Comments On`bayesian Backfitting' Bayesian Function Estimation

I warmly congratulate the authors on this paper. I am sure they will succeed in broadening acceptance of the Bayesian paradigm in inference in regression, by providing this well-written and accessible treatment of the use of Markov chain Monte Carlo (MCMC) in tting the important class of (generalised) additive models. The paper promotes several ideas. It can be interesting and revealing to examine Bayesian analogues of familiar frequentist models and procedures; MCMC is important in Bayesian inference; Gibbs sampling is a convenient general recipe for MCMC; if that isn't available, try Metropolis-Hastings; Gibbs sampling is a close analogue of backktting. None of these points are individually very original, of course! But it is very appealing to see their combination applied to additive models, with a number of practical details worked out to produce an eecient methodology, especially since Splus software to implement the resulting methodology is provided. My comments focus on some of these practical details, and some other relations and connections to the proposed methods. The reader who comes to this work from a background of backktting in (generalised) additive models rather than experience of inference about functions and surfaces in the Bayesian paradigm might get an impression that this paper is close to the start-of-the-art in Bayesian function estimation. In fact, the authors do not claim this; researchers have been investigating and using Bayesian models and MCMC calculations for more complicated situations than this, almost from the earliest days of MCMC in statistics. One could even claim that a driving force in the broader acceptance of practical Bayesian 1