical analysis shows that
the resulting algorithm is numerically stable, faster than all existing methods
and achieves state-of-the-art accuracy. For illustration, we apply it to an
important and challenging problem in the analysis of extremal data.