Adaptive Bayesian Regression Splines in Semiparametric Generalized Linear Models

Abstract

This paper presents a fully Bayesian approach to regression splines with automatic knot selection in generalized semiparametric models for fundamentally non-Gaussian responses. In a basis function representation of the regression spline we use a B-spline basis. The reversible jump Markov chain Monte Carlo method allows for simultaneous estimation both of the number of knots and the knot placement, together with the unknown basis coefficients determining the shape of the spline. Since the spline can be represented as design matrix times unknown (basis) coefficients, it is straightforward to include additionally a vector of covariates with fixed effects, yielding a semiparametric model. The method is illustrated with data sets from the literature for curve estimation in generalized linear models, the Tokyo rainfall data and the coal mining disaster data, and by a credit-scoring problem for generalized semiparametric models.