Discussion of `Bayesian Treed Generalized Linear Models’ by H. A. Chipman, E. I. George and R. E. McCulloch

In this stimulating paper, the authors have successfully exploited Markov chain Monte Carlo methods to explore the space of graphs for CART-like trees in which the terminal nodes represent generalized linear models (GLMs). Integration over the parameters of the terminal GLMs, in order to compute the marginal likelihood (probability of data given the model) for the MCMC search, is accomplished using the Laplace approximation. Hyper-parameters (such as those governing the GLM parameters) are either set by hand or fixed after a brief empirical search.