Generalized Bayesian MARS: Robust, Quantile, and Flexible-Likelihood Regression

• Activity Number: 134 - Bayesian Modeling
• Type: Contributed
• Date/Time: Monday, August 9, 2021 : 1:30 PM to 3:20 PM
• Sponsor: Section on Statistical Computing
• Author(s): Kellin Rumsey* and Devin Francom and Andy Shen
• Companies: Los Alamos National Laboratories and Los Alamos National Laboratories and Los Alamos National Laboratories
• Keywords: nonparametric regression; nonlinear regression; quantile regression; sensitivity analysis; splines

Abstract

The multivariate adaptive regression spline (MARS) approach of Friedman (1991) and its Bayesian counterpart (Denison et al. 1998) are popular approaches for nonparametric regression. In Bayesian MARS, the posterior distribution is explored using reversible jump Markov chain Monte Carlo, which relies on a Gaussian likelihood for computational efficiency. By introducing a set of latent variables, a large class of marginal likelihood func- tions, including the t, Laplace, logistic, variance-gamma, Horseshoe, Asymmetric Laplace and Normal-Wald (NW; also called Normal inverse-Gaussian), can be obtained. The resulting model achieves the scalability, flexibility and parsimony of BMARS but can be applied in a wide variety of applications not suitable to Gaussian BMARS, all without extensive tuning, making generalized BMARS a powerful tool for modern regression. We derive the necessary equations for efficient reversible jump MCMC in this generalized case, and demonstrate utility for a variety of cases including robust, quantile and NW regression.