Perfect simulation for Bayesian curve and surface

This paper demonstrates how to apply the method of coupling from the past (CFTP) (Propp and Wilson, 1996) to the problem of sampling orthogonal predictors in the Bayes linear model. Using CFTP we construct a coupled Gibbs sampler that automatically determines its own run time to convergence. This removes the need to rely on approximate convergence diagnostics as in usual Markov chain Monte Carlo methods. We apply this approach to a number of curve and surface t-ting examples using wavelets and Demmler-Reinsch smoothing splines where there are as many predictors as data points. The Demmler-Reinsch splines are derived from an orthogonalization of some standard smoothing models. Somewhat remarkably, this procedure actually adds to the transparency of the model. This proves especially beneecial in a Bayesian perspective as it facilitates the quantiication of prior beliefs on the model space. A striking result in all of our examples is the rapid convergence of the Gibbs sampler. Possessing this knowledge allows us to repeatedly regenerate the sampler following known convergence and hence generate independent and identically distributed samples from the posterior model space.