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 additive models with Gaussian or multicategorical responses. 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. Two applications on the determinants of undernutrition in developing countries and the health status of trees show how contour probabilities may be used in practice to assist the analyst in the model building process.