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