• Bayesian inference;
  • free knots;
  • Markov chain Monte Carlo;
  • surface regression;
  • splines


Methods for choosing a fixed set of knot locations in additive spline models are fairly well established in the statistical literature. The curse of dimensionality makes it nontrivial to extend these methods to nonadditive surface models, especially when there are more than a couple of covariates. We propose a multivariate Gaussian surface regression model that combines both additive splines and interactive splines, and a highly efficient Markov chain Monte Carlo algorithm that updates all the knot locations jointly. We use shrinkage prior to avoid overfitting with different estimated shrinkage factors for the additive and surface part of the model, and also different shrinkage parameters for the different response variables. Simulated data and an application to firm leverage data show that the approach is computationally efficient, and that allowing for freely estimated knot locations can offer a substantial improvement in out-of-sample predictive performance.