We propose general procedures for posterior sampling from additive and generalized additive models. The procedure is a stochastic generalization of the well-known backfitting algorithm for fitting additive models. One chooses a linear operator (“smoother”) for each predictor, and the algorithm requires only the application of the operator and its square root. The procedure is general and modular, and we describe its application to nonparametric, semiparametric and mixed models.
"Bayesian backfitting (with comments and a rejoinder by the authors." Statist. Sci. 15 (3) 196 - 223, August 2000. https://doi.org/10.1214/ss/1009212815