  The normal inverse Gaussian (NIG) and generalized asymmetric Laplace (GAL)
distributions can be seen as skewed and semi-heavy-tailed extensions of the
Gaussian distribution. Models driven by these more flexible noise distributions
are then regarded as flexible extensions of simpler Gaussian models.
Inferential procedures tend to overestimate the degree of non-Gaussianity in
the data and therefore we propose controlling the flexibility of these
non-Gaussian models by adding sensible priors in the inferential framework that
contract the model towards Gaussianity. In our venture to derive sensible
priors, we also propose a new intuitive parameterization of the non-Gaussian
models and discuss how to implement them efficiently in $Stan$. The methods are
derived for a generic class of non-Gaussian models that include spatial
Mat\'ern fields, autoregressive models for time series, and simultaneous
autoregressive models for aerial data. The results are illustrated with a
simulation study and geostatistics application, where priors that penalize
model complexity were shown to lead to more robust estimation and give
preference to the Gaussian model, while at the same time allowing for
non-Gaussianity if there is sufficient evidence in the data.
