Simultaneous probability statements for Bayesian P-splines

P-splines are a popular approach for fitting nonlinear effects of
continuous covariates in semiparametric regression models.
Recently, a Bayesian version for P-splines has been developed on
the basis of Markov chain Monte Carlo simulation techniques for
inference. In this work we adopt and generalize the concept of
Bayesian contour probabilities to Bayesian P-splines within a
generalized additive models framework. More specifically, we aim at
computing the maximum credible level (sometimes called Bayesian
p-value) for which a particular parameter vector of interest lies
within the corresponding highest posterior density (HPD) region. We
are particularly interested in parameter vectors that correspond to
a constant, linear or more generally a polynomial fit. As an
alternative to HPD regions simultaneous credible intervals could be
used to define pseudo contour probabilities. Efficient algorithms
for computing contour and pseudo contour probabilities are
developed. The performance of the approach is assessed through
simulation studies and applications to data for the Munich rental
guide and on undernutrition in Zambia and Tanzania.