Additive cubic spline regression with dirichlet process mixture errors

The goal of this paper is to develop a fully Bayesian nonparametric analysis of re-gression models for continuous and categorical outcomes. We do so by presenting mod-els in which covariate (or regression) effects are modeled additively by cubic splines, and the error distribution (for the latent outcomes in the case of categorical data) is modeled by a Dirichlet process mixture prior. One innovation of this paper is the use of a relatively unexplored basis in which the spline coefficients have the attractive fea-ture of being the unknown function ordinates at the knots. We exploit this feature to develop a proper prior distribution on the coefficients that involves the first and second differences of the ordinates, quantities about which one may be expected to have some prior knowledge. We also discuss the problem of comparing models with different num-bers of knots or different error distributions. For this purpose, we provide algorithms for computing the marginal likelihood for both the continuous and categorical models within the framework of Chib (1995) as extended to DPM models by Basu and Chib (2003). We illustrate the techniques with simulated and real data. Key words: additive regression; Bayesian nonparametric regression; Bayes factors