  This work has been motivated by the challenge of the 2017 conference on
Extreme-Value Analysis (EVA2017), with the goal of predicting daily
precipitation quantiles at the $99.8\%$ level for each month at observed and
unobserved locations. We here develop a Bayesian generalized additive modeling
framework tailored to estimate complex trends in marginal extremes observed
over space and time. Our approach is based on a set of regression equations
linked to the exceedance probability above a high threshold and to the size of
the excess, the latter being modeled using the generalized Pareto (GP)
distribution suggested by Extreme-Value Theory. Latent random effects are
modeled additively and semi-parametrically using Gaussian process priors, which
provides high flexibility and interpretability. Fast and accurate estimation of
posterior distributions may be performed thanks to the Integrated Nested
Laplace approximation (INLA), efficiently implemented in the R-INLA software,
which we also use for determining a nonstationary threshold based on a model
for the body of the distribution. We show that the GP distribution meets the
theoretical requirements of INLA, and we then develop a penalized complexity
prior specification for the tail index, which is a crucial parameter for
extrapolating tail event probabilities. This prior concentrates mass close to a
light exponential tail while allowing heavier tails by penalizing the distance
to the exponential distribution. We illustrate this methodology through the
modeling of spatial and seasonal trends in daily precipitation data provided by
the EVA2017 challenge. Capitalizing on R-INLA's fast computation capacities and
large distributed computing resources, we conduct an extensive cross-validation
study to select model parameters governing the smoothness of trends. Our
results outperform simple benchmarks and are comparable to the best-scoring
approach.
