Generalized Bayesian MARS: Tools for Stochastic Computer Model Emulation\ast

The multivariate adaptive regression spline (MARS) approach of Friedman [J. H. Friedman, Ann. Statist., 19 (1991), pp. 1--67] and its Bayesian counterpart [D. Francom et al., Statist. Sinica, 28 (2018), pp. 791--816] are effective approaches for the emulation of computer models. The traditional assumption of Gaussian errors limits the usefulness of MARS, and many popular alternatives, when dealing with stochastic computer models. We propose a generalized Bayesian MARS (GBMARS) framework which admits the broad class of generalized hyperbolic distributions as the induced likelihood function. This allows us to develop tools for the emulation of stochastic simulators which are parsimonious, scalable, and interpretable and require minimal tuning, while providing powerful predictive and uncertainty quantification capabilities. GBMARS is capable of robust regression with t distributions, quantile regression with asymmetric Laplace distributions, and a general form of ``Normal-Wald"" regression in which the shape of the error distribution and the structure of the mean function are learned simultaneously. We demonstrate the effectiveness of GBMARS on various stochastic computer models, and we show that it compares favorably to several popular alternatives.