arXiv:2108.11863v4 Announce Type: replace Abstract: We propose Multivariate L\'evy Adaptive B-Spline Regression (MLABS), a Bayesian nonparametric regression model designed to capture nonlinear effects, interactions, and locally varying features of multivariate functions. MLABS represents the regression function as a random sum of low-order tensor-product B-spline basis functions under a finite L\'evy random measure prior. The resulting representation adapts to the complexity and local structure of the regression function while favoring a sparse basis representation. For posterior inference, we develop a reversible-jump Markov chain Monte Carlo (MCMC) algorithm with Bayesian backfitting, collapsed trans-dimensional updates, and residual-guided proposals. These computational strategies are designed to improve posterior exploration across basis configurations of varying dimensions and structures. We also establish approximation bounds for functions with low-order interactions and heterogeneous smoothness. These bounds relate approximation accuracy to the number of basis terms. Simulation studies show that MLABS achieves competitive predictive performance across a wide range of regression settings, including high-dimensional problems. MLABS also demonstrates robust predictive performance across benchmark real datasets. In an environmental-mixture application, it attains predictive accuracy comparable to Bayesian kernel machine regression at substantially lower computational cost.