Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models

The potential important role of the prior distribution of the roughness penalty parameter in the resulting smoothness of Bayesian Psplines
models is considered. The recommended specification for that distribution yields models that can lack flexibility in specific
circumstances. In such instances, these are shown to correspond to a frequentist P-splines model with a predefined and severe
roughness penalty parameter, an obviously undesirable feature. It is shown that the specification of a hyperprior distribution for one
parameter of that prior distribution provides the desired flexibility. Alternatively, a mixture prior can also be used. An extension of
these two models by enabling adaptive penalties is provided. The posterior of all the proposed models can be quickly explored using
the convenient Gibbs sampler.