Bayesian spline-type smoothing in generalized regression models

Abstract

Spline smoothing in non- or semiparametric regression models is usually based on the roughness penalty approach. For regression with normal errors, the spline smoother also has a Bayesian justification: Placing a smoothness prior over the regression function, it is the mean of the posterior given the data. For non-normal regression this equivalence is lost, but the spline smoother can still be viewed as the posterior mode. In this paper, we provide a full Bayesian approach to spline-type smoothing. The focus is on generalized additive models, however the models can be extended to other non-normal regression models. Our approach uses Markov Chain Monte Carlo methods to simulate samples from the posterior. Thus it is possible to estimate characteristics like the mean, median, moments, and quantiles of the posterior, or interesting functionals of the regression function. Also, this provides an alternative for the choice of smoothing parameters. For comparison, our approach is applied to real-data examples analyzed previously by the roughness penalty approach.